#####################################################################
##ParkingStatistics.txt: Save this file as  ParkingStatistics.txt    #
## To use it, stay in the                                            #
##same directory, get into Maple (by typing: maple <Enter> )         #
##and then type:  read `ParkingStatistics.txt`<Enter>                #
##Then follow the instructions given there                           #
##                                                                   #
##Written by Yukun Yao and Doron Zeilberger, Rutgers University ,    #
# yao at math dor rutgers dot edu, DoronZeil at gmail dot com        #
######################################################################
 
#Created: May 2018
 
print(`Created: May 2018`):
print(` This is ParkingStatistics.txt `):
print(`It is  accompanies the article `):
print(`  An Experimental Mathematics Approach to the Area Statistic of Parking Functions`):
print(`by Yukun Yao and Doron Zeilberger`):

print(`and also available from Yao's and Zeilberger's websites`):
print(``):
print(`Please report bugs to DoronZeil at gmail dot com `):
print(``):
 print(`The most current version of this  package and paper`):
 print(` are  available from`):
 print(`http://sites.math.rutgers.edu/~zeilberg/  .`):
 print(`http://sites.math.rutgers.edu/~yao/  .`):

print(`---------------------------------------`):
 print(`For a list of the AREA procedures type ezraArea();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the Supporting procedures type ezra1();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the Pre-Computed procedures type ezraPC();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

print(`---------------------------------------`):
 print(`For a list of the MAIN procedures type ezra();, for help with`):
 print(`a specific procedure, type ezra(procedure_name);   .`):
 print(``):
print(`---------------------------------------`):

with(combinat):

ezraArea:=proc()

if args=NULL then
 print(` The area procedures are: BZ11area, CentralMomentArea, FactorialMomentArea, FactorialMomentGeneralArea,FMareaStory,`):
 print(` LimScMomentArea, LC, RawMomentArea, PnxArea `):
 print(``):

else
ezra(args):
fi:

end:

ezra1:=proc()

if args=NULL then
 print(` The supporting procedures are: BZ1, BZ2, BZ2fast, CM,EBexListFlajolet, EBexListLochard, RM,GP,Pnz, PUx, Qn  `):
 print(` FindPoly  , plotDist, , plotDistU,plotHist, PolyAK, SCenM, Wn `):
 print(``):

else
ezra(args):
fi:

end:

ezraPC:=proc()

if args=NULL then
 print(` The pre-computed procedures are: CentralMomentPC, LimScMomentPC, FactorialMomentGeneralPC, FactorialMomentPC,`):
 print(` ProveScaledParkingPC, RawMomentPC,`):
 print(``):

else
ezra(args):
fi:

end:

ezra:=proc()

if args=NULL then
 print(`The main procedures are: AlpahSeq, CentralMoment,  FactorialMoment,Fk, FactorialMomentGeneral, LimScMoment, `):
print(` Pn, PnS, Pnx, ProveScaledParking, PU,  RawMoment,  RPF, RPF1 `):
 print(` `):



elif nops([args])=1 and op(1,[args])=AlphaSeq then
print(`AlphaSeq(f,x,N): Given a probability generating function`):
print( `f(x) (a polynomial, rational function and possibly beyond) `):
print(` returns a list, of length N, whose `):
print(` (i) First entry is the average `):
print(` (ii): Second entry is the variance `):
print(` for i=3...N, the i-th entry is the so-called alpha-coefficients `):
print(` that is the i-th moment about the mean divided by the `):
print(` variance to the power i/2 (For example, i=3 is the skewness `):
print( ` and i=4 is the Kurtosis) `):
print(` If f(1) is not 1, than it first normalizes it by dividing `):
print(`by f(1) (if it is not 0) .`):
print(` For example, try: `):
print(` AlphaSeq(((1+x)/2)^100,x,4); `):

elif nops([args])=1 and op(1,[args])=BZ1 then
 print(`BZ1(a,N): Fk(a,1,N) according to Kung-Yan`):

elif nops([args])=1 and op(1,[args])=BZ11area then
print(`BZ11area(n): the explicit expression for the expectation of the area of a 1parking function (i.e. a usual parking function.`):
print(`Try:`):
print(`[seq(BZ11area(i),i=1..10)];`):

elif nops([args])=1 and op(1,[args])=BZ2 then
 print(`BZ2(a,N): Fk(a,2,N) according to Kung-Yan`):

elif nops([args])=1 and op(1,[args])=BZ2fast then
 print(`BZ2fast(a,N): a MUCH faster version of BZ2(a,N), using FactorialMoment(a,2,S1,n). Try:`):
 print(`BZ2fast(1,100);`):

elif nops([args])=1 and op(1,[args])=EBexListFlajolet  then
print(`EBexListFlajolet(k): input k, output a list of length k such that the kth entry is EBexK(k) according to Flajolet et. al.`):
print(`recurrence given on p. 85 of Svante Janson's article "Brownian areas and Wright's constant" `):
print(`Try: `):
print(`EBexListFlajolet(20)`):

elif nops([args])=1 and op(1,[args])=EBexListLouchard  then
print(`EBexListLouchard(k): input k, output a list of length k such that the kth entry is EBexK(k) according to Luchard's`):
print(`recurrence given on p. 84 of Svante Janson's article "Brownian areas and Wright's constant" `):
print(`Try: `):
print(`EBexListLouchard(20)`):


elif nops([args])=1 and op(1,[args])=Fk then
print(`Fk(a,k,N): The first N terms of the sequence of the k-th factorial moment of a-parking functions. Try:`):
print(`Fk(1,2,10):`):

elif nops([args])=1 and op(1,[args])=FMareaStory then
print(`FMareaStory(N,A1,n): Inputs a positive integer N, and symbols A1 and n,  and outputs an article with explicit expressions for the`):
print(`first N factorial moments of the r.v. "area" defined on parking functions (the area is n*(n+1)/2-sum)`):
print(`in terms of n and the expectation, denoted by A1 `):
print(`including the leading asymptotics, and proving that in the limit they converge to the moments of`):
print(`the (continuous) so-called Airy distribution (area under Brownian excursion). Try:`):
print(`FMareaStory(4,A1,n);`):

elif nops([args])=1 and op(1,[args])=plotDist then
print(`plotDist(f,x,K): Given a prob. gen. function f(x) that has a `):
print(`Taylor series`):
print(`for a discrete r.v.`):
print(`plots its normalized version (X-mu)/sig between mu-K1*sig and`):
print(` mu+K2*sig `):
print(` For example, try: `):
print(` plotDist((1+x)^40,x,5,5); `):


elif nops([args])=1 and op(1,[args])=plotDistU then
print(`plotDistU(f,x,K): Like plotDist(f,x,K), but UNCONSTRAINED`):
print(` For example, try: `):
print(` plotDistU((1+x)^40,x,5,5); `):

elif nops([args])=1 and op(1,[args])=plotHist then
print(`plotHist(f,x): Given a polynomial f, plots the histograms of its coefficients`):
print(`For example, try:`):
print(`plotHist((1+x)^40,x);`):


elif nops([args])=1 and op(1,[args])=GP then
print(`GP(L,x): inputs a list of pairs [a,b], a symbol x, and a non-neg. integer d, and outputs`):
print(` a polynomial, let's call it P, such that for each pair [a,b] in L, P(a)=b, and the degree is<=nops(L)-3`):
print(`Try: `):
print(`GP([seq([i,i^3],i=1..10)],x);`):

elif nops([args])=1 and op(1,[args])=LC then
print(`LC(P,A1,n): inputs a polynomial P in A1 and n, of degree 1 in A1 and outputs the leading asymptotics if A1=n^(3/2)*sqrt(Pi/2). Try`):
print(`LC(FactorialMomentArea(1,2,A1,n),A1,n);`):

elif nops([args])=1 and op(1,[args])=Pn then
print(` Pn(n,a) the number of a-parking functions of length n for numeric n,a. Try `):
print(`Pn(5,1);`):

elif nops([args])=1 and op(1,[args])=PnS then
print(`PnS(n,a): inputs a pos. integer . Outputs the expression for a-parking functions of length n for symbolic a . Try: `):
print(` PnS(5,a); `):


elif nops([args])=1 and op(1,[args])=Pnx then
print(`Pnx(n,a,x): the weight enumerator of a-parking functions according to the weight`):
print(` x^(sum of parking function). For example, try: `):
print(` Pnx(10,1,x); `):

elif nops([args])=1 and op(1,[args])=PnxArea then
print(`PnxArea(n,a,x): Inputs positive integers n and a, and a variable x, outputs the weight enumerator of a-parking functions according to the AREA`):
print(`x^(binomial(n+1/2)-sum) of parking function). For example, try:`):
print(` PnxArea(10,1,x); `):

elif nops([args])=1 and op(1,[args])=Pnz then
print(`Pnz(n,a,z,K): the  polynomial, in z, whose coefficient of z^i is`):
print(`the numerator of the i-th binomial moments of a-parking functions of length n, up to i=K. Try: `):
print(` Pnz(10,1,z,4); `):

elif nops([args])=1 and op(1,[args])=ProveScaledParking then
print(`ProveScaledParking(K): rigorously proves that the limit of the first K sca;ed moments  of the area statistics on parking functions `):
print(`coincides with the scaled moments of the "area under Brownian Excursion" continous random variable. Try:`):
print(`ProveScaledParking(4);`):

elif nops([args])=1 and op(1,[args])=ProveScaledParkingPC then
print(`ProveScaledParkingPC(K):  Like eProveScaledParking(K), but much faster, using pre-computed data.  K must be <=20. Try: `).
print(` ProveScaledParkingPC(20); `):

elif nops([args])=1 and op(1,[args])=PU then
print(`PU(U) the number of U-parking functions of length n for a list U . Try:`):
print(` PU([1,2,3,4]); `):

elif nops([args])=1 and op(1,[args])=PUx then
print(`PUx(U,x) the weight-enuerators of U-parking functions of length n for a list U `):

elif nops([args])=1 and op(1,[args])=Qn then
print(`Qn(n,a) the number of ways of arranging n employees and a bosses, using the Zeilberger recurrence`):



elif nargs=1 and args[1]=FactorialMoment then
print(`FactorialMoment(a,k,S1,n): inputs a positive integer a, a positive integer k, and symbols S1 and n`):
print(` Outputs a polynomial f(S1,n) that is an exact expression for the k-th factorial moment of the SUM statistics`):
print(`of a-parking functions pf length n, in terms of the expectatation, called S1, and n, for SYMBOLIC. Try:`):
print(` FactorialMoment(1,4,S1,n);`):

elif nargs=1 and args[1]=FactorialMomentArea then
print(`FactorialMomentArea(a,k,x,y): find a polynomial f(x,y) such that f(ExpectatioOfArea,n)=k-th FactorialMoment of Area.`):
print(`F or example, to get the formula for the fourth factorial moment of the random variable AREA (aka INCONVENIENCE) defined on 3-parking functions. `):
print(`of length n, in terms of n and the expectation,Try:`):
print(`FactorialMomentArea(3,4,A1,n);`):

elif nargs=1 and args[1]=FM then
print(` FM(E1,n): input the symbols E1 and n, returns the pre-computed list of kth factorial moments’ polynomial F(x,y) for 1-parking function (1<=k<=20). Try:`):
print(` FM(E1,n)[3];`):

elif nargs=1 and args[1]=RawMoment then
print(` RawMoment(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) `):
print(`is the raw moment (i.e. E(X^k))  Try:`):
print(` RawMoment(1,3,x,y);`):

elif nargs=1 and args[1]=RawMomentArea then
print(` RawMomentArea(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) `):
print(`is the raw moment (i.e. E(X^k)), according to AREA.  Try:`):
print(` RawMomentArea(1,3,x,y);`):

elif nargs=1 and args[1]=RawMomentPC then
print(` RawMomentPC(k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) is the raw moment according to SUM.`):
print(` (i.e. E(X^k)) by 2nd Stirling number and FM when 1<=k<=20 and a=1. Same output as RawMoment(1,k,x,y), but much faster for k<=20. `):
print(`since  it uses pre-computed data.Try:`):
print(` RawMomentPC(3,x,y);`):

elif nargs=1 and args[1]=RM then
print(` RM(x,y): input the symbol x and y, return the pre-computed list of kth raw moments’ polynomial F(x,y) for 1-parking function (1<=k<=20). Try:`):
print(` RM(x,y)[3];`):

elif nargs=1 and args[1]=CentralMoment then
print(` CentralMoment(a,k,S1,n): finds a polynomial expression of the form in the variables S1 and n`):
print(`of degree 1 in S1 , for the k-th Central moment of a-parking functions according to SUM, where S1 is the expectation. Try:`):
print(` CentralMoment(1,3,S1,n);`):

elif nargs=1 and args[1]=CentralMomentArea then
print(` CentralMomentArea(a,k,A1,n): finds a polynomial expression of the form in the variables A1 and n`):
print(`of degree 1 in A1 , for the k-th Central moment of a-parking functions according to AREA, where A1 is the expectation. Try:`):
print(` CentralMomentArea(1,3,A1,n);`):

elif nargs=1 and args[1]=CentralMomentPC then
print(` CentralMomentPC(k,x,y): find a polynomial f(x,y) such that f(Expectation,n)=k-th Central  moment `):
print(`for 1-parking function and k<=20, using pre-computed data. Try:`):
print(` CentralMomentPC(3,x,y);`):

elif nargs=1 and args[1]=CM then
print(` CM(x,y): input the symbol x and y, return the pre-computed list of kth central moments’ polynomial F(x,y) for 1-parking function (1<=k<=20). Try:`):
print(` CM(x,y)[3];`):

elif nargs=1 and args[1]=FindPoly then
print(` FindPoly(L,d,x): find a polynomial f(x) up to degree d such that f(i)=L[i]  Try:`):
print(` FindPoly([1,4,9,16,25,36,49],3,x);`):

elif nargs=1 and args[1]=PolyAK then
print(` PolyAK(k,d,i,j): Find the coefficient of x^i*y^j in FactorialMoment(a,k, x,y) which is a polynomial in a up to degree d. Return fail if the coefficient is not a polynomial of degree up to d.  Try:`):
print(` PolyAK(3,10,1,1);`):


elif nargs=1 and args[1]=LimScMoment then
print(`LimScMoment(K): the list of coefficients of the asymptotic expression of scaled kth moment for 1-parking function. (i.e., `):
print(` According to SUM, for k from 1 to K`):
print(`Try: `):
print(`LimScMoment(4);`):

elif nargs=1 and args[1]=LimScMomentArea then
print(`LimScMomentArea(K): the list of coefficients of the asymptotic expression of scaled kth moment for 1-parking function. (i.e., `):
print(` according to AREA, from k=1 to k=K.`):
print(`Try: `):
print(`LimScMomentArea(4);`):


elif nargs=1 and args[1]=LimScMomentPC then
print(` LimScMomentPC(K):  Like LimScMomen(K), but much faster, using pre-computed output. Try: `):
print(` LimScMomentPC(10);`):


elif nargs=1 and args[1]=FactorialMomentGeneral then
print(` FactorialMomentGeneral(a,k,S1,n): inputs the symbols a,n, and E1 (for a-parking function).`): 
print(`Return the polynomial in the symbols a, S1,n for kth factorial moment of the SUM statistics defined over a-parking functions`):
print(`of length n.  It is in terms of SYMBOLIC n, a, and S1, where S1 is the expectation. Try:`):
print(` FactorialMomentGeneral(a,2,S1,n);`):

elif nargs=1 and args[1]=FactorialMomentGeneralArea then
print(` FactorialMomentGeneralArea(a,k,A1,n): inputs the symbols a,n, and E1 (for a-parking function).`): 
print(`Return the polynomial in the symbols a, A1,n for kth factorial moment of the AREA statistics defined over a-parking functions`):
print(`of length n.  It is in terms of SYMBOLIC n, a, and S1, where S1 is the expectation. Try:`):
print(` FactorialMomentGeneralArea(a,2,S1,n);`):

elif nargs=1 and args[1]=FactorialMomentGeneralPC then
print(` FactorialMomentGeneralPC(a,x,y,k): `):
print(`Like FactorialMomentGeneral(a,x,y,k), but much faster, using pre-computed output. k must be <=12. Try: `):
print(` FactorialMomentGeneralPC(a,x,y,12);`):

elif nargs=1 and args[1]=RPF then
print(`RPF(n,a): a random a-parking function of length n. Try:`):
print(`RPF(30,1); `):

elif nargs=1 and args[1]=SCenM then
print(` SCenM(L): inputs the list of moments L, outputs the list of scaled centralized moments `):

elif nargs=1 and args[1]=RPF1 then
print(`RPF1(n): a random 1-partking function of length n using the algorithm in the Diaconis-Hicks paper "Probablistic Parking Functions", `):
print(`arxXiv:1611.09821v1 [math.PR], p.4 : Try:`) :
print(` add(x[RPF1(5)],i=1..10*(5+1)^(5-1)); `):

elif nargs=1 and args[1]=Wn then
print(`Wn(n): the expectation of total height over all rooted labelled trees with n vertices, according to the`):
print(` famous formula of Riordan and Sloane that initiated the OEIS. Try: Wn(10); `):

else
print(`There is no ezra for`,args):

fi:
 
end:


###########From HISTABRUT
#AveAndMoms(f,x,N): Given a probability generating function
#f(x) (a polynomial, rational function and possibly beyond)
#returns a list whose first entry is the average 
#(alias expectation, alias mean)
#followed by the variance, the third moment (about the mean) and
#so on, until the N-th moment (about the mean).
#If f(1) is not 1, than it first normalizes it by dividing
#by f(1) (if it is not 0) .
#For example, try:
#AveAndMoms(((1+x)/2)^100,x,4);
AveAndMoms:=proc(f,x,N) local mu,F,memu1,gu,i:
mu:=simplify(subs(x=1,f)):

if mu=0 then
print(f, `is neither a prob. generating function nor can it be made so`):
RETURN(FAIL):
fi:

F:=f/mu:


memu1:=simplify(subs(x=1,diff(F,x))):

gu:=[memu1]:

F:=F/x^memu1:

F:=x*diff(F,x):
for i from 2 to N do
 F:=x*diff(F,x):
 gu:=[op(gu),simplify(subs(x=1,F))]:
od:

gu:

end:


#AlphaSeq(f,x,N): Given a probability generating function
#f(x) (a polynomial, rational function and possibly beyond)
#returns a list, of length N, whose 
#(i) First entry is the average
#(ii): Second entry is the variance
#for i=3...N, the i-th entry is the so-called alpha-coefficients
#that is the i-th moment about the mean divided by the
#variance to the power i/2 (For example, i=3 is the skewness
#and i=4 is the Kurtosis)
#If f(1) is not 1, than it first normalizes it by dividing
#by f(1) (if it is not 0) .
#For example, try:
#AlphaSeq(((1+x)/2)^100,x,4);
AlphaSeq:=proc(f,x,N) local gu,i:
 gu:=AveAndMoms(f,x,N):
 if gu=FAIL then
  RETURN(gu):
 fi:

if gu[2]=0 then
print(`The variance is 0`):
  RETURN(FAIL):
fi:

[gu[1],gu[2],seq(gu[i]/gu[2]^(i/2),i=3..N)]:

end:

#plotDist(f,x,K): Given a prob. gen. function f(x) that has a 
#Taylor series
#for a discrete r.v.
#plots its normalized version (X-mu)/sig between mu-K1*sig and
#mu+K2*sig
#For example, try:
#plotDist((1+x)^40,x,5,5);
plotDist:=proc(f,x,K1,K2) local mu,f1,lu,gu,sig,i,j1:
f1:=f/subs(x=1,f):
mu:=subs(x=1,diff(f1,x)):
gu:=f1/x^mu:
sig:=sqrt(subs(x=1,x*diff(x*diff(gu,x),x))):

lu:=taylor(f1,x=0,trunc(mu)+K2*trunc(sig)+10):

lu:=[seq([i,coeff(lu,x,i)],i=max(0,trunc(mu-K1*sig))..trunc(mu+K2*sig))]:

lu:=evalf([seq([(lu[j1][1]-mu)/sig,lu[j1][2]*sig],j1=1..nops(lu))]):

plot(lu,scaling=constrained):

end:

#plotDistU(f,x,K): Given a prob. gen. function f(x) that has a . NOT constrained
#Taylor series
#for a discrete r.v.
#plots its normalized version (X-mu)/sig between mu-K1*sig and
#mu+K2*sig
#For example, try:
#plotDistU((1+x)^40,x,5,5);
plotDistU:=proc(f,x,K1,K2) local mu,f1,lu,gu,sig,i,j1:
f1:=f/subs(x=1,f):
mu:=subs(x=1,diff(f1,x)):
gu:=f1/x^mu:
sig:=sqrt(subs(x=1,x*diff(x*diff(gu,x),x))):

lu:=taylor(f1,x=0,trunc(mu)+K2*trunc(sig)+10):

lu:=[seq([i,coeff(lu,x,i)],i=max(0,trunc(mu-K1*sig))..trunc(mu+K2*sig))]:

lu:=evalf([seq([(lu[j1][1]-mu)/sig,lu[j1][2]*sig],j1=1..nops(lu))]):

plot(lu):

end:


#plotHist(f,x): Given a polynomial f, plots the histograms of its coefficients
#For example, try:
#plotHist((1+x)^40,x);
plotHist:=proc(f,x) local gu,mu,i:

gu:=[seq(coeff(f,x,i),i=0..degree(f,x))]:
mu:=[[0,0],[0,gu[1]]]:

for i from 1 to nops(gu)-1 do
mu:=[op(mu), [i,gu[i]], [i,gu[i+1]]]:
od:
mu:=[op(mu),[nops(gu),0]]:

plot(mu):

end:



###########End HISTABRUT



###start GP
#GP1(L,x,d): inputs a list of pairs [a,b], a symbol x, and a non-neg. integer d, such that d+2<nops(L), outputs
# a polynomial, let's call it P, such that for each pair [a,b] in L, P(a)=b
GP1:=proc(L,x,d) local eq,var,a,i,P:
if nops(L)<=d+2 then
 print(L, `should have at least`, d+3, `data points. `):
 RETURN(FAIL):
fi:

P:=add(a[i]*x^i,i=0..d):
var:={seq(a[i],i=0..d)}:
eq:={seq(subs(x=L[i][1],P)=L[i][2], i=1..d+2)}:
var:=solve(eq,var):
if var=NULL then
  RETURN(FAIL):
fi:

P:=subs(var,P):

if {seq(subs(x=L[i][1],P)-L[i][2], i=d+3..nops(L))}<>{0} then
  print(P, `did not work out`):
  RETURN(FAIL):
fi:

P:

end:



#GP(L,x): inputs a list of pairs [a,b], a symbol x, and a non-neg. integer d, and outputs
# a polynomial, let's call it P, such that for each pair [a,b] in L, P(a)=b, and the degree is<=nops(L)-3
#Try:
#GF([seq([i,i^3],i=1..10)],x);
GP:=proc(L,x) local d,gu:

for d from 0 to nops(L)-3 do
gu:=GP1(L,x,d):
 if gu<>FAIL then
  RETURN(gu):
 fi:
od:

FAIL:
end:

###end GP
##begin Random Parking Function
#tn(n,a) the number of forests of rooted trees  where each component has the form [root, SetOfEdges]
#n employees and a bosses, i.e. the number of ordered labelled forests
tn:=proc(n,a) local k:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

add(binomial(n,k)*tn(n-k,a+k-1),k=0..n):
end:

#RW(a,b): a random walk from [0,0] to [a,b] that ends at [a,b]. Try:
#RW(2,4);
RW:=proc(a,b) local gu,ro:
if a=0 and b=0 then
 RETURN([]):
fi:

ro:=Roul([a,b]):
if ro=1 then
 gu:=RW(a-1,b):
 RETURN([op(gu),0]):
else
  gu:=RW(a,b-1):
 RETURN([op(gu),1]):
fi:
end:

#RSU(n,k): a random k-subset  of {1, ..., n}. Try: RSU(5,3);
RSU:=proc(n,k) local gu,mu,i:
gu:=RW(n-k,k):

mu:={}:

for i from 1 to n do
 if gu[i]=1 then
   mu:=mu union {i}:
 fi:
od:

mu:

end:

#Roul(L): inputs a list L and outputs i from 1 to nops(L) with prob. L[i]/convert(L,`+`); Try
#Roul([1,2,1]);
Roul:=proc(L) local i,su,r,i1:
su:=convert(L,`+`):

r:=rand(1..su)():

for i from 1 while add(L[i1],i1=1..i-1)<r do od:
i-1:
end:


#RPF(n,a): a random a-parking function of length n. Try
#RPF(5,1);
RPF:=proc(n,a) local k,P1,P,S,i:

if n=0 then
 RETURN([]):
fi:

if n=1 and a=1 then
 RETURN([1]):
fi:

k:=Roul([seq(binomial(n,k)*tn(n-k,a+k-1),k=0..n)])-1:


S:=RSU(n,k):


P1:=RPF(n-k,a+k-1):


P:=[]:
for i from 1 to n do

 if member(i,S) then
  P:=[op(P),1]:
 else
  P:=[op(P),P1[1]+1]:
  P1:=[op(2..nops(P1),P1)]:
 fi:

od:

P:

end:

#IsParking(L): is L a parking function
IsParking:=proc(L) local L1,i:
L1:=sort(L):
for i from 1 to nops(L1) do
 if L1[i]>i then
   RETURN(false):
 fi:
od:
true:
end:

#RPF1(n): a random 1-partking function of length n using the algorithm in the Diaconis-Hicks paper "Probablistic Parking Functions", 
#arxXiv:1611.09821v1 [math.PR], p.4 : Try:
#RPF1(10);
RPF1:=proc(n) local ra,L,i:
ra:=rand(1..n+1):
L:=[seq(ra(),i=1..n)]:

while not IsParking(L) do 
 L:=[seq(L[i]+1,i=1..n)]:
 L:=subs(n+2=1,L):
od:
L:

end:



##End Random Parking Function

#Pn(n,a) the number of a-parking functions of length n for numeric n,a
Pn:=proc(n,a) local k:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

add(binomial(n,k)*Pn(n-k,a+k-1),k=0..n):
end:

#Pnx(n,a,x): the weight enumerator of a- parking functions of length n according to the weight
#x^(sum of parking function). For example, try:
#Pnx(10,1,x);
Pnx:=proc(n,a,x) local k:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

expand(x^n*add(binomial(n,k)*Pnx(n-k,a+k-1,x),k=0..n)):
end:



#Pnz(n,a,z,K): the  polynomial, in z, whose coefficient of z^i is
#the numerator of the i-th binomial moments of a- parking functions of length n up to i=K. Try
#Pnz(10,1,z,4);
Pnz:=proc(n,a,z,K) local k,gu,i:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

gu:=expand(add(binomial(n,i)*z^i,i=0..K)*add(binomial(n,k)*Pnz(n-k,a+k-1,z,K),k=0..n)):
gu:=add(coeff(gu,z,i)*z^i,i=0..K):
end:


#Fk(a,k,N): The first N terms of the sequence of the k-th factorial moment. Try:
#Fk(1,1,2,10):
Fk:=proc(a,k,N) local gu0,gu,n1,z:
gu0:=[seq(Pn(n1,a),n1=1..N)]:
gu:=[seq(Pnz(n1,a,z,k),n1=1..N)]:
[seq(coeff(gu[n1],z,k)*k!/gu0[n1],n1=1..N)]:
end:


#BZ1(a,N): Fk(a,1,N) according to the bnei zonot
BZ1:=proc(a,N) local n1,j:
[seq(n1*(a+n1+1)/2 -1/2*add(binomial(n1,j)*j!/(a+n1)^(j-1),j=1..n1),n1=1..N)]:
end:


#BZ2(a,N): Fk(a,1,N) according to the bnei zonot
BZ2:=proc(a,N) local n1,j:
[seq(n1*(n1-1)*(a+n1+1)^2/4 + 1/3*n1*(a+n1+1)*(a+n1-1)
-n1*(a+n1+1)/2*add(binomial(n1,j)*j!/(a+n1)^(j-1),j=1..n1)
+add(binomial(n1,j)*j!/(a+n1)^(j-1)*(1/6*j^3-1/6*j+1/2),j=1..n1),n1=1..N)]:
end:

#BZ2fast(a,N):  a faster version of BZ2(a,N), using the truly explicit expression found in the Yao-Zeilberger article. Try
#BZ2fast(2,200);
BZ2fast:=proc(a,N) local S1,n,n1,j,mu,gu:
gu:=FactorialMoment(a,2,S1,n):
mu:=BZ1(a,N):
[seq(subs({n=n1,S1=mu[n1]},gu),n1=1..N)]:
end:


#PnS(n,a): inputs a pos. integer . Outputs the expression for a-parking functions of length n for symbolic a and b. 
#PnS(5,a);
PnS:=proc(n,a) local k,gu,a1:
option remember:

if n=0 then
  RETURN(1):
fi:

gu:=expand(add(binomial(n,k)*subs(a=a+k-1,PnS(n-k,a)),k=1..n)):

factor(subs(a1=a,sum(gu,a=1..a1))):
end:



#PU(U) the number of U-parking functions of length n for a list U . Try:
#PU([1,2,3,4]);
PU:=proc(U) local k,i,n:
option remember:
n:=nops(U):

if nops(U)=0 then
  RETURN(1):
fi:

if U[1]=0 then
  RETURN(0):
fi:

add(binomial(n,k)*PU([seq(U[i]-1,i=k+1..n)] ),k=0..n):
end:

#PUx(U,x) the weight-enuerators of U-parking functions of length n for a list U 
PUx:=proc(U,x) local k,i,n:
option remember:
n:=nops(U):

if nops(U)=0 then
  RETURN(1):
fi:

if U[1]=0 then
  RETURN(0):
fi:

expand(x^nops(U)*add(binomial(n,k)*PUx([seq(U[i]-1,i=k+1..n)],x ),k=0..n)):
end:


#Qn(n,a) the number of ways of arranging n employees and a bosses
Qn:=proc(n,a) local k:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

add(binomial(n,k)*a^k*Qn(n-k,k),k=0..n):
end:


#Pnax(n,a,x): the sum of x^(sum of all entries in the parking function) over the set of a-parking functions of length n by recurrence relation.
Pnax:=proc(n,a,x) local k:
option remember:
if n=0 then
  return 1:
fi:
if n>0 and a=0 then
  return 0:
fi:
return expand(x^n*add(binomial(n,k)*Pnax(n-k,a+k-1,x),k=0..n)):
end:





#FactorialMoment(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) with the observation that the degree of x is at most 1.
FactorialMoment:=proc(a,k,x,y) local d,z,A0,A1,Ak,i,j,sol,F,eq,var,n,c:
option remember:
if k=0 then
  return 1:
fi:
d:=2*k:
A0:=[seq(Pnax(n,a,1),n=1..2*d+5)]:
A1:=[seq(subs(z=1,diff(Pnax(n,a,z),z))/A0[n],n=1..2*d+5)]:
Ak:=[seq(subs(z=1,diff(Pnax(n,a,z),z$k))/A0[n],n=1..2*d+5)]:
var:={seq(seq(c[i,j],i=0..1),j=0..d)}:
eq:={}:
F:=add(add(c[i,j]*x^i*y^j,i=0..1),j=0..d):
for n from 1 to 2*d+5 do
  eq:=eq union {subs({x=A1[n],y=n},F)-Ak[n]}:
od:
sol:=solve(eq, var):
if {sol} <> {} then
  return subs(sol,F):
else
  return(FAIL):
fi:
end:

#FactorialMomentPC(k,x,y): 
#Pre-computed k-th Factorial moment of 1-parking functions in terms of expectation ,x, and y (the length)
#input the symbol x and y, return the pre-computed list of kth factorial moments’ polynomial F(x,y) for 1-parking function (1<=k<=20).
FactorialMomentPC:=proc(k,x,y)  local gu:
gu:=[x, (1/3)*x-y+(10/3)*x*y-(13/6)*y^2+x*y^2-(5/4)*y^3-(1/4)*y^4, -(25/96)*x+(57/32)*y-(221/96)*x*y+(95/192)*y^2+(833/96)*x*y^2-(1193/192)*y^3+(175/32)*x*y^3-(1159/192)*y^4+(3/4)*x*y^4-(135/64)*y^5-(1/4)*y^6, -(3/16)*y^8+(50161/30240)*y^2-(58127/2520)*x*y^2+(167831/10080)*y^3+(2431/216)*x*y^3-(15019/5040)*y^4+(959/48)*x*y^4-(129377/7560)*y^5+(95/16)*x*y^5-(20495/2016)*y^6+(1/2)*x*y^6-(75/32)*y^7+(6733/15120)*x-(26381/5040)*y+(2613/560)*x*y, -(1/8)*y^10-(3398635/258048)*y^8+(5/16)*x*y^8-(275/128)*y^9-(41130167/2322432)*y^6+(59285/2048)*x*y^6-(4284437/129024)*y^7+(1025/192)*x*y^7-(65722591/1161216)*y^3-(409511/4536)*x*y^3+(191607281/3870720)*y^4-(51282689/1161216)*x*y^4+(46236097/829440)*y^5+(159115/3456)*x*y^5-(6904999/5806080)*x+(40409543/1935360)*y-(828259/51840)*x*y-(155709937/11612160)*y^2+(17464103/215040)*x*y^2, (2074834901/27675648)*x*y+(100699179631/830269440)*y^7-(28954629203/276756480)*y+(254702911/1013760)*y^6-(113340755041/830269440)*y^5-(35024021341/103783680)*y^4+(502998577/912384)*x*y^3+(150531505633/1660538880)*y^2+(67281288463/276756480)*y^3-(1460735/96768)*x*y^6+(204605/6144)*x*y^8-(725779255/13837824)*y^8-(4996009/322560)*x*y^4+(3599544499/830269440)*x-(11003402401/29652480)*x*y^2-(30467435/82944)*x*y^5+(856345/9216)*x*y^7-(22525577/439296)*y^9+(275/64)*x*y^9-(414265/28672)*y^10+(3/16)*x*y^10-(225/128)*y^11-(5/64)*y^12, -(6833898713/340623360)*x+(71529387961/113541120)*y-(10441366935521/15941173248)*y^2-(303089223676853/239117598720)*y^3+(1642294505383/711659520)*y^4+(488998720591/29889699840)*y^5-(216533900449297/119558799360)*y^6-(52438004010923/119558799360)*x*y+(10450119276011/4981616640)*x*y^2-(107596899540943/29889699840)*x*y^3+(15555224031859/11955879936)*x*y^4+(46084753689647/59779399680)*y^8+(98235949507/39852933120)*y^7-(28013454103/253034496)*y^10+(1071176999567/5693276160)*y^9-(16996032517/253034496)*y^11-(2081809/147456)*y^12-(1365/1024)*y^13-(3/64)*y^14+(82593679/884736)*x*y^8-(504638135/663552)*x*y^7+(122077315/884736)*x*y^9+(816571/24576)*x*y^10+(1645/512)*x*y^11+(7/64)*x*y^12+(469351634779/218972160)*x*y^5-(29253545033/29859840)*x*y^6, (12687624276700783/112634352230400)*x-(245/256)*y^15-(7/256)*y^16-(166220367528043151/37544784076800)*y+(875/384)*x*y^13+(1/16)*x*y^14+(3537037122201387953/675806113382400)*y^2+(37037548489146917/4770396094464)*y^3-(35005494317090733647/2027418340147200)*y^4+(545392298802858337/119259902361600)*y^5+(164128922704689599/13516122267648)*y^6+(617294929065628529/202741834014720)*x*y-(1102249342896429479/77977628467200)*x*y^2+(1403075170702396481/53353114214400)*x*y^3-(1180921437099101/74724249600)*x*y^4-(167039682858883117/28963119144960)*y^8-(5058998393742679999/1013709170073600)*y^7+(3718920478052887/2145416232960)*y^10+(7977385614935171/7240779786240)*y^9+(1325465818777523/6436248698880)*y^11-(80231492079239/429083246592)*y^12-(182162407/2342912)*y^13-(466939/36864)*y^14-(723949611953/209018880)*x*y^8+(11039009485183/3448811520)*x*y^7-(26423363/24576)*x*y^9+(181196215/663552)*x*y^10+(37712923/221184)*x*y^11+(184681/6144)*x*y^12-(515324366494999/44834549760)*x*y^5+(574512734211109/44834549760)*x*y^6, -(14157836725798171176751/19030700152848384000)*x-(3831578119/46858240)*y^15-(348715/32768)*y^16+(225545766553394326375247/6343566717616128000)*y+(18270787/98304)*x*y^13+(414571/16384)*x*y^14-(352930053011189070257189/7612280061139353600)*y^2-(13296282028755779268361/243983335292928000)*y^3+(22550442859512184759039/157278513659904000)*y^4-(255487033750405478578313/3806140030569676800)*y^5-(346856247771667316952439/4229044478410752000)*y^6-(26052891280165212769/1070102347776000)*x*y+(1051145875782844349888509/9515350076424192000)*x*y^2-(102613077689961226354961/475767503821209600)*x*y^3+(97161489956723352426559/576687883419648000)*x*y^4+(795/512)*x*y^15+(9/256)*x*y^16-(675/1024)*y^17-(1/64)*y^18+(63887867379647888387029/1903070015284838400)*y^8+(48025856822490850687361/679667862601728000)*y^7-(1263289457523656271367/98860780014796800)*y^10-(7045687984011661194391/253742668704645120)*y^9+(11005268331924359/2477713784832)*y^11+(97324717728863153/31384374607872)*y^12+(45238298235631931/366151037091840)*y^13-(65603068545708973/244100691394560)*y^14+(260596351730846557/6121410527232)*x*y^8+(4354914313662551/637646929920)*x*y^7-(49517438321011/98099527680)*x*y^9-(89411172921739/11890851840)*x*y^10-(1717922526391/1557135360)*x*y^11+(2024714293277/4152360960)*x*y^12+(60569360876732664433/1138199769907200)*x*y^5-(5061352082606489207/38258815795200)*x*y^6, 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gu[k]:
end:


#RawMoment(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) 
#where E(n,a,k) is the raw moment (i.e. E(X^k)) by 2nd Stirling number and 
RawMoment:=proc(a,k,x,y) local i:
option remember:
add(Stirling2(k,i)*FactorialMoment(a,i,x,y),i=0..k):
end:

#RawMomentPC(k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) is the raw moment (i.e. E(X^k)) by 2nd Stirling number and FM when 1<=k<=20 and a=1.
RawMomentPC:=proc(k,x,y) local i,L:
L:=[seq(FactorialMomentPC(i,x,y),i=1..k)]:
add(Stirling2(k,i)*L[i],i=1..k):
end:

#RM(x,y): input the symbol x and y, return the pre-computed list of kth raw moments’ polynomial F(x,y) for 1-parking function (1<=k<=20).
RM:=proc(x,y) 
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end:

#CentralMoment(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) 
#is the central moment (i.e. E[(X-mu)^k)] by binomial theorem and RMoment.
CentralMoment:=proc(a,k,x,y) local i:
expand(add(binomial(k,i)*RawMoment(a,i,x,y)*(-x)^(k-i),i=0..k)):
end:

#CentralMomentPC(k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) 
#where E(n,a,k) is the central moment (i.e. E[(X-mu)^k)] by binomial theorem and RM.
CentralMomentPC:=proc(k,x,y) local i,L:
if not (type(k, integer) and k>0 and type(x,symbol) and type(y,symbol) ) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

if k>20 then
 print(` so far we only have up to the 20-th central moment, use CentralMoment(k,x,y) instead, but it may take awhile.`):
 RETURN(FAIL):
fi:

L:=[1,op(RM(x,y))]:
expand(add(binomial(k,i)*L[i+1]*(-x)^(k-i),i=0..k)):
end:

#CM(x,y): input the symbol x and y, return the pre-computed list of kth central moments’ polynomial F(x,y) for 1-parking function (1<=k<=20).
CM:=proc(x,y) local k: 
[seq(CentralMomentPC(k,x,y),k=1..20)]:
end:


#FindPoly(L,d,x): find a polynomial f(x) up to degree d such that f(i)=L[i] 
FindPoly:=proc(L,d,x) local n, var, eq,f,a,i:
n:=nops(L):
if d>n-2 then
  return(FAIL):
fi:
var:={seq(a[i],i=0..d)}:
f:=add(a[i]*x^i,i=0..d):
eq:={seq(subs(x=i,f)-L[i],i=1..n)}:
var:=solve(eq, var):
if {var}<>{} then
  return subs(var, f):
else 
  return(FAIL):
fi:
end:

#PolyAK(a,k,d,i,j): Find the coefficient of x^i*y^j in FactorialMoment(a,k, x,y) which is a polynomial in a up to degree d. Return fail if the coefficient is not a polynomial of degree up to d. 
PolyAK:=proc(a,k,d,i,j) local L,n,b,L1,x,y:
n:=d+2:
L:=[seq(FactorialMoment(b,k,x,y),b=1..n)]:
if i=0 and j=0 then
  L1:=[seq(subs({x=0,y=0},L[b]),b=1..n)]:
elif i>0 and j=0 then
  L1:=[seq(subs(y=0,coeff(L[b],x^i)),b=1..n)]:
elif i=0 and j>0 then
  L1:=[seq(subs(x=0,coeff(L[b],y^j)),b=1..n)]:
elif i>0 and j>0 then
  L1:=[seq(coeff(coeff(L[b],y^j),x^i),b=1..n)]:
fi:
FindPoly(L1,d,a):
end:





#LimScMoment(K): the list of coefficients of the asymptotic expression of scaled kth moment for 1-parking function. (i.e., 
#the coefficient of n^(3*k/2), we already obtained that E(n,a) is asymptotic to n^2/2-sqrt(2*Pi)/4*n^(3/2).). EXACTLY (1<=k<=K).
LimScMoment:=proc(K) local gu,gu1,sig,x,k,L,n:
if not( type(K,integer) and K>=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

gu:=[seq(CentralMoment(1,k,x,n),k=1..K)]:
gu:=expand(subs(x=n^2/2-sqrt(2*Pi)/4*n^(3/2),gu)):
sig:=gu[2]:
sig:=limit(expand(sig/n^3),n=infinity):

L:=[0,1]:
for k from 3 to K do 
  gu1:=limit(expand(gu[k]/n^(3*k/2)),n=infinity):
  gu1:=gu1/sig^(k/2):
 L:=[op(L),gu1]:
od:
L:
end:

#LimScMomentPC(K): Like LimScMoment(K) but much faster, using pre-computed output
LimScMomentPC:=proc(K) local sig,x,k,L,n:
if not (type(K,integer) and K>1 and K<=20) then
print(`The argument must be an integer between 2 and K`):
 RETURN(FAIL):
fi:
sig:=CM(x,n)[2]:
sig:=expand(subs(x=n^2/2-sqrt(2*Pi)/4*n^(3/2),sig)):
sig:=limit(expand(sig/n^3),n=infinity):
L:=[op(1..K,CM(x,n))]:
L:=expand(subs(x=n^2/2-sqrt(2*Pi)/4*n^(3/2),L)):
for k from 1 to K do 
  L[k]:=limit(expand(L[k]/n^(3*k/2)),n=infinity):
  L[k]:=L[k]/sig^(k/2):
od:
[seq(L[k],k=1..K)]:
end:

#FactorialMomentGeneraOld(a,k,x,y):  Old version, VERY slow
#input the symbol a (for a-parking function), return the polynomial f(x,y) for kth factorial moment with coefficients which are polynomials of a. 
FactorialMomentGeneralOld:=proc(a,k,x,y) local d,i,j:
d:=3*k-2:
add(add(PolyAK(a,k,d,i,j)*x^i*y^j,i=0..1),j=0..2*k):
end:



#FacorialMomentGeneralPC(a,k,x,y):
#Like FactorialMomentGenearl(a,k,x,y) but much faster, using pre-computed output.
#k must be between 2 and 12:  
FactorialMomentGeneralPC:=proc(a,k,x,y) local gu:

if not (type(k,integer)  and k>1 and k<=12) then
 print(`The second argument, k, must be an integer between 2 and 12 `):
 RETURN(FAIL):
fi:

gu:=
[x, ((1/3)*a^3+(2/3)*a^2+(1/3)*a-1)*x+(-(1/6)*a^4-(1/3)*a^3-(1/4)*a^2-(1/6)*a-1/12)*y+(2*a+4/3)*x*y+(-(1/6)*a^3-(11/12)*a^2-(2/3)*a-5/12)*y^2+x*y^2+(-a-1/4)*y^3-(1/4)*y^4, ((5/96)*a^6+(19/48)*a^5+(19/24)*a^4-(1/2)*a^3-2*a^2-a+2)*x+(-(5/192)*a^7-(19/96)*a^6-(27/64)*a^5+(5/48)*a^4+(27/32)*a^3+(35/48)*a^2+(1/2)*a+1/4)*y+((35/32)*a^4+(79/24)*a^3+(145/48)*a^2-(131/24)*a-17/4)*x*y+(-(5/192)*a^6-(23/32)*a^5-(91/48)*a^4-(3/2)*a^3+(27/16)*a^2+(163/96)*a+5/4)*y^2+((1/2)*a^3+(143/32)*a^2+(87/16)*a-83/48)*x*y^2+(-(51/64)*a^4-(207/64)*a^3-(85/24)*a^2+(217/192)*a+11/48)*y^3+(3*a+79/32)*x*y^3+(-(1/4)*a^3-(199/64)*a^2-(43/16)*a+1/96)*y^4+(3/4)*x*y^4+(-(3/2)*a-39/64)*y^5-(1/4)*y^6, ((79/15120)*a^9+(11/126)*a^8+(331/720)*a^7+(457/720)*a^6-(199/120)*a^5-(281/60)*a^4+(3/5)*a^3+(22/3)*a^2+(11/3)*a-6)*x+(-(79/30240)*a^10-(11/252)*a^9-(703/3024)*a^8-(1537/4320)*a^7+(6763/10080)*a^6+(2559/1120)*a^5+(2573/5040)*a^4-(1509/560)*a^3-(2209/840)*a^2-(11/6)*a-109/120)*y+((79/336)*a^7+(151/72)*a^6+(4063/720)*a^5-(697/720)*a^4-(1327/72)*a^3-(15683/840)*a^2+(2603/140)*a+97/6)*x*y+(-(79/30240)*a^9-(1199/7560)*a^8-(5357/4320)*a^7-(13373/4320)*a^6+(8759/15120)*a^5+(28309/3024)*a^4+(8179/840)*a^3-(3539/1008)*a^2-(3923/720)*a-73/16)*y^2+((5/48)*a^6+(161/48)*a^5+(49/4)*a^4+(1781/144)*a^3-(2963/144)*a^2-(599/18)*a+6833/2520)*x*y^2+(-(19/112)*a^7-(78949/30240)*a^6-(17729/2160)*a^5-(34115/6048)*a^4+(396269/30240)*a^3+(582689/30240)*a^2+(9851/30240)*a+731/1260)*y^3+((35/16)*a^4+(613/48)*a^3+(59/3)*a^2-(433/48)*a-6199/432)*x*y^3+(-(5/96)*a^6-(87/32)*a^5-(7909/720)*a^4-(82153/6048)*a^3+(58799/6048)*a^2+(33419/2520)*a+10399/7560)*y^4+((1/2)*a^3+(127/16)*a^2+(49/4)*a-17/24)*x*y^4+(-(43/32)*a^4-(145/16)*a^3-(12557/1008)*a^2+(3299/1008)*a+74917/30240)*y^5+(3*a+47/16)*x*y^5+(-(1/4)*a^3-(155/32)*a^2-(85/16)*a+121/504)*y^6+(1/2)*x*y^6+(-(3/2)*a-27/32)*y^7-(3/16)*y^8, 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gu[k]:
end:








  


###starts AREA procedures

#PnxArea(n,a,x): Inputs positive integers n and a, and a variable x, outputs the weight enumerator of a-parking functions according to the AREA
#x^(binomial(n+1/2)-sum) of parking function). For example, try:
#PnxArea(10,1,x);
PnxArea:=proc(n,a,x) local gu:
gu:=Pnx(n,a,x):
sort(expand(subs(x=1/x,gu)*x^( (2*a+n-1)*n/2 ))):
end:

#PnxAreaDirect(n,a,x): the weight enumerator of a- parking functions of length n according to the weight
#x^(sum of parking function). For example, try:
#PnxAreaDirect(10,1,x);
PnxAreaDirect:=proc(n,a,x) local k:
option remember:


if n=0 then
  RETURN(1):
fi:

if a<=0 then
 RETURN(0):
fi:

expand(add(binomial(n,k)*x^(k*(k+2*a-3)/2)*PnxAreaDirect(n-k,a+k-1,x),k=0..n)):
end:


#FactorialMomentArea(a,k,x,y): find a polynomial f(x,y) such that f(ExpectatioOfArea,n)=k-th FactorialMoment of Area. Try
#FactorialMomentArea(1,k,A1,n);
FactorialMomentArea:=proc(a,k,x,y) local d,z,A0,A1,Ak,i,j,sol,F,eq,var,n,c:
option remember:
if k=0 then
  return 1:
fi:
d:=2*k:
A0:=[seq(PnxArea(n,a,1),n=1..2*d+5)]:
A1:=[seq(subs(z=1,diff(PnxArea(n,a,z),z))/A0[n],n=1..2*d+5)]:
Ak:=[seq(subs(z=1,diff(PnxArea(n,a,z),z$k))/A0[n],n=1..2*d+5)]:
var:={seq(seq(c[i,j],i=0..1),j=0..d)}:
eq:={}:
F:=add(add(c[i,j]*x^i*y^j,i=0..1),j=0..d):
for n from 1 to 2*d+5 do
  eq:=eq union {subs({x=A1[n],y=n},F)-Ak[n]}:
od:
sol:=solve(eq, var):
if {sol} <> {} then
  return subs(sol,F):
else
  return(FAIL):
fi:
end:


#RawMomentArea(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k)  for a-parking statistics according to the area
RawMomentArea:=proc(a,k,x,y) local i:
option remember:
add(Stirling2(k,i)*FactorialMomentArea(a,i,x,y),i=0..k):
end:

#CentralMomentArea(a,k,x,y): find a polynomial f(x,y) such that f(E[n,a,1],n)=E(n,a,k) where E(n,a,k) 
#is the central moment (i.e. E[(X-mu)^k)] where X is AREA
CentralMomentArea:=proc(a,k,x,y) local i:
expand(add(binomial(k,i)*RawMomentArea(a,i,x,y)*(-x)^(k-i),i=0..k)):
end:

#LimScMomentArea(K): the list of coefficients of the asymptotic expression of scaled kth moment for 1-parking function. (i.e., 
#the coefficient of n^(3*k/2), we already obtained that E(n,a) is asymptotic to n^2/2-sqrt(2*Pi)/4*n^(3/2).). 
#According to AREA. 
LimScMomentArea:=proc(K) local gu,gu1,sig,x,k,L,n:
if not( type(K,integer) and K>=2) then
 print(`Bad input`):
 RETURN(FAIL):
fi:

gu:=[seq(CentralMomentArea(1,k,x,n),k=1..K)]:
gu:=expand(subs(x=sqrt(2*Pi)/4*n^(3/2),gu)):
sig:=gu[2]:
sig:=limit(expand(sig/n^3),n=infinity):

L:=[0,1]:
for k from 3 to K do 
  gu1:=limit(expand(gu[k]/n^(3*k/2)),n=infinity):
  gu1:=gu1/sig^(k/2):
 L:=[op(L),gu1]:
od:
L:
end:

ez:=proc(): print(`BetaList(k), gr(r) , bk(k) , EBexList(k), CM(L) , SCM(L)  `): end:

#begin from Svante Janson's article “Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas”.
#OmegaK(k): input k, calculate Omega_k based on the formula (9) in the paper 
#“Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas”.
OmegaK:=proc(k) local j:
option remember:
if k<0 or not type(k, integer) then
  return(FAIL):
elif k=0 then
  return -1:
elif k=1 then
  return 1/2:
fi:
return (3*k-4)*k*OmegaK(k-1)/2+add(binomial(k,j)*OmegaK(j)*OmegaK(k-j),j=1..k-1)/2:
end:

#OmegaList(k): input k, output a list of length k such that the nth element is OmegaK(n).
OmegaList:=proc(k) local i:
[seq(OmegaK(i),i=1..k)]:
end:

#EBexK(k): input k, output the expectation of (B_ex^k).
EBexK:=proc(k)
2*sqrt(Pi)*OmegaK(k)/GAMMA((3*k-1)/2)/2^(3*k/2):
end:

#EBexListFlajolet(k): input k, output a list of length k such that the nth entry is EBexK(n).
EBexListFlajolet:=proc(k) local i:
[seq(EBexK(i),i=1..k)]:
end:





gr:=proc(r)  local j:
option remember:
if r<1 then
 RETURN(FAIL):
fi:

12*r/(6*r-1)*GAMMA(3*r+1/2)/GAMMA(r+1/2) -add( binomial(r,j)*GAMMA(3*j+1/2)/GAMMA(j+1/2)*gr(r-j),j=1..r-1):
end:

bk:=proc(k) option remember:
if k<0 then
 RETURN(FAIL):
fi:
if k=0 then
 RETURN(1):
fi:

(36*sqrt(2))^(-k)*2*sqrt(Pi)/GAMMA((3*k-1)/2)*gr(k):
end:

#EBexListLouchard(k): input k, output a list of length k such that the nth entry is EBexK(n).
EBexListLouchard:=proc(k) local i:
[seq(bk(i),i=1..k)]:
end:


#CenM(L): inputs the list of moments L, outputs the list of centralized moments
CenM:=proc(L) local mu,k,r:
mu:=L[1]:
[seq((-mu)^k+add(binomial(k,r)*L[k-r]*(-mu)^r,r=0..k-1),k=1..nops(L))]:
end:

#SCenM(L): inputs the list of moments L, outputs the list of scaled centralized moments
SCenM:=proc(L) local gu,k:
gu:=CenM(L):
if nops(gu)<2 or gu[2]=0 then
 RETURN(FAIL):
fi:

[0,seq(gu[k]/gu[2]^(k/2),k=2..nops(gu))]:

end:
#end stuff from Svante Janson's article “Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas”.

#ProveScaledParking(K): rigorously proves that the limit of the first K sca;ed moments  of the area statistics on parking functions 
#coincides with the scaled moments of the "area under Brownian Excursion" continous random variable. Try:
#ProveScaledParking(4);

ProveScaledParking:=proc(K) local gu,mu,i:

gu:=LimScMoment(K):
print(`The first `, K, `rigorously derived limits of the SUM statistics are`):
print(``):
print(gu):
print(``):
gu:=[seq(gu[i]*(-1)^i,i=1..nops(gu))]:
print(``):
print(`hence The first `, K, `rigorously derived limits of the AREA statistics are`):
print(``):
print(gu):
print(``):

print(`The first scaled moments of the "area under Brownian Excursion" continuous random variable (often called Airy distribution) `):
print(``):
mu:=SCenM(EBexListLouchard(K)):
print(``):
print(mu):
print():
if simplify(mu-gu)=[0$nops(gu)] then
 print(`They are indeed the same`):
else
 print(`Too bad, we disproved it.`):
fi:
evalb(simplify(mu-gu)=[0$nops(gu)]):
end:

#ProveScaledParkingPC(K): same as ProveScaledParking(K) but using pre-computed data
ProveScaledParkingPC:=proc(K) local gu,mu,i:

if K>20 then
 print(`Not yet implemented`):
 RETURN(FAIL):
fi:

gu:=LimScMomentPC(K):
print(`The first `, K, `rigorously derived limits of the SUM statistics are`):
print(``):
print(gu):
print(``):
gu:=[seq(gu[i]*(-1)^i,i=1..nops(gu))]:
print(``):
print(`hence The first `, K, `rigorously derived limits of the AREA statistics are`):
print(``):
print(gu):
print(``):

print(`The first scaled moments of the "area under Brownian Excursion" continuous random variable (often called Airy distribution) `):
print(``):
mu:=SCenM(EBexListLouchard(K)):
print(``):
print(mu):
print():
if simplify(mu-gu)=[0$nops(gu)] then
 print(`They are indeed the same`):
else
 print(`Too bad, we disproved it.`):
fi:
evalb(simplify(mu-gu)=[0$nops(gu)]):
end:




#Wn(n): the expectation of total height over all rooted labelled trees with n vertices, according to the
#famous formula of Riordan and Sloane that initiated the OEIS. Try: Wn(10);
Wn:=proc(n) local k:
 n!/n^(n-1)*add(n^k/k!,k=0..n-2):
end:


#BZ1area(a,N): FkArea(a,1,N) according to the bnei zonot
BZ1area:=proc(a,N) local n1,j:
[seq(n1*(a-2)/2 + 1/2*add(binomial(n1,j)*j!/(a+n1)^(j-1),j=1..n1),n1=1..N)]:
end:

#BZ11area(n): the explicit expression for the expectation of the area of a 1parking function (i.e. a usual parking function.
#Try:
#[seq(BZ11area(i),i=1..10)];
BZ11area:=proc(n) local k:
 1/2*(n+1)!/(n+1)^(n)*add((n+1)^k/k!,k=0..n-1)-n/2:
end:

#LC(P,A1,n): inputs a polynomial P in A1 and n, of degree 1 in A1 and outputs the leading asymptotics if A1=n^(3/2)*sqrt(Pi/2). Try
#LC(FactorialMomentArea(1,2,A1,n),A1,n);
LC:=proc(P,A1,n) local gu0,gu1,d0,d1:
gu0:=coeff(P,A1,0):
gu1:=coeff(P,A1,1):
d0:=degree(gu0,n):
d1:=degree(gu1,n):

if d1+3/2>d0 then
 RETURN(coeff(gu1,n,d1)*sqrt(Pi*2)/4*n^(d1+3/2)):
else
 RETURN(coeff(gu0,n,d0)*n^d0):
fi:
end:


#FMareaStory(N): Inputs a positive integer N and outputs an article with explicit expressions for the
#first N factorial moments of the r.v. "area" defined on parking functions (the area is n*(n+1)/2-sum)
#including the leading asymptotics, and proving that in the limit they converge to the moments of
#the (continuous) so-called Airy distribution (area under Brownian excursion). Try:
#FMareaStory(4,A1,n);
FMareaStory:=proc(N,A1,n) local gu,x,i,lu,t0,gu1,k,k1,A:
t0:=time():
lu:=EBexListFlajolet(N):
print(``):
print(`Truly Explicit (Polynomial!) expressions for the first`, N, `factorial moments of the Area of Parking Functions of length n`):
print(`in terms of n and the expectation , and proving (in purely elementary means!) that, up to these moments, the limiting distribution`):
print(`is the Airy distribution`):
print(``):
print(`By Shalosh B. Ekhad and Yukun Yao's Computer`):
print(``):
print(`Consider the sample space of all parking functions of length n, and let A be the random variable`):
print(`area, that is defined as n*(n+1)/2-SumOfParkingFunction `):
print(``):
print(`In other words `):
print(``):
print(n*(n+1)/2-Sum(x[i],i=1..n)):
print(``):
print(`It follows from many of the known bijections to labelled trees that the Expectation,let's call it A1, is given expclictly by the expression`):
print(``):
print(A1=1/2*(n+1)!/(n+1)^(n)*Sum((n+1)^k/k!,k=0..n-1)-n/2):
print(``):
print(`via the the Riordan-Sloane famous expression for the expectation of the sum of the total height for rooted labelled trees`):
print(``):
print(n!/(n)^(n-1)*Sum(n^k/k!,k=0..n-2)):
print(``):
print(`It also follows from the natural recurrence involving the more general parking function`):
print(`and it was also derived (via a complicated argument) by J. Kung and C. Yan.`):
print(``):
print(`Thanks to Ramanujan (and Watson, see the Riordan-Sloane paper), we know that A1 is asymptotic to`, sqrt(2*Pi)/4*n^(3/2) )  :
print(``):
print(`Note that the expectation of the Airy distribution is`):
print(``):
print(lu[1]):
print(``):
gu:=[A1]:

for k1 from 2 to N do
print(``):
gu1:=FactorialMomentArea(1,k1,A1,n):
#gu1:=collect(gu1,A):
gu1:=sort(coeff(gu1,A1,0))+sort(coeff(gu1,A1,1))*A1:
print(`The `, k1, `-th factorial moment of the area, in terms of A1 and n is given by`):
print(``):
print(gu1):
print(``):
print(`and in Maple format `):
print(``):
lprint(gu1):
gu:=[op(gu),gu1]:
print(``):
print(`and in latex `):
print(``):
print(latex(gu1)):
print(``):
print(`This is asymptotic to`):
print(``):
print(LC(gu1,A1,n)):
print(``):
print(`compare this to the`, k1, `-th moment of the Airy distribution `,lu[k1]):
print(``):
od:
print(``):
print(`------------------------------------------------------------------------`):
print(``):
print(`To sum up, the expressions for the first`,N, `factorial moments are`):
print(``):
lprint(gu):
print(``):
print(`and, at least for k from 1 to`, N, ` the k-th moment is asymptotic to the corresponding moment of the Airy distribution times`):
print(``):
print(n^(3/2*k) ):
print(``):
print(`this proves, that up to the`, N, `-th moment , the limiting distribution is indeed the Airy distribution. `):
print(``):
print(`---------------------------------------------------------`):
print(``):
print(`This ends this article, that took`, time()-t0, `seconds to produce. `):

end:







#LIF(L,x): Lagrange interpolation formula
LIF:=proc(L,x) local i,mu,gu,mu1:
mu:=mul(x-L[i][1],i=1..nops(L)):

gu:=0:

for i from 1 to nops(L) do
mu1:=normal(mu/(x-L[i][1])):
gu:=expand(gu+L[i][2]*mu1/subs(x=L[i][1],mu1)):
od:

end:

#FactorialMomentGeneral(a,k,x,y): 
#input the symbol a (for a-parking function), return the polynomial f(x,y) for kth factorial moment with coefficients which are polynomials of a. 
FactorialMomentGeneral:=proc(a,k,x,y) local a1:
LIF([seq([a1,FactorialMoment(a1,k,x,y)],a1=1..3*k)],a):
end:


#FactorialMomentGeneralArea(a,k,x,y): 
#input the symbol a (for a-parking function), return the polynomial f(x,y) for kth factorial moment with coefficients which are polynomials of a. 
FactorialMomentGeneralArea:=proc(a,k,x,y) local a1:
LIF([seq([a1,FactorialMomentArea(a1,k,x,y)],a1=1..3*k)],a):
end: