read Cheyne:
read AlgFunEq:
read oCheyne2:
print(`Explicit Expressions, in n, for the First Two Moments of the r.v. Number of occurrences of the pattern 213`):
print(`in the set of 123-permutations of length n, and a linear recurrence for the the third moment`):
print(``):
print(`By Shalosh B. Ekhad `):
print(``):
gu:=CH35:
gu:=t*diff(gu,t):
print(``):
print(`An explicit expression for the expectation (times C(n)), in other total number of occurrences of the pattern 213 in AV(123) is`):
print(`where C(n) denotes the Catalan number C(n)=(2*n)!/(n!*(n+1)! is: `):
print(``):
ka:=NakhesC(subs(t=1,gu),n,C):
print(ka):
print(`and in Maple-input format`):
print(``):
lprint(ka):
print(``):
print(`This reproduces, in a somewhat nicer form, the expression for b_n given in Cor. 24 of Cheyne Homberger's article`):
print(` ElJC 19(3) (2012), #P43 `):
gu:=t*diff(gu,t):
print(``):
print(`An explicit expression (very possibly beyond the scope of mere humans) `):
print(` for the second moment (times C(n)), i.e. the sum of (#occurrences of the pattern 213(pi))^2 over pi in Av_n(123) `):
print(`where C(n) denotes the Catalan number C(n)=(2*n)!/(n!*(n+1)! is: `):
print(``):
ka:=NakhesC(subs(t=1,gu),n,C):
print(ka):
print(`and in Maple-input format`):
print(``):
lprint(ka):
gu:=t*diff(gu,t):
print(``):
print(`We could not find an explicit expression for the third moment (times C(n)), i.e. the sum of (#occurrences of the pattern 213(pi))^3 over pi in Av_n(123) `):
print(`but we found the next best thing, a linear recurrence equation with polynomial coefficients`):
print(`We bet that no human would ever be able to find it!`):
print(``):
print(`Calling this sequence M3(n) `):
print(``):
gu:=subs(t=1,gu):
gu:=[op(2..nops(gu),gu)]:
ope:=Findrec(gu,n,N,10):
print(ope):
print(``):
print(add(factor(coeff(ope,N,i))*M3(n+i),i=0..degree(ope,N))=0):
print(`and in Maple-input format`):
print(``):
lprint(add(factor(coeff(ope,N,i))*M3(n+i),i=0..degree(ope,N))=0):
print(`This ends this article, that took`, time(), `seconds. but relying on pre-computed data that took much longer `):
quit: