##The following output file contains 5 computer-generated articles,
#describing all the "Bona classes": (see article http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/abnormal.html )
#using permutations of length up to 9, for the r.v. "number of occurrences of the pattern pat" for
#all patterns of length k, for k from 2 to 6. It is almost certain the the sequence
#number of distinct Bona classes for 123-avoiding permutations" for k, staring with k=1 is
#[1,2,3,6,12,32]. Can you guess a pattern?
#-------------------------------
Empirical Study of all Sequences "Sum of Occurrences of a Pattern" over all \
123-avoiding permutations for all patterns of length , 2
for permutations up to length , 9
By Shalosh B. Ekhad
If you go up to permutations of length, 9, there are , 2,
different 123-Bona classes
The sequence, [0, 0, 1, 6, 29, 130, 562, 2380, 9949, 41226]
belongs to the following patterns, {[1, 2]}
The sequence, [0, 0, 1, 9, 55, 290, 1418, 6629, 30091, 133806]
belongs to the following patterns, {[2, 1]}
#-------------------------------start k=3
Empirical Study of all Sequences "Sum of Occurrences of a Pattern" over all \
123-avoiding permutations for all patterns of length , 3
for permutations up to length , 9
By Shalosh B. Ekhad
If you go up to permutations of length, 9, there are , 3,
different 123-Bona classes
The sequence, [0, 0, 0, 1, 9, 57, 312, 1578, 7599, 35401]
belongs to the following patterns, {[1, 3, 2], [2, 1, 3]}
The sequence, [0, 0, 0, 1, 11, 81, 500, 2794, 14649, 73489]
belongs to the following patterns, {[2, 3, 1], [3, 1, 2]}
The sequence, [0, 0, 0, 1, 16, 144, 1016, 6271, 35584, 190628]
belongs to the following patterns, {[3, 2, 1]}
#-------------------------------start k=4
Empirical Study of all Sequences "Sum of Occurrences of a Pattern" over all \
123-avoiding permutations for all patterns of length , 4
for permutations up to length , 9
By Shalosh B. Ekhad
If you go up to permutations of length, 9, there are , 6,
different 123-Bona classes
The sequence, [0, 0, 0, 0, 1, 12, 94, 608, 3525, 19044]
belongs to the following patterns,
{[1, 4, 3, 2], [2, 1, 4, 3], [2, 4, 1, 3], [3, 1, 4, 2], [3, 2, 1, 4]}
The sequence, [0, 0, 0, 0, 1, 13, 112, 796, 5041, 29539]
belongs to the following patterns, {[3, 4, 1, 2]}
The sequence, [0, 0, 0, 0, 1, 15, 139, 1028, 6660, 39549]
belongs to the following patterns,
{[2, 4, 3, 1], [3, 2, 4, 1], [4, 1, 3, 2], [4, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 1, 16, 160, 1276, 8875, 56284]
belongs to the following patterns, {[4, 2, 3, 1]}
The sequence, [0, 0, 0, 0, 1, 18, 186, 1490, 10299, 64644]
belongs to the following patterns, {[3, 4, 2, 1], [4, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 1, 25, 310, 2811, 21321, 144085]
belongs to the following patterns, {[4, 3, 2, 1]}
#-------------------------------start k=5
Empirical Study of all Sequences "Sum of Occurrences of a Pattern" over all \
123-avoiding permutations for all patterns of length , 5
for permutations up to length , 9
By Shalosh B. Ekhad
If you go up to permutations of length, 9, there are , 12,
different 123-Bona classes
The sequence, [0, 0, 0, 0, 0, 1, 15, 140, 1045, 6835]
belongs to the following patterns, {[1, 5, 4, 3, 2], [2, 1, 5, 4, 3],
[2, 5, 1, 4, 3], [2, 5, 4, 1, 3], [3, 1, 5, 4, 2], [3, 2, 1, 5, 4],
[3, 2, 5, 1, 4], [3, 5, 2, 1, 4], [4, 1, 5, 3, 2], [4, 2, 1, 5, 3],
[4, 3, 1, 5, 2], [4, 3, 2, 1, 5]}
The sequence, [0, 0, 0, 0, 0, 1, 16, 157, 1220, 8249]
belongs to the following patterns, {[3, 5, 1, 4, 2], [4, 2, 5, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 1, 17, 180, 1512, 11016]
belongs to the following patterns,
{[3, 5, 4, 1, 2], [4, 3, 5, 1, 2], [4, 5, 1, 3, 2], [4, 5, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 1, 18, 195, 1650, 12021]
belongs to the following patterns,
{[3, 5, 2, 4, 1], [4, 2, 5, 3, 1], [5, 2, 4, 1, 3], [5, 3, 1, 4, 2]}
The sequence, [0, 0, 0, 0, 0, 1, 18, 201, 1779, 13644]
belongs to the following patterns, {[4, 5, 2, 3, 1], [5, 3, 4, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 1, 19, 212, 1825, 13435]
belongs to the following patterns, {[2, 5, 4, 3, 1], [3, 2, 5, 4, 1],
[4, 3, 2, 5, 1], [5, 1, 4, 3, 2], [5, 2, 1, 4, 3], [5, 3, 2, 1, 4]}
The sequence, [0, 0, 0, 0, 0, 1, 20, 232, 2070, 15781]
belongs to the following patterns, {[4, 5, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 1, 21, 259, 2447, 19607]
belongs to the following patterns, {[5, 2, 4, 3, 1], [5, 3, 2, 4, 1]}
The sequence, [0, 0, 0, 0, 0, 1, 23, 289, 2717, 21485]
belongs to the following patterns,
{[3, 5, 4, 2, 1], [4, 3, 5, 2, 1], [5, 4, 1, 3, 2], [5, 4, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 1, 23, 302, 2999, 25083]
belongs to the following patterns, {[5, 3, 4, 2, 1], [5, 4, 2, 3, 1]}
The sequence, [0, 0, 0, 0, 0, 1, 27, 372, 3739, 31201]
belongs to the following patterns, {[4, 5, 3, 2, 1], [5, 4, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 1, 36, 587, 6636, 60545]
belongs to the following patterns, {[5, 4, 3, 2, 1]}
#-------------------------------start k=6
Empirical Study of all Sequences "Sum of Occurrences of a Pattern" over all \
123-avoiding permutations for all patterns of length , 6
for permutations up to length , 9
By Shalosh B. Ekhad
If you go up to permutations of length, 9, there are , 32,
different 123-Bona classes
The sequence, [0, 0, 0, 0, 0, 0, 1, 18, 195, 1650]
belongs to the following patterns, {[1, 6, 5, 4, 3, 2], [2, 1, 6, 5, 4, 3],
[2, 6, 1, 5, 4, 3], [2, 6, 5, 1, 4, 3], [2, 6, 5, 4, 1, 3],
[3, 1, 6, 5, 4, 2], [3, 2, 1, 6, 5, 4], [3, 2, 6, 1, 5, 4],
[3, 2, 6, 5, 1, 4], [3, 6, 2, 1, 5, 4], [3, 6, 2, 5, 1, 4],
[3, 6, 5, 2, 1, 4], [4, 1, 6, 5, 3, 2], [4, 2, 1, 6, 5, 3],
[4, 2, 6, 1, 5, 3], [4, 3, 1, 6, 5, 2], [4, 3, 2, 1, 6, 5],
[4, 3, 2, 6, 1, 5], [4, 3, 6, 2, 1, 5], [4, 6, 3, 1, 5, 2],
[4, 6, 3, 2, 1, 5], [5, 1, 6, 4, 3, 2], [5, 2, 1, 6, 4, 3],
[5, 2, 6, 4, 1, 3], [5, 3, 1, 6, 4, 2], [5, 3, 2, 1, 6, 4],
[5, 4, 1, 6, 3, 2], [5, 4, 2, 1, 6, 3], [5, 4, 3, 1, 6, 2],
[5, 4, 3, 2, 1, 6]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 19, 215, 1886]
belongs to the following patterns, {[3, 6, 1, 5, 4, 2], [4, 2, 6, 5, 1, 3],
[4, 3, 6, 1, 5, 2], [4, 6, 2, 1, 5, 3], [5, 2, 6, 1, 4, 3],
[5, 3, 2, 6, 1, 4]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 20, 235, 2122]
belongs to the following patterns, {[3, 6, 5, 1, 4, 2], [4, 6, 1, 5, 3, 2],
[5, 3, 6, 2, 1, 4], [5, 4, 2, 6, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 20, 237, 2166]
belongs to the following patterns, {[4, 6, 2, 5, 1, 3], [5, 3, 6, 1, 4, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 20, 242, 2281]
belongs to the following patterns, {[4, 6, 3, 5, 1, 2], [5, 3, 6, 4, 1, 2],
[5, 6, 2, 4, 1, 3], [5, 6, 3, 1, 4, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 20, 246, 2382]
belongs to the following patterns, {[5, 6, 3, 4, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 21, 260, 2473]
belongs to the following patterns, {[3, 6, 5, 2, 4, 1], [5, 4, 2, 6, 3, 1],
[6, 3, 5, 2, 1, 4], [6, 4, 1, 5, 3, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 21, 263, 2540]
belongs to the following patterns, {[3, 6, 5, 4, 1, 2], [5, 4, 3, 6, 1, 2],
[5, 6, 1, 4, 3, 2], [5, 6, 3, 2, 1, 4]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 21, 264, 2561]
belongs to the following patterns, {[4, 3, 6, 5, 1, 2], [5, 6, 2, 1, 4, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 21, 265, 2585]
belongs to the following patterns, {[4, 6, 5, 2, 1, 3], [5, 4, 6, 1, 3, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 22, 280, 2709]
belongs to the following patterns, {[3, 6, 2, 5, 4, 1], [4, 2, 6, 5, 3, 1],
[4, 3, 6, 2, 5, 1], [4, 6, 3, 2, 5, 1], [5, 2, 6, 4, 3, 1],
[5, 3, 2, 6, 4, 1], [6, 2, 5, 1, 4, 3], [6, 2, 5, 4, 1, 3],
[6, 3, 1, 5, 4, 2], [6, 3, 2, 5, 1, 4], [6, 4, 2, 1, 5, 3],
[6, 4, 3, 1, 5, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 22, 283, 2780]
belongs to the following patterns, {[4, 6, 2, 5, 3, 1], [5, 3, 6, 2, 4, 1],
[6, 3, 5, 1, 4, 2], [6, 4, 2, 5, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 22, 285, 2823]
belongs to the following patterns, {[4, 6, 5, 1, 3, 2], [5, 4, 6, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 22, 290, 2956]
belongs to the following patterns, {[4, 6, 5, 2, 3, 1], [5, 4, 6, 2, 3, 1],
[6, 4, 5, 1, 3, 2], [6, 4, 5, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 23, 300, 2945]
belongs to the following patterns, {[2, 6, 5, 4, 3, 1], [3, 2, 6, 5, 4, 1],
[4, 3, 2, 6, 5, 1], [5, 4, 3, 2, 6, 1], [6, 1, 5, 4, 3, 2],
[6, 2, 1, 5, 4, 3], [6, 3, 2, 1, 5, 4], [6, 4, 3, 2, 1, 5]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 23, 312, 3242]
belongs to the following patterns, {[5, 6, 2, 4, 3, 1], [5, 6, 3, 2, 4, 1],
[6, 3, 5, 4, 1, 2], [6, 4, 3, 5, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 23, 315, 3329]
belongs to the following patterns, {[6, 4, 5, 2, 3, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 24, 333, 3498]
belongs to the following patterns, {[6, 3, 5, 2, 4, 1], [6, 4, 2, 5, 3, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 25, 346, 3572]
belongs to the following patterns, {[4, 6, 5, 3, 1, 2], [5, 4, 6, 3, 1, 2],
[5, 6, 4, 1, 3, 2], [5, 6, 4, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 25, 357, 3853]
belongs to the following patterns, {[5, 6, 3, 4, 2, 1], [6, 5, 3, 4, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 25, 358, 3863]
belongs to the following patterns, {[5, 6, 4, 2, 3, 1], [6, 4, 5, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 26, 370, 3899]
belongs to the following patterns, {[4, 6, 3, 5, 2, 1], [5, 3, 6, 4, 2, 1],
[6, 5, 2, 4, 1, 3], [6, 5, 3, 1, 4, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 26, 381, 4156]
belongs to the following patterns,
{[6, 2, 5, 4, 3, 1], [6, 3, 2, 5, 4, 1], [6, 4, 3, 2, 5, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 28, 413, 4442]
belongs to the following patterns, {[4, 3, 6, 5, 2, 1], [6, 5, 2, 1, 4, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 28, 414, 4463]
belongs to the following patterns, {[3, 6, 5, 4, 2, 1], [5, 4, 3, 6, 2, 1],
[6, 5, 1, 4, 3, 2], [6, 5, 3, 2, 1, 4]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 29, 438, 4817]
belongs to the following patterns, {[5, 6, 4, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 29, 459, 5322]
belongs to the following patterns, {[6, 3, 5, 4, 2, 1], [6, 4, 3, 5, 2, 1],
[6, 5, 2, 4, 3, 1], [6, 5, 3, 2, 4, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 30, 495, 5970]
belongs to the following patterns, {[6, 5, 3, 4, 2, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 32, 535, 6454]
belongs to the following patterns, {[6, 4, 5, 3, 2, 1], [6, 5, 4, 2, 3, 1]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 33, 538, 6233]
belongs to the following patterns, {[4, 6, 5, 3, 2, 1], [5, 4, 6, 3, 2, 1],
[6, 5, 4, 1, 3, 2], [6, 5, 4, 2, 1, 3]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 38, 673, 8282]
belongs to the following patterns, {[5, 6, 4, 3, 2, 1], [6, 5, 4, 3, 1, 2]}
The sequence, [0, 0, 0, 0, 0, 0, 1, 49, 1015, 13972]
belongs to the following patterns, {[6, 5, 4, 3, 2, 1]}
This took, 3403.153, seconds.