Identifier
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 5
[1,2] => 3
[2,1] => 3
[3] => 1
[1,1,1,1] => 14
[1,1,2] => 9
[1,2,1] => 10
[1,3] => 4
[2,1,1] => 9
[2,2] => 6
[3,1] => 4
[4] => 1
[1,1,1,1,1] => 42
[1,1,1,2] => 28
[1,1,2,1] => 32
[1,1,3] => 14
[1,2,1,1] => 32
[1,2,2] => 22
[1,3,1] => 17
[1,4] => 5
[2,1,1,1] => 28
[2,1,2] => 19
[2,2,1] => 22
[2,3] => 10
[3,1,1] => 14
[3,2] => 10
[4,1] => 5
[5] => 1
[1,1,1,1,1,1] => 132
[1,1,1,1,2] => 90
[1,1,1,2,1] => 104
[1,1,1,3] => 48
[1,1,2,1,1] => 107
[1,1,2,2] => 75
[1,1,3,1] => 62
[1,1,4] => 20
[1,2,1,1,1] => 104
[1,2,1,2] => 72
[1,2,2,1] => 84
[1,2,3] => 40
[1,3,1,1] => 62
[1,3,2] => 45
[1,4,1] => 26
[1,5] => 6
[2,1,1,1,1] => 90
[2,1,1,2] => 62
[2,1,2,1] => 72
[2,1,3] => 34
[2,2,1,1] => 75
[2,2,2] => 53
[2,3,1] => 45
[2,4] => 15
[3,1,1,1] => 48
[3,1,2] => 34
[3,2,1] => 40
[3,3] => 20
[4,1,1] => 20
[4,2] => 15
[5,1] => 6
[6] => 1
[1,1,1,1,1,1,1] => 429
[1,1,1,1,1,2] => 297
[1,1,1,1,2,1] => 345
[1,1,1,1,3] => 165
[1,1,1,2,1,1] => 359
[1,1,1,2,2] => 255
[1,1,1,3,1] => 219
[1,1,1,4] => 75
[1,1,2,1,1,1] => 359
[1,1,2,1,2] => 252
[1,1,2,2,1] => 295
[1,1,2,3] => 145
[1,1,3,1,1] => 233
[1,1,3,2] => 171
[1,1,4,1] => 107
[1,1,5] => 27
[1,2,1,1,1,1] => 345
[1,2,1,1,2] => 241
[1,2,1,2,1] => 281
[1,2,1,3] => 137
[1,2,2,1,1] => 295
[1,2,2,2] => 211
[1,2,3,1] => 185
[1,2,4] => 65
[1,3,1,1,1] => 219
[1,3,1,2] => 157
[1,3,2,1] => 185
[1,3,3] => 95
[1,4,1,1] => 107
[1,4,2] => 81
[1,5,1] => 37
[1,6] => 7
[2,1,1,1,1,1] => 297
[2,1,1,1,2] => 207
[2,1,1,2,1] => 241
[2,1,1,3] => 117
[2,1,2,1,1] => 252
[2,1,2,2] => 180
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Description
Number of parabolic noncrossing partitions indexed by the composition.
Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.
Also the number of elements in the $\nu$-Tamari lattice with $\nu = \nu_\alpha = 1^{\alpha_1} 0^{\alpha_1} \cdots 1^{\alpha_k} 0^{\alpha_k}$, the bounce path indexed by the composition $\alpha$. These elements are Dyck paths weakly above the bounce path $\nu_\alpha$.
References
[1] Mühle, H., Williams, N. Tamari Lattices for Parabolic Quotients of the Symmetric Group arXiv:1804.02761
[2] Bergeron, N., Ceballos, C., Pilaud, V. Hopf dreams arXiv:1807.03044
[2] Bergeron, N., Ceballos, C., Pilaud, V. Hopf dreams arXiv:1807.03044
Code
def contains(A,B):
Aa = A.to_area_sequence()
Bb = B.to_area_sequence()
return all( Aa[i] >= Bb[i] for i in range(len(Aa)) )
def statistic(L):
n = sum(list(L))
Bp = []
for a in L:
Bp += ([1] * a) + ([0] * a)
Bp = DyckWord(Bp)
return sum(1 for D in DyckWords(n) if contains(D, Bp))
Created
Dec 12, 2018 at 11:36 by Wenjie Fang
Updated
Dec 13, 2018 at 19:02 by Martin Rubey
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