*****************************************************************************
* www.FindStat.org - The Combinatorial Statistic Finder *
* *
* Copyright (C) 2019 The FindStatCrew <info@findstat.org> *
* *
* This information is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. *
*****************************************************************************
-----------------------------------------------------------------------------
Statistic identifier: St001312
-----------------------------------------------------------------------------
Collection: Integer compositions
-----------------------------------------------------------------------------
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 $\nu_\alpha$.
-----------------------------------------------------------------------------
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]]
-----------------------------------------------------------------------------
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))
-----------------------------------------------------------------------------
Statistic 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
[2,1,3,1] => 157
[2,1,4] => 55
[2,2,1,1,1] => 255
[2,2,1,2] => 180
[2,2,2,1] => 211
[2,2,3] => 105
[2,3,1,1] => 171
[2,3,2] => 126
[2,4,1] => 81
[2,5] => 21
[3,1,1,1,1] => 165
[3,1,1,2] => 117
[3,1,2,1] => 137
[3,1,3] => 69
[3,2,1,1] => 145
[3,2,2] => 105
[3,3,1] => 95
[3,4] => 35
[4,1,1,1] => 75
[4,1,2] => 55
[4,2,1] => 65
[4,3] => 35
[5,1,1] => 27
[5,2] => 21
[6,1] => 7
[7] => 1
[1,1,1,1,1,1,1,1] => 1430
[1,1,1,1,1,1,2] => 1001
[1,1,1,1,1,2,1] => 1166
[1,1,1,1,1,3] => 572
[1,1,1,1,2,1,1] => 1220
[1,1,1,1,2,2] => 875
[1,1,1,1,3,1] => 770
[1,1,1,1,4] => 275
[1,1,1,2,1,1,1] => 1234
[1,1,1,2,1,2] => 875
[1,1,1,2,2,1] => 1026
[1,1,1,2,3] => 516
[1,1,1,3,1,1] => 842
[1,1,1,3,2] => 623
[1,1,1,4,1] => 410
[1,1,1,5] => 110
[1,1,2,1,1,1,1] => 1220
[1,1,2,1,1,2] => 861
[1,1,2,1,2,1] => 1006
[1,1,2,1,3] => 502
[1,1,2,2,1,1] => 1060
[1,1,2,2,2] => 765
[1,1,2,3,1] => 685
[1,1,2,4] => 250
[1,1,3,1,1,1] => 842
[1,1,3,1,2] => 609
[1,1,3,2,1] => 718
[1,1,3,3] => 376
[1,1,4,1,1] => 450
[1,1,4,2] => 343
[1,1,5,1] => 170
[1,1,6] => 35
[1,2,1,1,1,1,1] => 1166
[1,2,1,1,1,2] => 821
[1,2,1,1,2,1] => 958
[1,2,1,1,3] => 476
[1,2,1,2,1,1] => 1006
[1,2,1,2,2] => 725
[1,2,1,3,1] => 646
[1,2,1,4] => 235
[1,2,2,1,1,1] => 1026
[1,2,2,1,2] => 731
[1,2,2,2,1] => 858
[1,2,2,3] => 436
[1,2,3,1,1] => 718
[1,2,3,2] => 533
[1,2,4,1] => 358
[1,2,5] => 98
[1,3,1,1,1,1] => 770
[1,3,1,1,2] => 551
[1,3,1,2,1] => 646
[1,3,1,3] => 332
[1,3,2,1,1] => 685
[1,3,2,2] => 500
[1,3,3,1] => 460
[1,3,4] => 175
[1,4,1,1,1] => 410
[1,4,1,2] => 303
[1,4,2,1] => 358
[1,4,3] => 196
[1,5,1,1] => 170
[1,5,2] => 133
[1,6,1] => 50
[1,7] => 8
[2,1,1,1,1,1,1] => 1001
[2,1,1,1,1,2] => 704
[2,1,1,1,2,1] => 821
[2,1,1,1,3] => 407
[2,1,1,2,1,1] => 861
[2,1,1,2,2] => 620
[2,1,1,3,1] => 551
[2,1,1,4] => 200
[2,1,2,1,1,1] => 875
[2,1,2,1,2] => 623
[2,1,2,2,1] => 731
[2,1,2,3] => 371
[2,1,3,1,1] => 609
[2,1,3,2] => 452
[2,1,4,1] => 303
[2,1,5] => 83
[2,2,1,1,1,1] => 875
[2,2,1,1,2] => 620
[2,2,1,2,1] => 725
[2,2,1,3] => 365
[2,2,2,1,1] => 765
[2,2,2,2] => 554
[2,2,3,1] => 500
[2,2,4] => 185
[2,3,1,1,1] => 623
[2,3,1,2] => 452
[2,3,2,1] => 533
[2,3,3] => 281
[2,4,1,1] => 343
[2,4,2] => 262
[2,5,1] => 133
[2,6] => 28
[3,1,1,1,1,1] => 572
[3,1,1,1,2] => 407
[3,1,1,2,1] => 476
[3,1,1,3] => 242
[3,1,2,1,1] => 502
[3,1,2,2] => 365
[3,1,3,1] => 332
[3,1,4] => 125
[3,2,1,1,1] => 516
[3,2,1,2] => 371
[3,2,2,1] => 436
[3,2,3] => 226
[3,3,1,1] => 376
[3,3,2] => 281
[3,4,1] => 196
[3,5] => 56
[4,1,1,1,1] => 275
[4,1,1,2] => 200
[4,1,2,1] => 235
[4,1,3] => 125
[4,2,1,1] => 250
[4,2,2] => 185
[4,3,1] => 175
[4,4] => 70
[5,1,1,1] => 110
[5,1,2] => 83
[5,2,1] => 98
[5,3] => 56
[6,1,1] => 35
[6,2] => 28
[7,1] => 8
[8] => 1
[1,1,1,1,1,1,1,1,1] => 4862
[1,1,1,1,1,1,1,2] => 3432
[1,1,1,1,1,1,2,1] => 4004
[1,1,1,1,1,1,3] => 2002
[1,1,1,1,1,2,1,1] => 4202
[1,1,1,1,1,2,2] => 3036
[1,1,1,1,1,3,1] => 2717
[1,1,1,1,1,4] => 1001
[1,1,1,1,2,1,1,1] => 4274
[1,1,1,1,2,1,2] => 3054
[1,1,1,1,2,2,1] => 3584
[1,1,1,1,2,3] => 1834
[1,1,1,1,3,1,1] => 3014
[1,1,1,1,3,2] => 2244
[1,1,1,1,4,1] => 1529
[1,1,1,1,5] => 429
[1,1,1,2,1,1,1,1] => 4274
[1,1,1,2,1,1,2] => 3040
[1,1,1,2,1,2,1] => 3556
[1,1,1,2,1,3] => 1806
[1,1,1,2,2,1,1] => 3754
[1,1,1,2,2,2] => 2728
[1,1,1,2,3,1] => 2479
[1,1,1,2,4] => 931
[1,1,1,3,1,1,1] => 3098
[1,1,1,3,1,2] => 2256
[1,1,1,3,2,1] => 2660
[1,1,1,3,3] => 1414
[1,1,1,4,1,1] => 1754
[1,1,1,4,2] => 1344
[1,1,1,5,1] => 704
[1,1,1,6] => 154
[1,1,2,1,1,1,1,1] => 4202
[1,1,2,1,1,1,2] => 2982
[1,1,2,1,1,2,1] => 3484
[1,1,2,1,1,3] => 1762
[1,1,2,1,2,1,1] => 3667
[1,1,2,1,2,2] => 2661
[1,1,2,1,3,1] => 2407
[1,1,2,1,4] => 901
[1,1,2,2,1,1,1] => 3754
[1,1,2,2,1,2] => 2694
[1,1,2,2,2,1] => 3164
[1,1,2,2,3] => 1634
[1,1,2,3,1,1] => 2704
[1,1,2,3,2] => 2019
[1,1,2,4,1] => 1399
[1,1,2,5] => 399
[1,1,3,1,1,1,1] => 3014
[1,1,3,1,1,2] => 2172
[1,1,3,1,2,1] => 2548
[1,1,3,1,3] => 1330
[1,1,3,2,1,1] => 2704
[1,1,3,2,2] => 1986
[1,1,3,3,1] => 1849
[1,1,3,4] => 721
[1,1,4,1,1,1] => 1754
[1,1,4,1,2] => 1304
[1,1,4,2,1] => 1540
[1,1,4,3] => 854
[1,1,5,1,1] => 794
[1,1,5,2] => 624
[1,1,6,1] => 254
[1,1,7] => 44
[1,2,1,1,1,1,1,1] => 4004
[1,2,1,1,1,1,2] => 2838
[1,2,1,1,1,2,1] => 3314
[1,2,1,1,1,3] => 1672
[1,2,1,1,2,1,1] => 3484
[1,2,1,1,2,2] => 2526
[1,2,1,1,3,1] => 2279
[1,2,1,1,4] => 851
[1,2,1,2,1,1,1] => 3556
[1,2,1,2,1,2] => 2550
[1,2,1,2,2,1] => 2994
[1,2,1,2,3] => 1544
[1,2,1,3,1,1] => 2548
[1,2,1,3,2] => 1902
[1,2,1,4,1] => 1315
[1,2,1,5] => 375
[1,2,2,1,1,1,1] => 3584
[1,2,2,1,1,2] => 2558
[1,2,2,1,2,1] => 2994
[1,2,2,1,3] => 1532
[1,2,2,2,1,1] => 3164
[1,2,2,2,2] => 2306
[1,2,2,3,1] => 2109
[1,2,2,4] => 801
[1,2,3,1,1,1] => 2660
[1,2,3,1,2] => 1942
[1,2,3,2,1] => 2290
[1,2,3,3] => 1224
[1,2,4,1,1] => 1540
[1,2,4,2] => 1182
[1,2,5,1] => 630
[1,2,6] => 140
[1,3,1,1,1,1,1] => 2717
[1,3,1,1,1,2] => 1947
[1,3,1,1,2,1] => 2279
[1,3,1,1,3] => 1177
[1,3,1,2,1,1] => 2407
[1,3,1,2,2] => 1761
[1,3,1,3,1] => 1622
[1,3,1,4] => 626
[1,3,2,1,1,1] => 2479
[1,3,2,1,2] => 1794
[1,3,2,2,1] => 2109
[1,3,2,3] => 1109
[1,3,3,1,1] => 1849
[1,3,3,2] => 1389
[1,3,4,1] => 994
[1,3,5] => 294
[1,4,1,1,1,1] => 1529
[1,4,1,1,2] => 1119
[1,4,1,2,1] => 1315
[1,4,1,3] => 709
[1,4,2,1,1] => 1399
[1,4,2,2] => 1041
[1,4,3,1] => 994
[1,4,4] => 406
[1,5,1,1,1] => 704
[1,5,1,2] => 534
[1,5,2,1] => 630
[1,5,3] => 364
[1,6,1,1] => 254
[1,6,2] => 204
[1,7,1] => 65
[1,8] => 9
[2,1,1,1,1,1,1,1] => 3432
[2,1,1,1,1,1,2] => 2431
[2,1,1,1,1,2,1] => 2838
[2,1,1,1,1,3] => 1430
[2,1,1,1,2,1,1] => 2982
[2,1,1,1,2,2] => 2161
[2,1,1,1,3,1] => 1947
[2,1,1,1,4] => 726
[2,1,1,2,1,1,1] => 3040
[2,1,1,2,1,2] => 2179
[2,1,1,2,2,1] => 2558
[2,1,1,2,3] => 1318
[2,1,1,3,1,1] => 2172
[2,1,1,3,2] => 1621
[2,1,1,4,1] => 1119
[2,1,1,5] => 319
[2,1,2,1,1,1,1] => 3054
[2,1,2,1,1,2] => 2179
[2,1,2,1,2,1] => 2550
[2,1,2,1,3] => 1304
[2,1,2,2,1,1] => 2694
[2,1,2,2,2] => 1963
[2,1,2,3,1] => 1794
[2,1,2,4] => 681
[2,1,3,1,1,1] => 2256
[2,1,3,1,2] => 1647
[2,1,3,2,1] => 1942
[2,1,3,3] => 1038
[2,1,4,1,1] => 1304
[2,1,4,2] => 1001
[2,1,5,1] => 534
[2,1,6] => 119
[2,2,1,1,1,1,1] => 3036
[2,2,1,1,1,2] => 2161
[2,2,1,1,2,1] => 2526
[2,2,1,1,3] => 1286
[2,2,1,2,1,1] => 2661
[2,2,1,2,2] => 1936
[2,2,1,3,1] => 1761
[2,2,1,4] => 666
[2,2,2,1,1,1] => 2728
[2,2,2,1,2] => 1963
[2,2,2,2,1] => 2306
[2,2,2,3] => 1198
[2,2,3,1,1] => 1986
[2,2,3,2] => 1486
[2,2,4,1] => 1041
[2,2,5] => 301
[2,3,1,1,1,1] => 2244
[2,3,1,1,2] => 1621
[2,3,1,2,1] => 1902
[2,3,1,3] => 998
[2,3,2,1,1] => 2019
[2,3,2,2] => 1486
[2,3,3,1] => 1389
[2,3,4] => 546
[2,4,1,1,1] => 1344
[2,4,1,2] => 1001
[2,4,2,1] => 1182
[2,4,3] => 658
[2,5,1,1] => 624
[2,5,2] => 491
[2,6,1] => 204
[2,7] => 36
[3,1,1,1,1,1,1] => 2002
[3,1,1,1,1,2] => 1430
[3,1,1,1,2,1] => 1672
[3,1,1,1,3] => 858
[3,1,1,2,1,1] => 1762
[3,1,1,2,2] => 1286
[3,1,1,3,1] => 1177
[3,1,1,4] => 451
[3,1,2,1,1,1] => 1806
[3,1,2,1,2] => 1304
[3,1,2,2,1] => 1532
[3,1,2,3] => 802
[3,1,3,1,1] => 1330
[3,1,3,2] => 998
[3,1,4,1] => 709
[3,1,5] => 209
[3,2,1,1,1,1] => 1834
[3,2,1,1,2] => 1318
[3,2,1,2,1] => 1544
[3,2,1,3] => 802
[3,2,2,1,1] => 1634
[3,2,2,2] => 1198
[3,2,3,1] => 1109
[3,2,4] => 431
[3,3,1,1,1] => 1414
[3,3,1,2] => 1038
[3,3,2,1] => 1224
[3,3,3] => 662
[3,4,1,1] => 854
[3,4,2] => 658
[3,5,1] => 364
[3,6] => 84
[4,1,1,1,1,1] => 1001
[4,1,1,1,2] => 726
[4,1,1,2,1] => 851
[4,1,1,3] => 451
[4,1,2,1,1] => 901
[4,1,2,2] => 666
[4,1,3,1] => 626
[4,1,4] => 251
[4,2,1,1,1] => 931
[4,2,1,2] => 681
[4,2,2,1] => 801
[4,2,3] => 431
[4,3,1,1] => 721
[4,3,2] => 546
[4,4,1] => 406
[4,5] => 126
[5,1,1,1,1] => 429
[5,1,1,2] => 319
[5,1,2,1] => 375
[5,1,3] => 209
[5,2,1,1] => 399
[5,2,2] => 301
[5,3,1] => 294
[5,4] => 126
[6,1,1,1] => 154
[6,1,2] => 119
[6,2,1] => 140
[6,3] => 84
[7,1,1] => 44
[7,2] => 36
[8,1] => 9
[9] => 1
-----------------------------------------------------------------------------
Created: Dec 12, 2018 at 11:36 by Wenjie Fang
-----------------------------------------------------------------------------
Last Updated: Dec 13, 2018 at 19:02 by Martin Rubey