***************************************************************************** * www.FindStat.org - The Combinatorial Statistic Finder * * * * Copyright (C) 2019 The FindStatCrew * * * * 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