***************************************************************************** * 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: St001101 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. For a generating function $f$ the associated formal group law is the symmetric function $f(f^{(-1)}(x_1) + f^{(-1)}(x_2), \dots)$, see [1]. This statistic records the coefficient of the monomial symmetric function $m_\lambda$ times the product of the factorials of the parts of $\lambda$ in the formal group law for increasing trees, whose generating function is $f(x) = -\log(1-x)$, see [1, sec. 9.1] Fix a coloring of $\{1,2, \ldots, n\}$ so that $\lambda_i$ are colored with the $i$th color. This statistic gives the number of increasing trees on this colored set of vertices so that no leaf has the same color as its parent. (An increasing tree is a rooted tree on the vertex set $\{1,2, \ldots, n\}$ with the property that any child of $i$ is greater than $i$.) ----------------------------------------------------------------------------- References: [1] Taylor, J. Formal group laws and hypergraph colorings [[MathSciNet:3542357]] ----------------------------------------------------------------------------- Code: @cached_function def data(n): R. = PowerSeriesRing(SR, default_prec=n+1) f = -log(1-x) # increasing trees f_coefficients = f.list() f_rev = f.reverse() t = var('t') polynomials = (t*f_rev).exp().list() polynomials = [p.expand() for p in polynomials] return (f_coefficients, polynomials) def statistic(P): f_coefficients, polynomials = data(P.size()) p = SR(1) for i in P: p *= polynomials[i] p = p.expand() return (prod(factorial(e) for e in P) *sum(p.coefficient(t,n) * f_coefficients[n] * factorial(n) for n in range(p.degree(t)+1)).expand()) ----------------------------------------------------------------------------- Statistic values: [2] => 0 [1,1] => 1 [3] => 0 [2,1] => 1 [1,1,1] => 2 [4] => 0 [3,1] => 1 [2,2] => 3 [2,1,1] => 4 [1,1,1,1] => 6 [5] => 0 [4,1] => 1 [3,2] => 7 [3,1,1] => 8 [2,2,1] => 14 [2,1,1,1] => 18 [1,1,1,1,1] => 24 [6] => 0 [5,1] => 1 [4,2] => 15 [4,1,1] => 16 [3,3] => 31 [3,2,1] => 46 [3,1,1,1] => 54 [2,2,2] => 64 [2,2,1,1] => 78 [2,1,1,1,1] => 96 [1,1,1,1,1,1] => 120 [7] => 0 [6,1] => 1 [5,2] => 31 [5,1,1] => 32 [4,3] => 115 [4,2,1] => 146 [4,1,1,1] => 162 [3,3,1] => 230 [3,2,2] => 284 [3,2,1,1] => 330 [3,1,1,1,1] => 384 [2,2,2,1] => 426 [2,2,1,1,1] => 504 [2,1,1,1,1,1] => 600 [1,1,1,1,1,1,1] => 720 [8] => 0 [7,1] => 1 [6,2] => 63 [6,1,1] => 64 [5,3] => 391 [5,2,1] => 454 [5,1,1,1] => 486 [4,4] => 675 [4,3,1] => 1066 [4,2,2] => 1228 [4,2,1,1] => 1374 [4,1,1,1,1] => 1536 [3,3,2] => 1672 [3,3,1,1] => 1902 [3,2,2,1] => 2286 [3,2,1,1,1] => 2616 [3,1,1,1,1,1] => 3000 [2,2,2,2] => 2790 [2,2,2,1,1] => 3216 [2,2,1,1,1,1] => 3720 [2,1,1,1,1,1,1] => 4320 [1,1,1,1,1,1,1,1] => 5040 [9] => 0 [8,1] => 1 [7,2] => 127 [7,1,1] => 128 [6,3] => 1267 [6,2,1] => 1394 [6,1,1,1] => 1458 [5,4] => 3451 [5,3,1] => 4718 [5,2,2] => 5204 [5,2,1,1] => 5658 [5,1,1,1,1] => 6144 [4,4,1] => 6902 [4,3,2] => 9488 [4,3,1,1] => 10554 [4,2,2,1] => 12090 [4,2,1,1,1] => 13464 [4,1,1,1,1,1] => 15000 [3,3,3] => 11828 [3,3,2,1] => 15402 [3,3,1,1,1] => 17304 [3,2,2,2] => 18018 [3,2,2,1,1] => 20304 [3,2,1,1,1,1] => 22920 [3,1,1,1,1,1,1] => 25920 [2,2,2,2,1] => 24024 [2,2,2,1,1,1] => 27240 [2,2,1,1,1,1,1] => 30960 [2,1,1,1,1,1,1,1] => 35280 [1,1,1,1,1,1,1,1,1] => 40320 [10] => 0 [9,1] => 1 [8,2] => 255 [8,1,1] => 256 [7,3] => 3991 [7,2,1] => 4246 [7,1,1,1] => 4374 [6,4] => 16275 [6,3,1] => 20266 [6,2,2] => 21724 [6,2,1,1] => 23118 [6,1,1,1,1] => 24576 [5,5] => 25231 [5,4,1] => 41506 [5,3,2] => 52336 [5,3,1,1] => 57054 [5,2,2,1] => 63198 [5,2,1,1,1] => 68856 [5,1,1,1,1,1] => 75000 [4,4,2] => 69208 [4,4,1,1] => 76110 [4,3,3] => 81460 [4,3,2,1] => 101502 [4,3,1,1,1] => 112056 [4,2,2,2] => 114966 [4,2,2,1,1] => 127056 [4,2,1,1,1,1] => 140520 [4,1,1,1,1,1,1] => 155520 [3,3,3,1] => 122190 [3,3,2,2] => 139494 [3,3,2,1,1] => 154896 [3,3,1,1,1,1] => 172200 [3,2,2,2,1] => 177816 [3,2,2,1,1,1] => 198120 [3,2,1,1,1,1,1] => 221040 [3,1,1,1,1,1,1,1] => 246960 [2,2,2,2,2] => 205056 [2,2,2,2,1,1] => 229080 [2,2,2,1,1,1,1] => 256320 [2,2,1,1,1,1,1,1] => 287280 [2,1,1,1,1,1,1,1,1] => 322560 [1,1,1,1,1,1,1,1,1,1] => 362880 [11] => 0 [10,1] => 1 [9,2] => 511 [9,1,1] => 512 [8,3] => 12355 [8,2,1] => 12866 [8,1,1,1] => 13122 [7,4] => 72955 [7,3,1] => 85310 [7,2,2] => 89684 [7,2,1,1] => 93930 [7,1,1,1,1] => 98304 [6,5] => 164731 [6,4,1] => 237686 [6,3,2] => 282464 [6,3,1,1] => 302730 [6,2,2,1] => 327306 [6,2,1,1,1] => 350424 [6,1,1,1,1,1] => 375000 [5,5,1] => 329462 [5,4,2] => 484136 [5,4,1,1] => 525642 [5,3,3] => 547820 [5,3,2,1] => 657210 [5,3,1,1,1] => 714264 [5,2,2,2] => 726066 [5,2,2,1,1] => 789264 [5,2,1,1,1,1] => 858120 [5,1,1,1,1,1,1] => 933120 [4,4,3] => 677636 [4,4,2,1] => 822954 [4,4,1,1,1] => 899064 [4,3,3,1] => 951546 [4,3,2,2] => 1063602 [4,3,2,1,1] => 1165104 [4,3,1,1,1,1] => 1277160 [4,2,2,2,1] => 1305624 [4,2,2,1,1,1] => 1432680 [4,2,1,1,1,1,1] => 1573200 [4,1,1,1,1,1,1,1] => 1728720 [3,3,3,2] => 1244034 [3,3,3,1,1] => 1366224 [3,3,2,2,1] => 1538424 [3,3,2,1,1,1] => 1693320 [3,3,1,1,1,1,1] => 1865520 [3,2,2,2,2] => 1736544 [3,2,2,2,1,1] => 1914360 [3,2,2,1,1,1,1] => 2112480 [3,2,1,1,1,1,1,1] => 2333520 [3,1,1,1,1,1,1,1,1] => 2580480 [2,2,2,2,2,1] => 2170680 [2,2,2,2,1,1,1] => 2399760 [2,2,2,1,1,1,1,1] => 2656080 [2,2,1,1,1,1,1,1,1] => 2943360 [2,1,1,1,1,1,1,1,1,1] => 3265920 [1,1,1,1,1,1,1,1,1,1,1] => 3628800 [12] => 0 [11,1] => 1 [10,2] => 1023 [10,1,1] => 1024 [9,3] => 37831 [9,2,1] => 38854 [9,1,1,1] => 39366 [8,4] => 316275 [8,3,1] => 354106 [8,2,2] => 367228 [8,2,1,1] => 380094 [8,1,1,1,1] => 393216 [7,5] => 999391 [7,4,1] => 1315666 [7,3,2] => 1499152 [7,3,1,1] => 1584462 [7,2,2,1] => 1682766 [7,2,1,1,1] => 1776696 [7,1,1,1,1,1] => 1875000 [6,6] => 1441923 [6,5,1] => 2441314 [6,4,2] => 3281608 [6,4,1,1] => 3519294 [6,3,3] => 3610372 [6,3,2,1] => 4195566 [6,3,1,1,1] => 4498296 [6,2,2,2] => 4545990 [6,2,2,1,1] => 4873296 [6,2,1,1,1,1] => 5223720 [6,1,1,1,1,1,1] => 5598720 [5,5,2] => 4223704 [5,5,1,1] => 4553166 [5,4,3] => 5485756 [5,4,2,1] => 6495534 [5,4,1,1,1] => 7021176 [5,3,3,1] => 7290942 [5,3,2,2] => 8005206 [5,3,2,1,1] => 8662416 [5,3,1,1,1,1] => 9376680 [5,2,2,2,1] => 9520536 [5,2,2,1,1,1] => 10309800 [5,2,1,1,1,1,1] => 11167920 [5,1,1,1,1,1,1,1] => 12101040 [4,4,4] => 6476644 [4,4,3,1] => 8724078 [4,4,2,2] => 9623142 [4,4,2,1,1] => 10446096 [4,4,1,1,1,1] => 11345160 [4,3,3,2] => 10942230 [4,3,3,1,1] => 11893776 [4,3,2,2,1] => 13170936 [4,3,2,1,1,1] => 14336040 [4,3,1,1,1,1,1] => 15613200 [4,2,2,2,2] => 14603616 [4,2,2,2,1,1] => 15909240 [4,2,2,1,1,1,1] => 17341920 [4,2,1,1,1,1,1,1] => 18915120 [4,1,1,1,1,1,1,1,1] => 20643840 [3,3,3,3] => 12497958 [3,3,3,2,1] => 15108216 [3,3,3,1,1,1] => 16474440 [3,3,2,2,2] => 16801536 [3,3,2,2,1,1] => 18339960 [3,3,2,1,1,1,1] => 20033280 [3,3,1,1,1,1,1,1] => 21898800 [3,2,2,2,2,1] => 20452440 [3,2,2,2,1,1,1] => 22366800 [3,2,2,1,1,1,1,1] => 24479280 [3,2,1,1,1,1,1,1,1] => 26812800 [3,1,1,1,1,1,1,1,1,1] => 29393280 [2,2,2,2,2,2] => 22852200 [2,2,2,2,2,1,1] => 25022880 [2,2,2,2,1,1,1,1] => 27422640 [2,2,2,1,1,1,1,1,1] => 30078720 [2,2,1,1,1,1,1,1,1,1] => 33022080 [2,1,1,1,1,1,1,1,1,1,1] => 36288000 [1,1,1,1,1,1,1,1,1,1,1,1] => 39916800 ----------------------------------------------------------------------------- Created: Feb 02, 2018 at 20:19 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 06, 2018 at 07:24 by Jair Taylor