***************************************************************************** * 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: St001097 ----------------------------------------------------------------------------- Collection: Integer partitions ----------------------------------------------------------------------------- Description: The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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 $m_\lambda$ in the formal group law for linear orders, with generating function $f(x) = x/(1-x)$, see [1, sec. 3.4]. This statistic gives the number of Smirnov arrangements of a set of letters with $\lambda_i$ of the $i$th letter, where a Smirnov word is a word with no repeated adjacent letters. e.g., [3,2,1] = > 10 since there are 10 Smirnov rearrangements of the word 'aaabbc': 'ababac', 'ababca', 'abacab', 'abacba', 'abcaba', 'acabab', 'acbaba', 'babaca', 'bacaba', 'cababa'. ----------------------------------------------------------------------------- References: [1] Taylor, J. Formal group laws and hypergraph colorings [[MathSciNet:3542357]] ----------------------------------------------------------------------------- Code: @cached_function def data(n): """ sage: data = data_linear_orders sage: n = 3; [(P, statistic(P)) for P in Partitions(n)] sage: findstat([(P, statistic(P)) for n in range(1,9) for P in Partitions(n)], depth=3) a new statistic on Cc0002: Integer partitions """ R. = PowerSeriesRing(SR, default_prec=n+1) f = x/(1-x) # linear orders 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 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] => 2 [3] => 0 [2,1] => 1 [1,1,1] => 6 [4] => 0 [3,1] => 0 [2,2] => 2 [2,1,1] => 6 [1,1,1,1] => 24 [5] => 0 [4,1] => 0 [3,2] => 1 [3,1,1] => 2 [2,2,1] => 12 [2,1,1,1] => 36 [1,1,1,1,1] => 120 [6] => 0 [5,1] => 0 [4,2] => 0 [4,1,1] => 0 [3,3] => 2 [3,2,1] => 10 [3,1,1,1] => 24 [2,2,2] => 30 [2,2,1,1] => 84 [2,1,1,1,1] => 240 [1,1,1,1,1,1] => 720 [7] => 0 [6,1] => 0 [5,2] => 0 [5,1,1] => 0 [4,3] => 1 [4,2,1] => 3 [4,1,1,1] => 6 [3,3,1] => 18 [3,2,2] => 38 [3,2,1,1] => 96 [3,1,1,1,1] => 240 [2,2,2,1] => 246 [2,2,1,1,1] => 660 [2,1,1,1,1,1] => 1800 [1,1,1,1,1,1,1] => 5040 [8] => 0 [7,1] => 0 [6,2] => 0 [6,1,1] => 0 [5,3] => 0 [5,2,1] => 0 [5,1,1,1] => 0 [4,4] => 2 [4,3,1] => 14 [4,2,2] => 24 [4,2,1,1] => 54 [4,1,1,1,1] => 120 [3,3,2] => 74 [3,3,1,1] => 184 [3,2,2,1] => 384 [3,2,1,1,1] => 960 [3,1,1,1,1,1] => 2400 [2,2,2,2] => 864 [2,2,2,1,1] => 2220 [2,2,1,1,1,1] => 5760 [2,1,1,1,1,1,1] => 15120 [1,1,1,1,1,1,1,1] => 40320 [9] => 0 [8,1] => 0 [7,2] => 0 [7,1,1] => 0 [6,3] => 0 [6,2,1] => 0 [6,1,1,1] => 0 [5,4] => 1 [5,3,1] => 4 [5,2,2] => 6 [5,2,1,1] => 12 [5,1,1,1,1] => 24 [4,4,1] => 24 [4,3,2] => 79 [4,3,1,1] => 186 [4,2,2,1] => 336 [4,2,1,1,1] => 780 [4,1,1,1,1,1] => 1800 [3,3,3] => 174 [3,3,2,1] => 836 [3,3,1,1,1] => 2040 [3,2,2,2] => 1686 [3,2,2,1,1] => 4140 [3,2,1,1,1,1] => 10200 [3,1,1,1,1,1,1] => 25200 [2,2,2,2,1] => 8760 [2,2,2,1,1,1] => 21960 [2,2,1,1,1,1,1] => 55440 [2,1,1,1,1,1,1,1] => 141120 [1,1,1,1,1,1,1,1,1] => 362880 [10] => 0 [9,1] => 0 [8,2] => 0 [8,1,1] => 0 [7,3] => 0 [7,2,1] => 0 [7,1,1,1] => 0 [6,4] => 0 [6,3,1] => 0 [6,2,2] => 0 [6,2,1,1] => 0 [6,1,1,1,1] => 0 [5,5] => 2 [5,4,1] => 18 [5,3,2] => 44 [5,3,1,1] => 96 [5,2,2,1] => 156 [5,2,1,1,1] => 336 [5,1,1,1,1,1] => 720 [4,4,2] => 138 [4,4,1,1] => 324 [4,3,3] => 248 [4,3,2,1] => 1074 [4,3,1,1,1] => 2520 [4,2,2,2] => 1974 [4,2,2,1,1] => 4620 [4,2,1,1,1,1] => 10800 [4,1,1,1,1,1,1] => 25200 [3,3,3,1] => 2184 [3,3,2,2] => 4204 [3,3,2,1,1] => 10080 [3,3,1,1,1,1] => 24240 [3,2,2,2,1] => 19740 [3,2,2,1,1,1] => 47760 [3,2,1,1,1,1,1] => 115920 [3,1,1,1,1,1,1,1] => 282240 [2,2,2,2,2] => 39480 [2,2,2,2,1,1] => 96480 [2,2,2,1,1,1,1] => 236880 [2,2,1,1,1,1,1,1] => 584640 [2,1,1,1,1,1,1,1,1] => 1451520 [1,1,1,1,1,1,1,1,1,1] => 3628800 [11] => 0 [10,1] => 0 [9,2] => 0 [9,1,1] => 0 [8,3] => 0 [8,2,1] => 0 [8,1,1,1] => 0 [7,4] => 0 [7,3,1] => 0 [7,2,2] => 0 [7,2,1,1] => 0 [7,1,1,1,1] => 0 [6,5] => 1 [6,4,1] => 5 [6,3,2] => 10 [6,3,1,1] => 20 [6,2,2,1] => 30 [6,2,1,1,1] => 60 [6,1,1,1,1,1] => 120 [5,5,1] => 30 [5,4,2] => 135 [5,4,1,1] => 306 [5,3,3] => 212 [5,3,2,1] => 816 [5,3,1,1,1] => 1824 [5,2,2,2] => 1386 [5,2,2,1,1] => 3084 [5,2,1,1,1,1] => 6840 [5,1,1,1,1,1,1] => 15120 [4,4,3] => 480 [4,4,2,1] => 2016 [4,4,1,1,1] => 4680 [4,3,3,1] => 3566 [4,3,2,2] => 6516 [4,3,2,1,1] => 15180 [4,3,1,1,1,1] => 35400 [4,2,2,2,1] => 27780 [4,2,2,1,1,1] => 64800 [4,2,1,1,1,1,1] => 151200 [4,1,1,1,1,1,1,1] => 352800 [3,3,3,2] => 12336 [3,3,3,1,1] => 29040 [3,3,2,2,1] => 54660 [3,3,2,1,1,1] => 129480 [3,3,1,1,1,1,1] => 307440 [3,2,2,2,2] => 103800 [3,2,2,2,1,1] => 247080 [3,2,2,1,1,1,1] => 589680 [3,2,1,1,1,1,1,1] => 1411200 [3,1,1,1,1,1,1,1,1] => 3386880 [2,2,2,2,2,1] => 478080 [2,2,2,2,1,1,1] => 1149120 [2,2,2,1,1,1,1,1] => 2772000 [2,2,1,1,1,1,1,1,1] => 6713280 [2,1,1,1,1,1,1,1,1,1] => 16329600 [1,1,1,1,1,1,1,1,1,1,1] => 39916800 [12] => 0 [11,1] => 0 [10,2] => 0 [10,1,1] => 0 [9,3] => 0 [9,2,1] => 0 [9,1,1,1] => 0 [8,4] => 0 [8,3,1] => 0 [8,2,2] => 0 [8,2,1,1] => 0 [8,1,1,1,1] => 0 [7,5] => 0 [7,4,1] => 0 [7,3,2] => 0 [7,3,1,1] => 0 [7,2,2,1] => 0 [7,2,1,1,1] => 0 [7,1,1,1,1,1] => 0 [6,6] => 2 [6,5,1] => 22 [6,4,2] => 70 [6,4,1,1] => 150 [6,3,3] => 100 [6,3,2,1] => 340 [6,3,1,1,1] => 720 [6,2,2,2] => 540 [6,2,2,1,1] => 1140 [6,2,1,1,1,1] => 2400 [6,1,1,1,1,1,1] => 5040 [5,5,2] => 222 [5,5,1,1] => 504 [5,4,3] => 588 [5,4,2,1] => 2322 [5,4,1,1,1] => 5256 [5,3,3,1] => 3712 [5,3,2,2] => 6432 [5,3,2,1,1] => 14496 [5,3,1,1,1,1] => 32640 [5,2,2,2,1] => 24996 [5,2,2,1,1,1] => 56160 [5,2,1,1,1,1,1] => 126000 [5,1,1,1,1,1,1,1] => 282240 [4,4,4] => 1092 [4,4,3,1] => 7524 [4,4,2,2] => 13464 [4,4,2,1,1] => 30960 [4,4,1,1,1,1] => 71280 [4,3,3,2] => 23254 [4,3,3,1,1] => 53640 [4,3,2,2,1] => 96900 [4,3,2,1,1,1] => 224160 [4,3,1,1,1,1,1] => 519120 [4,2,2,2,2] => 175440 [4,2,2,2,1,1] => 406440 [4,2,2,1,1,1,1] => 942480 [4,2,1,1,1,1,1,1] => 2187360 [4,1,1,1,1,1,1,1,1] => 5080320 [3,3,3,3] => 41304 [3,3,3,2,1] => 175440 [3,3,3,1,1,1] => 408960 [3,3,2,2,2] => 322860 [3,3,2,2,1,1] => 755040 [3,3,2,1,1,1,1] => 1769040 [3,3,1,1,1,1,1,1] => 4152960 [3,2,2,2,2,1] => 1403520 [3,2,2,2,1,1,1] => 3301200 [3,2,2,1,1,1,1,1] => 7781760 [3,2,1,1,1,1,1,1,1] => 18385920 [3,1,1,1,1,1,1,1,1,1] => 43545600 [2,2,2,2,2,2] => 2631600 [2,2,2,2,2,1,1] => 6219360 [2,2,2,2,1,1,1,1] => 14736960 [2,2,2,1,1,1,1,1,1] => 35017920 [2,2,1,1,1,1,1,1,1,1] => 83462400 [2,1,1,1,1,1,1,1,1,1,1] => 199584000 [1,1,1,1,1,1,1,1,1,1,1,1] => 479001600 ----------------------------------------------------------------------------- Created: Feb 02, 2018 at 19:51 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 04, 2018 at 19:59 by Jair Taylor