***************************************************************************** * 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: St001099 ----------------------------------------------------------------------------- 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 leaf labelled binary 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 leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2] Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color. ----------------------------------------------------------------------------- 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 = 1-sqrt(1-2*x) # labelled binary 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] => 2 [1,1,1] => 3 [4] => 0 [3,1] => 6 [2,2] => 10 [2,1,1] => 12 [1,1,1,1] => 15 [5] => 0 [4,1] => 24 [3,2] => 54 [3,1,1] => 60 [2,2,1] => 78 [2,1,1,1] => 90 [1,1,1,1,1] => 105 [6] => 0 [5,1] => 120 [4,2] => 336 [4,1,1] => 360 [3,3] => 450 [3,2,1] => 570 [3,1,1,1] => 630 [2,2,2] => 672 [2,2,1,1] => 750 [2,1,1,1,1] => 840 [1,1,1,1,1,1] => 945 [7] => 0 [6,1] => 720 [5,2] => 2400 [5,1,1] => 2520 [4,3] => 3960 [4,2,1] => 4680 [4,1,1,1] => 5040 [3,3,1] => 5670 [3,2,2] => 6360 [3,2,1,1] => 6930 [3,1,1,1,1] => 7560 [2,2,2,1] => 7860 [2,2,1,1,1] => 8610 [2,1,1,1,1,1] => 9450 [1,1,1,1,1,1,1] => 10395 [8] => 0 [7,1] => 5040 [6,2] => 19440 [6,1,1] => 20160 [5,3] => 37800 [5,2,1] => 42840 [5,1,1,1] => 45360 [4,4] => 46440 [4,3,1] => 60480 [4,2,2] => 65880 [4,2,1,1] => 70560 [4,1,1,1,1] => 75600 [3,3,2] => 75600 [3,3,1,1] => 81270 [3,2,2,1] => 89460 [3,2,1,1,1] => 96390 [3,1,1,1,1,1] => 103950 [2,2,2,2] => 98820 [2,2,2,1,1] => 106680 [2,2,1,1,1,1] => 115290 [2,1,1,1,1,1,1] => 124740 [1,1,1,1,1,1,1,1] => 135135 [9] => 0 [8,1] => 40320 [7,2] => 176400 [7,1,1] => 181440 [6,3] => 393120 [6,2,1] => 433440 [6,1,1,1] => 453600 [5,4] => 567000 [5,3,1] => 695520 [5,2,2] => 743400 [5,2,1,1] => 786240 [5,1,1,1,1] => 831600 [4,4,1] => 808920 [4,3,2] => 960120 [4,3,1,1] => 1020600 [4,2,2,1] => 1101240 [4,2,1,1,1] => 1171800 [4,1,1,1,1,1] => 1247400 [3,3,3] => 1065960 [3,3,2,1] => 1228500 [3,3,1,1,1] => 1309770 [3,2,2,2] => 1331820 [3,2,2,1,1] => 1421280 [3,2,1,1,1,1] => 1517670 [3,1,1,1,1,1,1] => 1621620 [2,2,2,2,1] => 1545180 [2,2,2,1,1,1] => 1651860 [2,2,1,1,1,1,1] => 1767150 [2,1,1,1,1,1,1,1] => 1891890 [1,1,1,1,1,1,1,1,1] => 2027025 [10] => 0 [9,1] => 362880 [8,2] => 1774080 [8,1,1] => 1814400 [7,3] => 4445280 [7,2,1] => 4808160 [7,1,1,1] => 4989600 [6,4] => 7318080 [6,3,1] => 8618400 [6,2,2] => 9092160 [6,2,1,1] => 9525600 [6,1,1,1,1] => 9979200 [5,5] => 8580600 [5,4,1] => 11362680 [5,3,2] => 13025880 [5,3,1,1] => 13721400 [5,2,2,1] => 14598360 [5,2,1,1,1] => 15384600 [5,1,1,1,1,1] => 16216200 [4,4,2] => 14636160 [4,4,1,1] => 15445080 [4,3,3] => 15898680 [4,3,2,1] => 17939880 [4,3,1,1,1] => 18960480 [4,2,2,2] => 19182240 [4,2,2,1,1] => 20283480 [4,2,1,1,1,1] => 21455280 [4,1,1,1,1,1,1] => 22702680 [3,3,3,1] => 19584180 [3,3,2,2] => 20975220 [3,3,2,1,1] => 22203720 [3,3,1,1,1,1] => 23513490 [3,2,2,2,1] => 23817780 [3,2,2,1,1,1] => 25239060 [3,2,1,1,1,1,1] => 26756730 [3,1,1,1,1,1,1,1] => 28378350 [2,2,2,2,2] => 25576320 [2,2,2,2,1,1] => 27121500 [2,2,2,1,1,1,1] => 28773360 [2,2,1,1,1,1,1,1] => 30540510 [2,1,1,1,1,1,1,1,1] => 32432400 [1,1,1,1,1,1,1,1,1,1] => 34459425 [11] => 0 [10,1] => 3628800 [9,2] => 19595520 [9,1,1] => 19958400 [8,3] => 54432000 [8,2,1] => 58060800 [8,1,1,1] => 59875200 [7,4] => 100336320 [7,3,1] => 114760800 [7,2,2] => 119931840 [7,2,1,1] => 124740000 [7,1,1,1,1] => 129729600 [6,5] => 134038800 [6,4,1] => 168512400 [6,3,2] => 188470800 [6,3,1,1] => 197089200 [6,2,2,1] => 207522000 [6,2,1,1,1] => 217047600 [6,1,1,1,1,1] => 227026800 [5,5,1] => 190852200 [5,4,2] => 234375120 [5,4,1,1] => 245737800 [5,3,3] => 250727400 [5,3,2,1] => 278170200 [5,3,1,1,1] => 291891600 [5,2,2,2] => 294341040 [5,2,2,1,1] => 308939400 [5,2,1,1,1,1] => 324324000 [5,1,1,1,1,1,1] => 340540200 [4,4,3] => 275244480 [4,4,2,1] => 306134640 [4,4,1,1,1] => 321579720 [4,3,3,1] => 329064120 [4,3,2,2] => 349045200 [4,3,2,1,1] => 366985080 [4,3,1,1,1,1] => 385945560 [4,2,2,2,1] => 389612160 [4,2,2,1,1,1] => 409895640 [4,2,1,1,1,1,1] => 431350920 [4,1,1,1,1,1,1,1] => 454053600 [3,3,3,2] => 376091100 [3,3,3,1,1] => 395675280 [3,3,2,2,1] => 420498540 [3,3,2,1,1,1] => 442702260 [3,3,1,1,1,1,1] => 466215750 [3,2,2,2,2] => 447158880 [3,2,2,2,1,1] => 470976660 [3,2,2,1,1,1,1] => 496215720 [3,2,1,1,1,1,1,1] => 522972450 [3,1,1,1,1,1,1,1,1] => 551350800 [2,2,2,2,2,1] => 501401880 [2,2,2,2,1,1,1] => 528523380 [2,2,2,1,1,1,1,1] => 557296740 [2,2,1,1,1,1,1,1,1] => 587837250 [2,1,1,1,1,1,1,1,1,1] => 620269650 [1,1,1,1,1,1,1,1,1,1,1] => 654729075 [12] => 0 [11,1] => 39916800 [10,2] => 235872000 [10,1,1] => 239500800 [9,3] => 718502400 [9,2,1] => 758419200 [9,1,1,1] => 778377600 [8,4] => 1462406400 [8,3,1] => 1636588800 [8,2,2] => 1698278400 [8,2,1,1] => 1756339200 [8,1,1,1,1] => 1816214400 ----------------------------------------------------------------------------- Created: Feb 02, 2018 at 20:13 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 06, 2018 at 07:21 by Jair Taylor