***************************************************************************** * 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: St001098 ----------------------------------------------------------------------------- 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 vertex labelled 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 vertex labelled trees, whose reversal of the generating function $f^{(-1)}(x) = x\exp(-x)$, see [1, sec. 3.3] Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. Then this statistic gives the number of ways of putting a rooted tree on this set of colored vertices so that no leaf is the same color as its parent. ----------------------------------------------------------------------------- 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_rev = x*exp(-x) # labelled trees f = f_rev.reverse() f_coefficients = f.list() 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] => 2 [3] => 0 [2,1] => 5 [1,1,1] => 9 [4] => 0 [3,1] => 16 [2,2] => 36 [2,1,1] => 46 [1,1,1,1] => 64 [5] => 0 [4,1] => 65 [3,2] => 236 [3,1,1] => 268 [2,2,1] => 405 [2,1,1,1] => 497 [1,1,1,1,1] => 625 [6] => 0 [5,1] => 326 [4,2] => 1646 [4,1,1] => 1776 [3,3] => 2658 [3,2,1] => 3682 [3,1,1,1] => 4218 [2,2,2] => 4722 [2,2,1,1] => 5532 [2,1,1,1,1] => 6526 [1,1,1,1,1,1] => 7776 [7] => 0 [6,1] => 1957 [5,2] => 12652 [5,1,1] => 13304 [4,3] => 28620 [4,2,1] => 35529 [4,1,1,1] => 39081 [3,3,1] => 48364 [3,2,2] => 57068 [3,2,1,1] => 64432 [3,1,1,1,1] => 72868 [2,2,2,1] => 77981 [2,2,1,1,1] => 89045 [2,1,1,1,1,1] => 102097 [1,1,1,1,1,1,1] => 117649 [8] => 0 [7,1] => 13700 [6,2] => 107814 [6,1,1] => 111728 [5,3] => 315486 [5,2,1] => 367724 [5,1,1,1] => 394332 [4,4] => 442880 [4,3,1] => 640330 [4,2,2] => 720268 [4,2,1,1] => 791326 [4,1,1,1,1] => 869488 [3,3,2] => 893304 [3,3,1,1] => 990032 [3,2,2,1] => 1139986 [3,2,1,1,1] => 1268850 [3,1,1,1,1,1] => 1414586 [2,2,2,2] => 1323608 [2,2,2,1,1] => 1479570 [2,2,1,1,1,1] => 1657660 [2,1,1,1,1,1,1] => 1861854 [1,1,1,1,1,1,1,1] => 2097152 [9] => 0 [8,1] => 109601 [7,2] => 1015352 [7,1,1] => 1042752 [6,3] => 3654000 [6,2,1] => 4095041 [6,1,1,1] => 4318497 [5,4] => 6659144 [5,3,1] => 8747056 [5,2,2] => 9549024 [5,2,1,1] => 10284472 [5,1,1,1,1] => 11073136 [4,4,1] => 11170353 [4,3,2] => 14293024 [4,3,1,1] => 15573684 [4,2,2,1] => 17351741 [4,2,1,1,1] => 18934393 [4,1,1,1,1,1] => 20673369 [3,3,3] => 16752744 [3,3,2,1] => 20567780 [3,3,1,1,1] => 22547844 [3,2,2,2] => 23169912 [3,2,2,1,1] => 25449884 [3,2,1,1,1,1] => 27987584 [3,1,1,1,1,1,1] => 30816756 [2,2,2,2,1] => 28854249 [2,2,2,1,1,1] => 31813389 [2,2,1,1,1,1,1] => 35128709 [2,1,1,1,1,1,1,1] => 38852417 [1,1,1,1,1,1,1,1,1] => 43046721 [10] => 0 [9,1] => 986410 [8,2] => 10506174 [8,1,1] => 10725376 [7,3] => 44918754 [7,2,1] => 49048662 [7,1,1,1] => 51134166 [6,4] => 101098560 [6,3,1] => 124671082 [6,2,2] => 133419804 [6,2,1,1] => 141609886 [6,1,1,1,1] => 150246880 [5,5] => 131400690 [5,4,1] => 194969340 [5,3,2] => 235686288 [5,3,1,1] => 253180400 [5,2,2,1] => 275721004 [5,2,1,1,1] => 296289948 [5,1,1,1,1,1] => 318436220 [4,4,2] => 283590654 [4,4,1,1] => 305931360 [4,3,3] => 320347536 [4,3,2,1] => 380721282 [4,3,1,1,1] => 411868650 [4,2,2,2] => 419381394 [4,2,2,1,1] => 454084876 [4,2,1,1,1,1] => 491953662 [4,1,1,1,1,1,1] => 533300400 [3,3,3,1] => 435037384 [3,3,2,2] => 481123104 [3,3,2,1,1] => 522258664 [3,3,1,1,1,1] => 567354352 [3,2,2,2,1] => 579502682 [3,2,2,1,1,1] => 630402450 [3,2,1,1,1,1,1] => 686377618 [3,1,1,1,1,1,1,1] => 748011130 [2,2,2,2,2] => 644609030 [2,2,2,2,1,1] => 702317528 [2,2,2,1,1,1,1] => 765944306 [2,2,1,1,1,1,1,1] => 836201724 [2,1,1,1,1,1,1,1,1] => 913906558 [1,1,1,1,1,1,1,1,1,1] => 1000000000 [11] => 0 [10,1] => 9864101 [9,2] => 118687532 [9,1,1] => 120660352 [8,3] => 588005676 [8,2,1] => 630578377 [8,1,1,1] => 652029129 [7,4] => 1579007720 [7,3,1] => 1863969724 [7,2,2] => 1967280808 [7,2,1,1] => 2065378132 [12] => 0 [11,1] => 108505112 [10,2] => 1455009206 [10,1,1] => 1474737408 ----------------------------------------------------------------------------- Created: Feb 02, 2018 at 20:09 by Martin Rubey ----------------------------------------------------------------------------- Last Updated: Feb 04, 2018 at 21:51 by Jair Taylor