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Identifier
Values
=>
Cc0002;cc-rep
[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
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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())
Created
Feb 02, 2018 at 20:09 by Martin Rubey
Updated
Feb 04, 2018 at 21:51 by Jair Taylor