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Statistic identifier: St001098
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Collection: Integer partitions
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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.
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References: [1] Taylor, J. Formal group laws and hypergraph colorings [[MathSciNet:3542357]]
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Code:
@cached_function
def data(n):
R.<x> = 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
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Created: Feb 02, 2018 at 20:09 by Martin Rubey
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Last Updated: Feb 04, 2018 at 21:51 by Jair Taylor