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Statistic identifier: St001101
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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 increasing 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 increasing trees, whose generating function is $f(x) = -\log(1-x)$, see [1, sec. 9.1]
Fix a coloring of $\{1,2, \ldots, n\}$ so that $\lambda_i$ are colored with the $i$th color. This statistic gives the number of increasing trees on this colored set of vertices so that no leaf has the same color as its parent. (An increasing tree is a rooted tree on the vertex set $\{1,2, \ldots, n\}$ with the property that any child of $i$ is greater than $i$.)
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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 = -log(1-x) # increasing 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] => 1
[1,1,1] => 2
[4] => 0
[3,1] => 1
[2,2] => 3
[2,1,1] => 4
[1,1,1,1] => 6
[5] => 0
[4,1] => 1
[3,2] => 7
[3,1,1] => 8
[2,2,1] => 14
[2,1,1,1] => 18
[1,1,1,1,1] => 24
[6] => 0
[5,1] => 1
[4,2] => 15
[4,1,1] => 16
[3,3] => 31
[3,2,1] => 46
[3,1,1,1] => 54
[2,2,2] => 64
[2,2,1,1] => 78
[2,1,1,1,1] => 96
[1,1,1,1,1,1] => 120
[7] => 0
[6,1] => 1
[5,2] => 31
[5,1,1] => 32
[4,3] => 115
[4,2,1] => 146
[4,1,1,1] => 162
[3,3,1] => 230
[3,2,2] => 284
[3,2,1,1] => 330
[3,1,1,1,1] => 384
[2,2,2,1] => 426
[2,2,1,1,1] => 504
[2,1,1,1,1,1] => 600
[1,1,1,1,1,1,1] => 720
[8] => 0
[7,1] => 1
[6,2] => 63
[6,1,1] => 64
[5,3] => 391
[5,2,1] => 454
[5,1,1,1] => 486
[4,4] => 675
[4,3,1] => 1066
[4,2,2] => 1228
[4,2,1,1] => 1374
[4,1,1,1,1] => 1536
[3,3,2] => 1672
[3,3,1,1] => 1902
[3,2,2,1] => 2286
[3,2,1,1,1] => 2616
[3,1,1,1,1,1] => 3000
[2,2,2,2] => 2790
[2,2,2,1,1] => 3216
[2,2,1,1,1,1] => 3720
[2,1,1,1,1,1,1] => 4320
[1,1,1,1,1,1,1,1] => 5040
[9] => 0
[8,1] => 1
[7,2] => 127
[7,1,1] => 128
[6,3] => 1267
[6,2,1] => 1394
[6,1,1,1] => 1458
[5,4] => 3451
[5,3,1] => 4718
[5,2,2] => 5204
[5,2,1,1] => 5658
[5,1,1,1,1] => 6144
[4,4,1] => 6902
[4,3,2] => 9488
[4,3,1,1] => 10554
[4,2,2,1] => 12090
[4,2,1,1,1] => 13464
[4,1,1,1,1,1] => 15000
[3,3,3] => 11828
[3,3,2,1] => 15402
[3,3,1,1,1] => 17304
[3,2,2,2] => 18018
[3,2,2,1,1] => 20304
[3,2,1,1,1,1] => 22920
[3,1,1,1,1,1,1] => 25920
[2,2,2,2,1] => 24024
[2,2,2,1,1,1] => 27240
[2,2,1,1,1,1,1] => 30960
[2,1,1,1,1,1,1,1] => 35280
[1,1,1,1,1,1,1,1,1] => 40320
[10] => 0
[9,1] => 1
[8,2] => 255
[8,1,1] => 256
[7,3] => 3991
[7,2,1] => 4246
[7,1,1,1] => 4374
[6,4] => 16275
[6,3,1] => 20266
[6,2,2] => 21724
[6,2,1,1] => 23118
[6,1,1,1,1] => 24576
[5,5] => 25231
[5,4,1] => 41506
[5,3,2] => 52336
[5,3,1,1] => 57054
[5,2,2,1] => 63198
[5,2,1,1,1] => 68856
[5,1,1,1,1,1] => 75000
[4,4,2] => 69208
[4,4,1,1] => 76110
[4,3,3] => 81460
[4,3,2,1] => 101502
[4,3,1,1,1] => 112056
[4,2,2,2] => 114966
[4,2,2,1,1] => 127056
[4,2,1,1,1,1] => 140520
[4,1,1,1,1,1,1] => 155520
[3,3,3,1] => 122190
[3,3,2,2] => 139494
[3,3,2,1,1] => 154896
[3,3,1,1,1,1] => 172200
[3,2,2,2,1] => 177816
[3,2,2,1,1,1] => 198120
[3,2,1,1,1,1,1] => 221040
[3,1,1,1,1,1,1,1] => 246960
[2,2,2,2,2] => 205056
[2,2,2,2,1,1] => 229080
[2,2,2,1,1,1,1] => 256320
[2,2,1,1,1,1,1,1] => 287280
[2,1,1,1,1,1,1,1,1] => 322560
[1,1,1,1,1,1,1,1,1,1] => 362880
[11] => 0
[10,1] => 1
[9,2] => 511
[9,1,1] => 512
[8,3] => 12355
[8,2,1] => 12866
[8,1,1,1] => 13122
[7,4] => 72955
[7,3,1] => 85310
[7,2,2] => 89684
[7,2,1,1] => 93930
[7,1,1,1,1] => 98304
[6,5] => 164731
[6,4,1] => 237686
[6,3,2] => 282464
[6,3,1,1] => 302730
[6,2,2,1] => 327306
[6,2,1,1,1] => 350424
[6,1,1,1,1,1] => 375000
[5,5,1] => 329462
[5,4,2] => 484136
[5,4,1,1] => 525642
[5,3,3] => 547820
[5,3,2,1] => 657210
[5,3,1,1,1] => 714264
[5,2,2,2] => 726066
[5,2,2,1,1] => 789264
[5,2,1,1,1,1] => 858120
[5,1,1,1,1,1,1] => 933120
[4,4,3] => 677636
[4,4,2,1] => 822954
[4,4,1,1,1] => 899064
[4,3,3,1] => 951546
[4,3,2,2] => 1063602
[4,3,2,1,1] => 1165104
[4,3,1,1,1,1] => 1277160
[4,2,2,2,1] => 1305624
[4,2,2,1,1,1] => 1432680
[4,2,1,1,1,1,1] => 1573200
[4,1,1,1,1,1,1,1] => 1728720
[3,3,3,2] => 1244034
[3,3,3,1,1] => 1366224
[3,3,2,2,1] => 1538424
[3,3,2,1,1,1] => 1693320
[3,3,1,1,1,1,1] => 1865520
[3,2,2,2,2] => 1736544
[3,2,2,2,1,1] => 1914360
[3,2,2,1,1,1,1] => 2112480
[3,2,1,1,1,1,1,1] => 2333520
[3,1,1,1,1,1,1,1,1] => 2580480
[2,2,2,2,2,1] => 2170680
[2,2,2,2,1,1,1] => 2399760
[2,2,2,1,1,1,1,1] => 2656080
[2,2,1,1,1,1,1,1,1] => 2943360
[2,1,1,1,1,1,1,1,1,1] => 3265920
[1,1,1,1,1,1,1,1,1,1,1] => 3628800
[12] => 0
[11,1] => 1
[10,2] => 1023
[10,1,1] => 1024
[9,3] => 37831
[9,2,1] => 38854
[9,1,1,1] => 39366
[8,4] => 316275
[8,3,1] => 354106
[8,2,2] => 367228
[8,2,1,1] => 380094
[8,1,1,1,1] => 393216
[7,5] => 999391
[7,4,1] => 1315666
[7,3,2] => 1499152
[7,3,1,1] => 1584462
[7,2,2,1] => 1682766
[7,2,1,1,1] => 1776696
[7,1,1,1,1,1] => 1875000
[6,6] => 1441923
[6,5,1] => 2441314
[6,4,2] => 3281608
[6,4,1,1] => 3519294
[6,3,3] => 3610372
[6,3,2,1] => 4195566
[6,3,1,1,1] => 4498296
[6,2,2,2] => 4545990
[6,2,2,1,1] => 4873296
[6,2,1,1,1,1] => 5223720
[6,1,1,1,1,1,1] => 5598720
[5,5,2] => 4223704
[5,5,1,1] => 4553166
[5,4,3] => 5485756
[5,4,2,1] => 6495534
[5,4,1,1,1] => 7021176
[5,3,3,1] => 7290942
[5,3,2,2] => 8005206
[5,3,2,1,1] => 8662416
[5,3,1,1,1,1] => 9376680
[5,2,2,2,1] => 9520536
[5,2,2,1,1,1] => 10309800
[5,2,1,1,1,1,1] => 11167920
[5,1,1,1,1,1,1,1] => 12101040
[4,4,4] => 6476644
[4,4,3,1] => 8724078
[4,4,2,2] => 9623142
[4,4,2,1,1] => 10446096
[4,4,1,1,1,1] => 11345160
[4,3,3,2] => 10942230
[4,3,3,1,1] => 11893776
[4,3,2,2,1] => 13170936
[4,3,2,1,1,1] => 14336040
[4,3,1,1,1,1,1] => 15613200
[4,2,2,2,2] => 14603616
[4,2,2,2,1,1] => 15909240
[4,2,2,1,1,1,1] => 17341920
[4,2,1,1,1,1,1,1] => 18915120
[4,1,1,1,1,1,1,1,1] => 20643840
[3,3,3,3] => 12497958
[3,3,3,2,1] => 15108216
[3,3,3,1,1,1] => 16474440
[3,3,2,2,2] => 16801536
[3,3,2,2,1,1] => 18339960
[3,3,2,1,1,1,1] => 20033280
[3,3,1,1,1,1,1,1] => 21898800
[3,2,2,2,2,1] => 20452440
[3,2,2,2,1,1,1] => 22366800
[3,2,2,1,1,1,1,1] => 24479280
[3,2,1,1,1,1,1,1,1] => 26812800
[3,1,1,1,1,1,1,1,1,1] => 29393280
[2,2,2,2,2,2] => 22852200
[2,2,2,2,2,1,1] => 25022880
[2,2,2,2,1,1,1,1] => 27422640
[2,2,2,1,1,1,1,1,1] => 30078720
[2,2,1,1,1,1,1,1,1,1] => 33022080
[2,1,1,1,1,1,1,1,1,1,1] => 36288000
[1,1,1,1,1,1,1,1,1,1,1,1] => 39916800
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Created: Feb 02, 2018 at 20:19 by Martin Rubey
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Last Updated: Feb 06, 2018 at 07:24 by Jair Taylor