Identifier
Identifier
Values
[2] generating graphics... => 0
[1,1] generating graphics... => 1
[3] generating graphics... => 0
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 2
[4] generating graphics... => 0
[3,1] generating graphics... => 1
[2,2] generating graphics... => 3
[2,1,1] generating graphics... => 4
[1,1,1,1] generating graphics... => 6
[5] generating graphics... => 0
[4,1] generating graphics... => 1
[3,2] generating graphics... => 7
[3,1,1] generating graphics... => 8
[2,2,1] generating graphics... => 14
[2,1,1,1] generating graphics... => 18
[1,1,1,1,1] generating graphics... => 24
[6] generating graphics... => 0
[5,1] generating graphics... => 1
[4,2] generating graphics... => 15
[4,1,1] generating graphics... => 16
[3,3] generating graphics... => 31
[3,2,1] generating graphics... => 46
[3,1,1,1] generating graphics... => 54
[2,2,2] generating graphics... => 64
[2,2,1,1] generating graphics... => 78
[2,1,1,1,1] generating graphics... => 96
[1,1,1,1,1,1] generating graphics... => 120
[7] generating graphics... => 0
[6,1] generating graphics... => 1
[5,2] generating graphics... => 31
[5,1,1] generating graphics... => 32
[4,3] generating graphics... => 115
[4,2,1] generating graphics... => 146
[4,1,1,1] generating graphics... => 162
[3,3,1] generating graphics... => 230
[3,2,2] generating graphics... => 284
[3,2,1,1] generating graphics... => 330
[3,1,1,1,1] generating graphics... => 384
[2,2,2,1] generating graphics... => 426
[2,2,1,1,1] generating graphics... => 504
[2,1,1,1,1,1] generating graphics... => 600
[1,1,1,1,1,1,1] generating graphics... => 720
[8] generating graphics... => 0
[7,1] generating graphics... => 1
[6,2] generating graphics... => 63
[6,1,1] generating graphics... => 64
[5,3] generating graphics... => 391
[5,2,1] generating graphics... => 454
[5,1,1,1] generating graphics... => 486
[4,4] generating graphics... => 675
[4,3,1] generating graphics... => 1066
[4,2,2] generating graphics... => 1228
[4,2,1,1] generating graphics... => 1374
[4,1,1,1,1] generating graphics... => 1536
[3,3,2] generating graphics... => 1672
[3,3,1,1] generating graphics... => 1902
[3,2,2,1] generating graphics... => 2286
[3,2,1,1,1] generating graphics... => 2616
[3,1,1,1,1,1] generating graphics... => 3000
[2,2,2,2] generating graphics... => 2790
[2,2,2,1,1] generating graphics... => 3216
[2,2,1,1,1,1] generating graphics... => 3720
[2,1,1,1,1,1,1] generating graphics... => 4320
[1,1,1,1,1,1,1,1] generating graphics... => 5040
[9] generating graphics... => 0
[8,1] generating graphics... => 1
[7,2] generating graphics... => 127
[7,1,1] generating graphics... => 128
[6,3] generating graphics... => 1267
[6,2,1] generating graphics... => 1394
[6,1,1,1] generating graphics... => 1458
[5,4] generating graphics... => 3451
[5,3,1] generating graphics... => 4718
[5,2,2] generating graphics... => 5204
[5,2,1,1] generating graphics... => 5658
[5,1,1,1,1] generating graphics... => 6144
[4,4,1] generating graphics... => 6902
[4,3,2] generating graphics... => 9488
[4,3,1,1] generating graphics... => 10554
[4,2,2,1] generating graphics... => 12090
[4,2,1,1,1] generating graphics... => 13464
[4,1,1,1,1,1] generating graphics... => 15000
[3,3,3] generating graphics... => 11828
[3,3,2,1] generating graphics... => 15402
[3,3,1,1,1] generating graphics... => 17304
[3,2,2,2] generating graphics... => 18018
[3,2,2,1,1] generating graphics... => 20304
[3,2,1,1,1,1] generating graphics... => 22920
[3,1,1,1,1,1,1] generating graphics... => 25920
[2,2,2,2,1] generating graphics... => 24024
[2,2,2,1,1,1] generating graphics... => 27240
[2,2,1,1,1,1,1] generating graphics... => 30960
[2,1,1,1,1,1,1,1] generating graphics... => 35280
[1,1,1,1,1,1,1,1,1] generating graphics... => 40320
[10] generating graphics... => 0
[9,1] generating graphics... => 1
[8,2] generating graphics... => 255
[8,1,1] generating graphics... => 256
[7,3] generating graphics... => 3991
[7,2,1] generating graphics... => 4246
[7,1,1,1] generating graphics... => 4374
[6,4] generating graphics... => 16275
[6,3,1] generating graphics... => 20266
[6,2,2] generating graphics... => 21724
[6,2,1,1] generating graphics... => 23118
[6,1,1,1,1] generating graphics... => 24576
[5,5] generating graphics... => 25231
[5,4,1] generating graphics... => 41506
[5,3,2] generating graphics... => 52336
[5,3,1,1] generating graphics... => 57054
[5,2,2,1] generating graphics... => 63198
[5,2,1,1,1] generating graphics... => 68856
[5,1,1,1,1,1] generating graphics... => 75000
[4,4,2] generating graphics... => 69208
[4,4,1,1] generating graphics... => 76110
[4,3,3] generating graphics... => 81460
[4,3,2,1] generating graphics... => 101502
[4,3,1,1,1] generating graphics... => 112056
[4,2,2,2] generating graphics... => 114966
[4,2,2,1,1] generating graphics... => 127056
[4,2,1,1,1,1] generating graphics... => 140520
[4,1,1,1,1,1,1] generating graphics... => 155520
[3,3,3,1] generating graphics... => 122190
[3,3,2,2] generating graphics... => 139494
[3,3,2,1,1] generating graphics... => 154896
[3,3,1,1,1,1] generating graphics... => 172200
[3,2,2,2,1] generating graphics... => 177816
[3,2,2,1,1,1] generating graphics... => 198120
[3,2,1,1,1,1,1] generating graphics... => 221040
[3,1,1,1,1,1,1,1] generating graphics... => 246960
[2,2,2,2,2] generating graphics... => 205056
[2,2,2,2,1,1] generating graphics... => 229080
[2,2,2,1,1,1,1] generating graphics... => 256320
[2,2,1,1,1,1,1,1] generating graphics... => 287280
[2,1,1,1,1,1,1,1,1] generating graphics... => 322560
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 362880
[11] generating graphics... => 0
[10,1] generating graphics... => 1
[9,2] generating graphics... => 511
[9,1,1] generating graphics... => 512
[8,3] generating graphics... => 12355
[8,2,1] generating graphics... => 12866
[8,1,1,1] generating graphics... => 13122
[7,4] generating graphics... => 72955
[7,3,1] generating graphics... => 85310
[7,2,2] generating graphics... => 89684
[7,2,1,1] generating graphics... => 93930
[7,1,1,1,1] generating graphics... => 98304
[6,5] generating graphics... => 164731
[6,4,1] generating graphics... => 237686
[6,3,2] generating graphics... => 282464
[6,3,1,1] generating graphics... => 302730
[6,2,2,1] generating graphics... => 327306
[6,2,1,1,1] generating graphics... => 350424
[6,1,1,1,1,1] generating graphics... => 375000
[5,5,1] generating graphics... => 329462
[5,4,2] generating graphics... => 484136
[5,4,1,1] generating graphics... => 525642
[5,3,3] generating graphics... => 547820
[5,3,2,1] generating graphics... => 657210
[5,3,1,1,1] generating graphics... => 714264
[5,2,2,2] generating graphics... => 726066
[5,2,2,1,1] generating graphics... => 789264
[5,2,1,1,1,1] generating graphics... => 858120
[5,1,1,1,1,1,1] generating graphics... => 933120
[4,4,3] generating graphics... => 677636
[4,4,2,1] generating graphics... => 822954
[4,4,1,1,1] generating graphics... => 899064
[4,3,3,1] generating graphics... => 951546
[4,3,2,2] generating graphics... => 1063602
[4,3,2,1,1] generating graphics... => 1165104
[4,3,1,1,1,1] generating graphics... => 1277160
[4,2,2,2,1] generating graphics... => 1305624
[4,2,2,1,1,1] generating graphics... => 1432680
[4,2,1,1,1,1,1] generating graphics... => 1573200
[4,1,1,1,1,1,1,1] generating graphics... => 1728720
[3,3,3,2] generating graphics... => 1244034
[3,3,3,1,1] generating graphics... => 1366224
[3,3,2,2,1] generating graphics... => 1538424
[3,3,2,1,1,1] generating graphics... => 1693320
[3,3,1,1,1,1,1] generating graphics... => 1865520
[3,2,2,2,2] generating graphics... => 1736544
[3,2,2,2,1,1] generating graphics... => 1914360
[3,2,2,1,1,1,1] generating graphics... => 2112480
[3,2,1,1,1,1,1,1] generating graphics... => 2333520
[3,1,1,1,1,1,1,1,1] generating graphics... => 2580480
[2,2,2,2,2,1] generating graphics... => 2170680
[2,2,2,2,1,1,1] generating graphics... => 2399760
[2,2,2,1,1,1,1,1] generating graphics... => 2656080
[2,2,1,1,1,1,1,1,1] generating graphics... => 2943360
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 3265920
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 3628800
[12] generating graphics... => 0
[11,1] generating graphics... => 1
[10,2] generating graphics... => 1023
[10,1,1] generating graphics... => 1024
[9,3] generating graphics... => 37831
[9,2,1] generating graphics... => 38854
[9,1,1,1] generating graphics... => 39366
[8,4] generating graphics... => 316275
[8,3,1] generating graphics... => 354106
[8,2,2] generating graphics... => 367228
[8,2,1,1] generating graphics... => 380094
[8,1,1,1,1] generating graphics... => 393216
[7,5] generating graphics... => 999391
[7,4,1] generating graphics... => 1315666
[7,3,2] generating graphics... => 1499152
[7,3,1,1] generating graphics... => 1584462
[7,2,2,1] generating graphics... => 1682766
[7,2,1,1,1] generating graphics... => 1776696
[7,1,1,1,1,1] generating graphics... => 1875000
[6,6] generating graphics... => 1441923
[6,5,1] generating graphics... => 2441314
[6,4,2] generating graphics... => 3281608
[6,4,1,1] generating graphics... => 3519294
[6,3,3] generating graphics... => 3610372
[6,3,2,1] generating graphics... => 4195566
[6,3,1,1,1] generating graphics... => 4498296
[6,2,2,2] generating graphics... => 4545990
[6,2,2,1,1] generating graphics... => 4873296
[6,2,1,1,1,1] generating graphics... => 5223720
[6,1,1,1,1,1,1] generating graphics... => 5598720
[5,5,2] generating graphics... => 4223704
[5,5,1,1] generating graphics... => 4553166
[5,4,3] generating graphics... => 5485756
[5,4,2,1] generating graphics... => 6495534
[5,4,1,1,1] generating graphics... => 7021176
[5,3,3,1] generating graphics... => 7290942
[5,3,2,2] generating graphics... => 8005206
[5,3,2,1,1] generating graphics... => 8662416
[5,3,1,1,1,1] generating graphics... => 9376680
[5,2,2,2,1] generating graphics... => 9520536
[5,2,2,1,1,1] generating graphics... => 10309800
[5,2,1,1,1,1,1] generating graphics... => 11167920
[5,1,1,1,1,1,1,1] generating graphics... => 12101040
[4,4,4] generating graphics... => 6476644
[4,4,3,1] generating graphics... => 8724078
[4,4,2,2] generating graphics... => 9623142
[4,4,2,1,1] generating graphics... => 10446096
[4,4,1,1,1,1] generating graphics... => 11345160
[4,3,3,2] generating graphics... => 10942230
[4,3,3,1,1] generating graphics... => 11893776
[4,3,2,2,1] generating graphics... => 13170936
[4,3,2,1,1,1] generating graphics... => 14336040
[4,3,1,1,1,1,1] generating graphics... => 15613200
[4,2,2,2,2] generating graphics... => 14603616
[4,2,2,2,1,1] generating graphics... => 15909240
[4,2,2,1,1,1,1] generating graphics... => 17341920
[4,2,1,1,1,1,1,1] generating graphics... => 18915120
[4,1,1,1,1,1,1,1,1] generating graphics... => 20643840
[3,3,3,3] generating graphics... => 12497958
[3,3,3,2,1] generating graphics... => 15108216
[3,3,3,1,1,1] generating graphics... => 16474440
[3,3,2,2,2] generating graphics... => 16801536
[3,3,2,2,1,1] generating graphics... => 18339960
[3,3,2,1,1,1,1] generating graphics... => 20033280
[3,3,1,1,1,1,1,1] generating graphics... => 21898800
[3,2,2,2,2,1] generating graphics... => 20452440
[3,2,2,2,1,1,1] generating graphics... => 22366800
[3,2,2,1,1,1,1,1] generating graphics... => 24479280
[3,2,1,1,1,1,1,1,1] generating graphics... => 26812800
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 29393280
[2,2,2,2,2,2] generating graphics... => 22852200
[2,2,2,2,2,1,1] generating graphics... => 25022880
[2,2,2,2,1,1,1,1] generating graphics... => 27422640
[2,2,2,1,1,1,1,1,1] generating graphics... => 30078720
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 33022080
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 36288000
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 39916800
click to show generating function       
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 $i$.)
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 = -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())

Created
Feb 02, 2018 at 20:19 by Martin Rubey
Updated
Feb 06, 2018 at 07:24 by Jair Taylor