Identifier
Identifier
Values
[2] generating graphics... => 0
[1,1] generating graphics... => 1
[3] generating graphics... => 0
[2,1] generating graphics... => 2
[1,1,1] generating graphics... => 3
[4] generating graphics... => 0
[3,1] generating graphics... => 6
[2,2] generating graphics... => 10
[2,1,1] generating graphics... => 12
[1,1,1,1] generating graphics... => 15
[5] generating graphics... => 0
[4,1] generating graphics... => 24
[3,2] generating graphics... => 54
[3,1,1] generating graphics... => 60
[2,2,1] generating graphics... => 78
[2,1,1,1] generating graphics... => 90
[1,1,1,1,1] generating graphics... => 105
[6] generating graphics... => 0
[5,1] generating graphics... => 120
[4,2] generating graphics... => 336
[4,1,1] generating graphics... => 360
[3,3] generating graphics... => 450
[3,2,1] generating graphics... => 570
[3,1,1,1] generating graphics... => 630
[2,2,2] generating graphics... => 672
[2,2,1,1] generating graphics... => 750
[2,1,1,1,1] generating graphics... => 840
[1,1,1,1,1,1] generating graphics... => 945
[7] generating graphics... => 0
[6,1] generating graphics... => 720
[5,2] generating graphics... => 2400
[5,1,1] generating graphics... => 2520
[4,3] generating graphics... => 3960
[4,2,1] generating graphics... => 4680
[4,1,1,1] generating graphics... => 5040
[3,3,1] generating graphics... => 5670
[3,2,2] generating graphics... => 6360
[3,2,1,1] generating graphics... => 6930
[3,1,1,1,1] generating graphics... => 7560
[2,2,2,1] generating graphics... => 7860
[2,2,1,1,1] generating graphics... => 8610
[2,1,1,1,1,1] generating graphics... => 9450
[1,1,1,1,1,1,1] generating graphics... => 10395
[8] generating graphics... => 0
[7,1] generating graphics... => 5040
[6,2] generating graphics... => 19440
[6,1,1] generating graphics... => 20160
[5,3] generating graphics... => 37800
[5,2,1] generating graphics... => 42840
[5,1,1,1] generating graphics... => 45360
[4,4] generating graphics... => 46440
[4,3,1] generating graphics... => 60480
[4,2,2] generating graphics... => 65880
[4,2,1,1] generating graphics... => 70560
[4,1,1,1,1] generating graphics... => 75600
[3,3,2] generating graphics... => 75600
[3,3,1,1] generating graphics... => 81270
[3,2,2,1] generating graphics... => 89460
[3,2,1,1,1] generating graphics... => 96390
[3,1,1,1,1,1] generating graphics... => 103950
[2,2,2,2] generating graphics... => 98820
[2,2,2,1,1] generating graphics... => 106680
[2,2,1,1,1,1] generating graphics... => 115290
[2,1,1,1,1,1,1] generating graphics... => 124740
[1,1,1,1,1,1,1,1] generating graphics... => 135135
[9] generating graphics... => 0
[8,1] generating graphics... => 40320
[7,2] generating graphics... => 176400
[7,1,1] generating graphics... => 181440
[6,3] generating graphics... => 393120
[6,2,1] generating graphics... => 433440
[6,1,1,1] generating graphics... => 453600
[5,4] generating graphics... => 567000
[5,3,1] generating graphics... => 695520
[5,2,2] generating graphics... => 743400
[5,2,1,1] generating graphics... => 786240
[5,1,1,1,1] generating graphics... => 831600
[4,4,1] generating graphics... => 808920
[4,3,2] generating graphics... => 960120
[4,3,1,1] generating graphics... => 1020600
[4,2,2,1] generating graphics... => 1101240
[4,2,1,1,1] generating graphics... => 1171800
[4,1,1,1,1,1] generating graphics... => 1247400
[3,3,3] generating graphics... => 1065960
[3,3,2,1] generating graphics... => 1228500
[3,3,1,1,1] generating graphics... => 1309770
[3,2,2,2] generating graphics... => 1331820
[3,2,2,1,1] generating graphics... => 1421280
[3,2,1,1,1,1] generating graphics... => 1517670
[3,1,1,1,1,1,1] generating graphics... => 1621620
[2,2,2,2,1] generating graphics... => 1545180
[2,2,2,1,1,1] generating graphics... => 1651860
[2,2,1,1,1,1,1] generating graphics... => 1767150
[2,1,1,1,1,1,1,1] generating graphics... => 1891890
[1,1,1,1,1,1,1,1,1] generating graphics... => 2027025
[10] generating graphics... => 0
[9,1] generating graphics... => 362880
[8,2] generating graphics... => 1774080
[8,1,1] generating graphics... => 1814400
[7,3] generating graphics... => 4445280
[7,2,1] generating graphics... => 4808160
[7,1,1,1] generating graphics... => 4989600
[6,4] generating graphics... => 7318080
[6,3,1] generating graphics... => 8618400
[6,2,2] generating graphics... => 9092160
[6,2,1,1] generating graphics... => 9525600
[6,1,1,1,1] generating graphics... => 9979200
[5,5] generating graphics... => 8580600
[5,4,1] generating graphics... => 11362680
[5,3,2] generating graphics... => 13025880
[5,3,1,1] generating graphics... => 13721400
[5,2,2,1] generating graphics... => 14598360
[5,2,1,1,1] generating graphics... => 15384600
[5,1,1,1,1,1] generating graphics... => 16216200
[4,4,2] generating graphics... => 14636160
[4,4,1,1] generating graphics... => 15445080
[4,3,3] generating graphics... => 15898680
[4,3,2,1] generating graphics... => 17939880
[4,3,1,1,1] generating graphics... => 18960480
[4,2,2,2] generating graphics... => 19182240
[4,2,2,1,1] generating graphics... => 20283480
[4,2,1,1,1,1] generating graphics... => 21455280
[4,1,1,1,1,1,1] generating graphics... => 22702680
[3,3,3,1] generating graphics... => 19584180
[3,3,2,2] generating graphics... => 20975220
[3,3,2,1,1] generating graphics... => 22203720
[3,3,1,1,1,1] generating graphics... => 23513490
[3,2,2,2,1] generating graphics... => 23817780
[3,2,2,1,1,1] generating graphics... => 25239060
[3,2,1,1,1,1,1] generating graphics... => 26756730
[3,1,1,1,1,1,1,1] generating graphics... => 28378350
[2,2,2,2,2] generating graphics... => 25576320
[2,2,2,2,1,1] generating graphics... => 27121500
[2,2,2,1,1,1,1] generating graphics... => 28773360
[2,2,1,1,1,1,1,1] generating graphics... => 30540510
[2,1,1,1,1,1,1,1,1] generating graphics... => 32432400
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 34459425
[11] generating graphics... => 0
[10,1] generating graphics... => 3628800
[9,2] generating graphics... => 19595520
[9,1,1] generating graphics... => 19958400
[8,3] generating graphics... => 54432000
[8,2,1] generating graphics... => 58060800
[8,1,1,1] generating graphics... => 59875200
[7,4] generating graphics... => 100336320
[7,3,1] generating graphics... => 114760800
[7,2,2] generating graphics... => 119931840
[7,2,1,1] generating graphics... => 124740000
[7,1,1,1,1] generating graphics... => 129729600
[6,5] generating graphics... => 134038800
[6,4,1] generating graphics... => 168512400
[6,3,2] generating graphics... => 188470800
[6,3,1,1] generating graphics... => 197089200
[6,2,2,1] generating graphics... => 207522000
[6,2,1,1,1] generating graphics... => 217047600
[6,1,1,1,1,1] generating graphics... => 227026800
[5,5,1] generating graphics... => 190852200
[5,4,2] generating graphics... => 234375120
[5,4,1,1] generating graphics... => 245737800
[5,3,3] generating graphics... => 250727400
[5,3,2,1] generating graphics... => 278170200
[5,3,1,1,1] generating graphics... => 291891600
[5,2,2,2] generating graphics... => 294341040
[5,2,2,1,1] generating graphics... => 308939400
[5,2,1,1,1,1] generating graphics... => 324324000
[5,1,1,1,1,1,1] generating graphics... => 340540200
[4,4,3] generating graphics... => 275244480
[4,4,2,1] generating graphics... => 306134640
[4,4,1,1,1] generating graphics... => 321579720
[4,3,3,1] generating graphics... => 329064120
[4,3,2,2] generating graphics... => 349045200
[4,3,2,1,1] generating graphics... => 366985080
[4,3,1,1,1,1] generating graphics... => 385945560
[4,2,2,2,1] generating graphics... => 389612160
[4,2,2,1,1,1] generating graphics... => 409895640
[4,2,1,1,1,1,1] generating graphics... => 431350920
[4,1,1,1,1,1,1,1] generating graphics... => 454053600
[3,3,3,2] generating graphics... => 376091100
[3,3,3,1,1] generating graphics... => 395675280
[3,3,2,2,1] generating graphics... => 420498540
[3,3,2,1,1,1] generating graphics... => 442702260
[3,3,1,1,1,1,1] generating graphics... => 466215750
[3,2,2,2,2] generating graphics... => 447158880
[3,2,2,2,1,1] generating graphics... => 470976660
[3,2,2,1,1,1,1] generating graphics... => 496215720
[3,2,1,1,1,1,1,1] generating graphics... => 522972450
[3,1,1,1,1,1,1,1,1] generating graphics... => 551350800
[2,2,2,2,2,1] generating graphics... => 501401880
[2,2,2,2,1,1,1] generating graphics... => 528523380
[2,2,2,1,1,1,1,1] generating graphics... => 557296740
[2,2,1,1,1,1,1,1,1] generating graphics... => 587837250
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 620269650
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 654729075
[12] generating graphics... => 0
[11,1] generating graphics... => 39916800
[10,2] generating graphics... => 235872000
[10,1,1] generating graphics... => 239500800
[9,3] generating graphics... => 718502400
[9,2,1] generating graphics... => 758419200
[9,1,1,1] generating graphics... => 778377600
[8,4] generating graphics... => 1462406400
[8,3,1] generating graphics... => 1636588800
[8,2,2] generating graphics... => 1698278400
[8,2,1,1] generating graphics... => 1756339200
[8,1,1,1,1] generating graphics... => 1816214400
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 leaf labelled binary 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 leaf labelled binary trees, with generating function $f(x) = 1-\sqrt{1-2x}$, see [1, sec. 3.2]
Fix a set of distinguishable vertices and a coloring of the vertices so that $\lambda_i$ are colored $i$. This statistic gives the number of rooted binary trees with leaves labeled with this set of vertices and internal vertices unlabeled so that no pair of 'twin' leaves have the same color.
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 = 1-sqrt(1-2*x) # labelled binary 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 20:13 by Martin Rubey
Updated
Feb 06, 2018 at 07:21 by Jair Taylor