Identifier
Identifier
Values
[2] generating graphics... => 0
[1,1] generating graphics... => 2
[3] generating graphics... => 0
[2,1] generating graphics... => 1
[1,1,1] generating graphics... => 6
[4] generating graphics... => 0
[3,1] generating graphics... => 0
[2,2] generating graphics... => 2
[2,1,1] generating graphics... => 6
[1,1,1,1] generating graphics... => 24
[5] generating graphics... => 0
[4,1] generating graphics... => 0
[3,2] generating graphics... => 1
[3,1,1] generating graphics... => 2
[2,2,1] generating graphics... => 12
[2,1,1,1] generating graphics... => 36
[1,1,1,1,1] generating graphics... => 120
[6] generating graphics... => 0
[5,1] generating graphics... => 0
[4,2] generating graphics... => 0
[4,1,1] generating graphics... => 0
[3,3] generating graphics... => 2
[3,2,1] generating graphics... => 10
[3,1,1,1] generating graphics... => 24
[2,2,2] generating graphics... => 30
[2,2,1,1] generating graphics... => 84
[2,1,1,1,1] generating graphics... => 240
[1,1,1,1,1,1] generating graphics... => 720
[7] generating graphics... => 0
[6,1] generating graphics... => 0
[5,2] generating graphics... => 0
[5,1,1] generating graphics... => 0
[4,3] generating graphics... => 1
[4,2,1] generating graphics... => 3
[4,1,1,1] generating graphics... => 6
[3,3,1] generating graphics... => 18
[3,2,2] generating graphics... => 38
[3,2,1,1] generating graphics... => 96
[3,1,1,1,1] generating graphics... => 240
[2,2,2,1] generating graphics... => 246
[2,2,1,1,1] generating graphics... => 660
[2,1,1,1,1,1] generating graphics... => 1800
[1,1,1,1,1,1,1] generating graphics... => 5040
[8] generating graphics... => 0
[7,1] generating graphics... => 0
[6,2] generating graphics... => 0
[6,1,1] generating graphics... => 0
[5,3] generating graphics... => 0
[5,2,1] generating graphics... => 0
[5,1,1,1] generating graphics... => 0
[4,4] generating graphics... => 2
[4,3,1] generating graphics... => 14
[4,2,2] generating graphics... => 24
[4,2,1,1] generating graphics... => 54
[4,1,1,1,1] generating graphics... => 120
[3,3,2] generating graphics... => 74
[3,3,1,1] generating graphics... => 184
[3,2,2,1] generating graphics... => 384
[3,2,1,1,1] generating graphics... => 960
[3,1,1,1,1,1] generating graphics... => 2400
[2,2,2,2] generating graphics... => 864
[2,2,2,1,1] generating graphics... => 2220
[2,2,1,1,1,1] generating graphics... => 5760
[2,1,1,1,1,1,1] generating graphics... => 15120
[1,1,1,1,1,1,1,1] generating graphics... => 40320
[9] generating graphics... => 0
[8,1] generating graphics... => 0
[7,2] generating graphics... => 0
[7,1,1] generating graphics... => 0
[6,3] generating graphics... => 0
[6,2,1] generating graphics... => 0
[6,1,1,1] generating graphics... => 0
[5,4] generating graphics... => 1
[5,3,1] generating graphics... => 4
[5,2,2] generating graphics... => 6
[5,2,1,1] generating graphics... => 12
[5,1,1,1,1] generating graphics... => 24
[4,4,1] generating graphics... => 24
[4,3,2] generating graphics... => 79
[4,3,1,1] generating graphics... => 186
[4,2,2,1] generating graphics... => 336
[4,2,1,1,1] generating graphics... => 780
[4,1,1,1,1,1] generating graphics... => 1800
[3,3,3] generating graphics... => 174
[3,3,2,1] generating graphics... => 836
[3,3,1,1,1] generating graphics... => 2040
[3,2,2,2] generating graphics... => 1686
[3,2,2,1,1] generating graphics... => 4140
[3,2,1,1,1,1] generating graphics... => 10200
[3,1,1,1,1,1,1] generating graphics... => 25200
[2,2,2,2,1] generating graphics... => 8760
[2,2,2,1,1,1] generating graphics... => 21960
[2,2,1,1,1,1,1] generating graphics... => 55440
[2,1,1,1,1,1,1,1] generating graphics... => 141120
[1,1,1,1,1,1,1,1,1] generating graphics... => 362880
[10] generating graphics... => 0
[9,1] generating graphics... => 0
[8,2] generating graphics... => 0
[8,1,1] generating graphics... => 0
[7,3] generating graphics... => 0
[7,2,1] generating graphics... => 0
[7,1,1,1] generating graphics... => 0
[6,4] generating graphics... => 0
[6,3,1] generating graphics... => 0
[6,2,2] generating graphics... => 0
[6,2,1,1] generating graphics... => 0
[6,1,1,1,1] generating graphics... => 0
[5,5] generating graphics... => 2
[5,4,1] generating graphics... => 18
[5,3,2] generating graphics... => 44
[5,3,1,1] generating graphics... => 96
[5,2,2,1] generating graphics... => 156
[5,2,1,1,1] generating graphics... => 336
[5,1,1,1,1,1] generating graphics... => 720
[4,4,2] generating graphics... => 138
[4,4,1,1] generating graphics... => 324
[4,3,3] generating graphics... => 248
[4,3,2,1] generating graphics... => 1074
[4,3,1,1,1] generating graphics... => 2520
[4,2,2,2] generating graphics... => 1974
[4,2,2,1,1] generating graphics... => 4620
[4,2,1,1,1,1] generating graphics... => 10800
[4,1,1,1,1,1,1] generating graphics... => 25200
[3,3,3,1] generating graphics... => 2184
[3,3,2,2] generating graphics... => 4204
[3,3,2,1,1] generating graphics... => 10080
[3,3,1,1,1,1] generating graphics... => 24240
[3,2,2,2,1] generating graphics... => 19740
[3,2,2,1,1,1] generating graphics... => 47760
[3,2,1,1,1,1,1] generating graphics... => 115920
[3,1,1,1,1,1,1,1] generating graphics... => 282240
[2,2,2,2,2] generating graphics... => 39480
[2,2,2,2,1,1] generating graphics... => 96480
[2,2,2,1,1,1,1] generating graphics... => 236880
[2,2,1,1,1,1,1,1] generating graphics... => 584640
[2,1,1,1,1,1,1,1,1] generating graphics... => 1451520
[1,1,1,1,1,1,1,1,1,1] generating graphics... => 3628800
[11] generating graphics... => 0
[10,1] generating graphics... => 0
[9,2] generating graphics... => 0
[9,1,1] generating graphics... => 0
[8,3] generating graphics... => 0
[8,2,1] generating graphics... => 0
[8,1,1,1] generating graphics... => 0
[7,4] generating graphics... => 0
[7,3,1] generating graphics... => 0
[7,2,2] generating graphics... => 0
[7,2,1,1] generating graphics... => 0
[7,1,1,1,1] generating graphics... => 0
[6,5] generating graphics... => 1
[6,4,1] generating graphics... => 5
[6,3,2] generating graphics... => 10
[6,3,1,1] generating graphics... => 20
[6,2,2,1] generating graphics... => 30
[6,2,1,1,1] generating graphics... => 60
[6,1,1,1,1,1] generating graphics... => 120
[5,5,1] generating graphics... => 30
[5,4,2] generating graphics... => 135
[5,4,1,1] generating graphics... => 306
[5,3,3] generating graphics... => 212
[5,3,2,1] generating graphics... => 816
[5,3,1,1,1] generating graphics... => 1824
[5,2,2,2] generating graphics... => 1386
[5,2,2,1,1] generating graphics... => 3084
[5,2,1,1,1,1] generating graphics... => 6840
[5,1,1,1,1,1,1] generating graphics... => 15120
[4,4,3] generating graphics... => 480
[4,4,2,1] generating graphics... => 2016
[4,4,1,1,1] generating graphics... => 4680
[4,3,3,1] generating graphics... => 3566
[4,3,2,2] generating graphics... => 6516
[4,3,2,1,1] generating graphics... => 15180
[4,3,1,1,1,1] generating graphics... => 35400
[4,2,2,2,1] generating graphics... => 27780
[4,2,2,1,1,1] generating graphics... => 64800
[4,2,1,1,1,1,1] generating graphics... => 151200
[4,1,1,1,1,1,1,1] generating graphics... => 352800
[3,3,3,2] generating graphics... => 12336
[3,3,3,1,1] generating graphics... => 29040
[3,3,2,2,1] generating graphics... => 54660
[3,3,2,1,1,1] generating graphics... => 129480
[3,3,1,1,1,1,1] generating graphics... => 307440
[3,2,2,2,2] generating graphics... => 103800
[3,2,2,2,1,1] generating graphics... => 247080
[3,2,2,1,1,1,1] generating graphics... => 589680
[3,2,1,1,1,1,1,1] generating graphics... => 1411200
[3,1,1,1,1,1,1,1,1] generating graphics... => 3386880
[2,2,2,2,2,1] generating graphics... => 478080
[2,2,2,2,1,1,1] generating graphics... => 1149120
[2,2,2,1,1,1,1,1] generating graphics... => 2772000
[2,2,1,1,1,1,1,1,1] generating graphics... => 6713280
[2,1,1,1,1,1,1,1,1,1] generating graphics... => 16329600
[1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 39916800
[12] generating graphics... => 0
[11,1] generating graphics... => 0
[10,2] generating graphics... => 0
[10,1,1] generating graphics... => 0
[9,3] generating graphics... => 0
[9,2,1] generating graphics... => 0
[9,1,1,1] generating graphics... => 0
[8,4] generating graphics... => 0
[8,3,1] generating graphics... => 0
[8,2,2] generating graphics... => 0
[8,2,1,1] generating graphics... => 0
[8,1,1,1,1] generating graphics... => 0
[7,5] generating graphics... => 0
[7,4,1] generating graphics... => 0
[7,3,2] generating graphics... => 0
[7,3,1,1] generating graphics... => 0
[7,2,2,1] generating graphics... => 0
[7,2,1,1,1] generating graphics... => 0
[7,1,1,1,1,1] generating graphics... => 0
[6,6] generating graphics... => 2
[6,5,1] generating graphics... => 22
[6,4,2] generating graphics... => 70
[6,4,1,1] generating graphics... => 150
[6,3,3] generating graphics... => 100
[6,3,2,1] generating graphics... => 340
[6,3,1,1,1] generating graphics... => 720
[6,2,2,2] generating graphics... => 540
[6,2,2,1,1] generating graphics... => 1140
[6,2,1,1,1,1] generating graphics... => 2400
[6,1,1,1,1,1,1] generating graphics... => 5040
[5,5,2] generating graphics... => 222
[5,5,1,1] generating graphics... => 504
[5,4,3] generating graphics... => 588
[5,4,2,1] generating graphics... => 2322
[5,4,1,1,1] generating graphics... => 5256
[5,3,3,1] generating graphics... => 3712
[5,3,2,2] generating graphics... => 6432
[5,3,2,1,1] generating graphics... => 14496
[5,3,1,1,1,1] generating graphics... => 32640
[5,2,2,2,1] generating graphics... => 24996
[5,2,2,1,1,1] generating graphics... => 56160
[5,2,1,1,1,1,1] generating graphics... => 126000
[5,1,1,1,1,1,1,1] generating graphics... => 282240
[4,4,4] generating graphics... => 1092
[4,4,3,1] generating graphics... => 7524
[4,4,2,2] generating graphics... => 13464
[4,4,2,1,1] generating graphics... => 30960
[4,4,1,1,1,1] generating graphics... => 71280
[4,3,3,2] generating graphics... => 23254
[4,3,3,1,1] generating graphics... => 53640
[4,3,2,2,1] generating graphics... => 96900
[4,3,2,1,1,1] generating graphics... => 224160
[4,3,1,1,1,1,1] generating graphics... => 519120
[4,2,2,2,2] generating graphics... => 175440
[4,2,2,2,1,1] generating graphics... => 406440
[4,2,2,1,1,1,1] generating graphics... => 942480
[4,2,1,1,1,1,1,1] generating graphics... => 2187360
[4,1,1,1,1,1,1,1,1] generating graphics... => 5080320
[3,3,3,3] generating graphics... => 41304
[3,3,3,2,1] generating graphics... => 175440
[3,3,3,1,1,1] generating graphics... => 408960
[3,3,2,2,2] generating graphics... => 322860
[3,3,2,2,1,1] generating graphics... => 755040
[3,3,2,1,1,1,1] generating graphics... => 1769040
[3,3,1,1,1,1,1,1] generating graphics... => 4152960
[3,2,2,2,2,1] generating graphics... => 1403520
[3,2,2,2,1,1,1] generating graphics... => 3301200
[3,2,2,1,1,1,1,1] generating graphics... => 7781760
[3,2,1,1,1,1,1,1,1] generating graphics... => 18385920
[3,1,1,1,1,1,1,1,1,1] generating graphics... => 43545600
[2,2,2,2,2,2] generating graphics... => 2631600
[2,2,2,2,2,1,1] generating graphics... => 6219360
[2,2,2,2,1,1,1,1] generating graphics... => 14736960
[2,2,2,1,1,1,1,1,1] generating graphics... => 35017920
[2,2,1,1,1,1,1,1,1,1] generating graphics... => 83462400
[2,1,1,1,1,1,1,1,1,1,1] generating graphics... => 199584000
[1,1,1,1,1,1,1,1,1,1,1,1] generating graphics... => 479001600
click to show generating function       
Description
The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders.
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 $m_\lambda$ in the formal group law for linear orders, with generating function $f(x) = x/(1-x)$, see [1, sec. 3.4].
This statistic gives the number of Smirnov arrangements of a set of letters with $\lambda_i$ of the $i$th letter, where a Smirnov word is a word with no repeated adjacent letters. e.g., [3,2,1] = > 10 since there are 10 Smirnov rearrangements of the word 'aaabbc': 'ababac', 'ababca', 'abacab', 'abacba', 'abcaba', 'acabab', 'acbaba', 'babaca', 'bacaba', 'cababa'.
References
[1] Taylor, J. Formal group laws and hypergraph colorings MathSciNet:3542357
Code
@cached_function
def data(n):
    """
    sage: data = data_linear_orders
    sage: n = 3; [(P, statistic(P)) for P in Partitions(n)]

    sage: findstat([(P, statistic(P)) for n in range(1,9) for P in Partitions(n)], depth=3)
    a new statistic on Cc0002: Integer partitions
    """
    R. = PowerSeriesRing(SR, default_prec=n+1)
    f = x/(1-x) # linear orders
    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 sum(p.coefficient(t,n) * f_coefficients[n] * factorial(n)
               for n in range(p.degree(t)+1)).expand()

Created
Feb 02, 2018 at 19:51 by Martin Rubey
Updated
Feb 04, 2018 at 19:59 by Jair Taylor