- FindStat

Identifier

St001097: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>0
[1,1]=>2
[3]=>0
[2,1]=>1
[1,1,1]=>6
[4]=>0
[3,1]=>0
[2,2]=>2
[2,1,1]=>6
[1,1,1,1]=>24
[5]=>0
[4,1]=>0
[3,2]=>1
[3,1,1]=>2
[2,2,1]=>12
[2,1,1,1]=>36
[1,1,1,1,1]=>120
[6]=>0
[5,1]=>0
[4,2]=>0
[4,1,1]=>0
[3,3]=>2
[3,2,1]=>10
[3,1,1,1]=>24
[2,2,2]=>30
[2,2,1,1]=>84
[2,1,1,1,1]=>240
[1,1,1,1,1,1]=>720
[7]=>0
[6,1]=>0
[5,2]=>0
[5,1,1]=>0
[4,3]=>1
[4,2,1]=>3
[4,1,1,1]=>6
[3,3,1]=>18
[3,2,2]=>38
[3,2,1,1]=>96
[3,1,1,1,1]=>240
[2,2,2,1]=>246
[2,2,1,1,1]=>660
[2,1,1,1,1,1]=>1800
[1,1,1,1,1,1,1]=>5040
[8]=>0
[7,1]=>0
[6,2]=>0
[6,1,1]=>0
[5,3]=>0
[5,2,1]=>0
[5,1,1,1]=>0
[4,4]=>2
[4,3,1]=>14
[4,2,2]=>24
[4,2,1,1]=>54
[4,1,1,1,1]=>120
[3,3,2]=>74
[3,3,1,1]=>184
[3,2,2,1]=>384
[3,2,1,1,1]=>960
[3,1,1,1,1,1]=>2400
[2,2,2,2]=>864
[2,2,2,1,1]=>2220
[2,2,1,1,1,1]=>5760
[2,1,1,1,1,1,1]=>15120
[1,1,1,1,1,1,1,1]=>40320
[9]=>0
[8,1]=>0
[7,2]=>0
[7,1,1]=>0
[6,3]=>0
[6,2,1]=>0
[6,1,1,1]=>0
[5,4]=>1
[5,3,1]=>4
[5,2,2]=>6
[5,2,1,1]=>12
[5,1,1,1,1]=>24
[4,4,1]=>24
[4,3,2]=>79
[4,3,1,1]=>186
[4,2,2,1]=>336
[4,2,1,1,1]=>780
[4,1,1,1,1,1]=>1800
[3,3,3]=>174
[3,3,2,1]=>836
[3,3,1,1,1]=>2040
[3,2,2,2]=>1686
[3,2,2,1,1]=>4140
[3,2,1,1,1,1]=>10200
[3,1,1,1,1,1,1]=>25200
[2,2,2,2,1]=>8760
[2,2,2,1,1,1]=>21960
[2,2,1,1,1,1,1]=>55440
[2,1,1,1,1,1,1,1]=>141120
[1,1,1,1,1,1,1,1,1]=>362880
[10]=>0
[9,1]=>0
[8,2]=>0
[8,1,1]=>0
[7,3]=>0
[7,2,1]=>0
[7,1,1,1]=>0
[6,4]=>0
[6,3,1]=>0
[6,2,2]=>0
[6,2,1,1]=>0
[6,1,1,1,1]=>0
[5,5]=>2
[5,4,1]=>18
[5,3,2]=>44
[5,3,1,1]=>96
[5,2,2,1]=>156
[5,2,1,1,1]=>336
[5,1,1,1,1,1]=>720
[4,4,2]=>138
[4,4,1,1]=>324
[4,3,3]=>248
[4,3,2,1]=>1074
[4,3,1,1,1]=>2520
[4,2,2,2]=>1974
[4,2,2,1,1]=>4620
[4,2,1,1,1,1]=>10800
[4,1,1,1,1,1,1]=>25200
[3,3,3,1]=>2184
[3,3,2,2]=>4204
[3,3,2,1,1]=>10080
[3,3,1,1,1,1]=>24240
[3,2,2,2,1]=>19740
[3,2,2,1,1,1]=>47760
[3,2,1,1,1,1,1]=>115920
[3,1,1,1,1,1,1,1]=>282240
[2,2,2,2,2]=>39480
[2,2,2,2,1,1]=>96480
[2,2,2,1,1,1,1]=>236880
[2,2,1,1,1,1,1,1]=>584640
[2,1,1,1,1,1,1,1,1]=>1451520
[1,1,1,1,1,1,1,1,1,1]=>3628800
[11]=>0
[10,1]=>0
[9,2]=>0
[9,1,1]=>0
[8,3]=>0
[8,2,1]=>0
[8,1,1,1]=>0
[7,4]=>0
[7,3,1]=>0
[7,2,2]=>0
[7,2,1,1]=>0
[7,1,1,1,1]=>0
[6,5]=>1
[6,4,1]=>5
[6,3,2]=>10
[6,3,1,1]=>20
[6,2,2,1]=>30
[6,2,1,1,1]=>60
[6,1,1,1,1,1]=>120
[5,5,1]=>30
[5,4,2]=>135
[5,4,1,1]=>306
[5,3,3]=>212
[5,3,2,1]=>816
[5,3,1,1,1]=>1824
[5,2,2,2]=>1386
[5,2,2,1,1]=>3084
[5,2,1,1,1,1]=>6840
[5,1,1,1,1,1,1]=>15120
[4,4,3]=>480
[4,4,2,1]=>2016
[4,4,1,1,1]=>4680
[4,3,3,1]=>3566
[4,3,2,2]=>6516
[4,3,2,1,1]=>15180
[4,3,1,1,1,1]=>35400
[4,2,2,2,1]=>27780
[4,2,2,1,1,1]=>64800
[4,2,1,1,1,1,1]=>151200
[4,1,1,1,1,1,1,1]=>352800
[3,3,3,2]=>12336
[3,3,3,1,1]=>29040
[3,3,2,2,1]=>54660
[3,3,2,1,1,1]=>129480
[3,3,1,1,1,1,1]=>307440
[3,2,2,2,2]=>103800
[3,2,2,2,1,1]=>247080
[3,2,2,1,1,1,1]=>589680
[3,2,1,1,1,1,1,1]=>1411200
[3,1,1,1,1,1,1,1,1]=>3386880
[2,2,2,2,2,1]=>478080
[2,2,2,2,1,1,1]=>1149120
[2,2,2,1,1,1,1,1]=>2772000
[2,2,1,1,1,1,1,1,1]=>6713280
[2,1,1,1,1,1,1,1,1,1]=>16329600
[1,1,1,1,1,1,1,1,1,1,1]=>39916800
[12]=>0
[11,1]=>0
[10,2]=>0
[10,1,1]=>0
[9,3]=>0
[9,2,1]=>0
[9,1,1,1]=>0
[8,4]=>0
[8,3,1]=>0
[8,2,2]=>0
[8,2,1,1]=>0
[8,1,1,1,1]=>0
[7,5]=>0
[7,4,1]=>0
[7,3,2]=>0
[7,3,1,1]=>0
[7,2,2,1]=>0
[7,2,1,1,1]=>0
[7,1,1,1,1,1]=>0
[6,6]=>2
[6,5,1]=>22
[6,4,2]=>70
[6,4,1,1]=>150
[6,3,3]=>100
[6,3,2,1]=>340
[6,3,1,1,1]=>720
[6,2,2,2]=>540
[6,2,2,1,1]=>1140
[6,2,1,1,1,1]=>2400
[6,1,1,1,1,1,1]=>5040
[5,5,2]=>222
[5,5,1,1]=>504
[5,4,3]=>588
[5,4,2,1]=>2322
[5,4,1,1,1]=>5256
[5,3,3,1]=>3712
[5,3,2,2]=>6432
[5,3,2,1,1]=>14496
[5,3,1,1,1,1]=>32640
[5,2,2,2,1]=>24996
[5,2,2,1,1,1]=>56160
[5,2,1,1,1,1,1]=>126000
[5,1,1,1,1,1,1,1]=>282240
[4,4,4]=>1092
[4,4,3,1]=>7524
[4,4,2,2]=>13464
[4,4,2,1,1]=>30960
[4,4,1,1,1,1]=>71280
[4,3,3,2]=>23254
[4,3,3,1,1]=>53640
[4,3,2,2,1]=>96900
[4,3,2,1,1,1]=>224160
[4,3,1,1,1,1,1]=>519120
[4,2,2,2,2]=>175440
[4,2,2,2,1,1]=>406440
[4,2,2,1,1,1,1]=>942480
[4,2,1,1,1,1,1,1]=>2187360
[4,1,1,1,1,1,1,1,1]=>5080320
[3,3,3,3]=>41304
[3,3,3,2,1]=>175440
[3,3,3,1,1,1]=>408960
[3,3,2,2,2]=>322860
[3,3,2,2,1,1]=>755040
[3,3,2,1,1,1,1]=>1769040
[3,3,1,1,1,1,1,1]=>4152960
[3,2,2,2,2,1]=>1403520
[3,2,2,2,1,1,1]=>3301200
[3,2,2,1,1,1,1,1]=>7781760
[3,2,1,1,1,1,1,1,1]=>18385920
[3,1,1,1,1,1,1,1,1,1]=>43545600
[2,2,2,2,2,2]=>2631600
[2,2,2,2,2,1,1]=>6219360
[2,2,2,2,1,1,1,1]=>14736960
[2,2,2,1,1,1,1,1,1]=>35017920
[2,2,1,1,1,1,1,1,1,1]=>83462400
[2,1,1,1,1,1,1,1,1,1,1]=>199584000
[1,1,1,1,1,1,1,1,1,1,1,1]=>479001600
                    

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$F_{2} = 1 + q^{2}$

$F_{3} = 1 + q + q^{6}$

$F_{4} = 2 + q^{2} + q^{6} + q^{24}$

$F_{5} = 2 + q + q^{2} + q^{12} + q^{36} + q^{120}$

$F_{6} = 4 + q^{2} + q^{10} + q^{24} + q^{30} + q^{84} + q^{240} + q^{720}$

$F_{7} = 4 + q + q^{3} + q^{6} + q^{18} + q^{38} + q^{96} + q^{240} + q^{246} + q^{660} + q^{1800} + q^{5040}$

$F_{8} = 7 + q^{2} + q^{14} + q^{24} + q^{54} + q^{74} + q^{120} + q^{184} + q^{384} + q^{864} + q^{960} + q^{2220} + q^{2400} + q^{5760} + q^{15120} + q^{40320}$

$F_{9} = 7 + q + q^{4} + q^{6} + q^{12} + 2\ q^{24} + q^{79} + q^{174} + q^{186} + q^{336} + q^{780} + q^{836} + q^{1686} + q^{1800} + q^{2040} + q^{4140} + q^{8760} + q^{10200} + q^{21960} + q^{25200} + q^{55440} + q^{141120} + q^{362880}$

$F_{10} = 12 + q^{2} + q^{18} + q^{44} + q^{96} + q^{138} + q^{156} + q^{248} + q^{324} + q^{336} + q^{720} + q^{1074} + q^{1974} + q^{2184} + q^{2520} + q^{4204} + q^{4620} + q^{10080} + q^{10800} + q^{19740} + q^{24240} + q^{25200} + q^{39480} + q^{47760} + q^{96480} + q^{115920} + q^{236880} + q^{282240} + q^{584640} + q^{1451520} + q^{3628800}$

$F_{11} = 12 + q + q^{5} + q^{10} + q^{20} + 2\ q^{30} + q^{60} + q^{120} + q^{135} + q^{212} + q^{306} + q^{480} + q^{816} + q^{1386} + q^{1824} + q^{2016} + q^{3084} + q^{3566} + q^{4680} + q^{6516} + q^{6840} + q^{12336} + q^{15120} + q^{15180} + q^{27780} + q^{29040} + q^{35400} + q^{54660} + q^{64800} + q^{103800} + q^{129480} + q^{151200} + q^{247080} + q^{307440} + q^{352800} + q^{478080} + q^{589680} + q^{1149120} + q^{1411200} + q^{2772000} + q^{3386880} + q^{6713280} + q^{16329600} + q^{39916800}$

$F_{12} = 19 + q^{2} + q^{22} + q^{70} + q^{100} + q^{150} + q^{222} + q^{340} + q^{504} + q^{540} + q^{588} + q^{720} + q^{1092} + q^{1140} + q^{2322} + q^{2400} + q^{3712} + q^{5040} + q^{5256} + q^{6432} + q^{7524} + q^{13464} + q^{14496} + q^{23254} + q^{24996} + q^{30960} + q^{32640} + q^{41304} + q^{53640} + q^{56160} + q^{71280} + q^{96900} + q^{126000} + 2\ q^{175440} + q^{224160} + q^{282240} + q^{322860} + q^{406440} + q^{408960} + q^{519120} + q^{755040} + q^{942480} + q^{1403520} + q^{1769040} + q^{2187360} + q^{2631600} + q^{3301200} + q^{4152960} + q^{5080320} + q^{6219360} + q^{7781760} + q^{14736960} + q^{18385920} + q^{35017920} + q^{43545600} + q^{83462400} + q^{199584000} + q^{479001600}$

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