- FindStat

Identifier

St001099: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [2]=>0
[1,1]=>1
[3]=>0
[2,1]=>2
[1,1,1]=>3
[4]=>0
[3,1]=>6
[2,2]=>10
[2,1,1]=>12
[1,1,1,1]=>15
[5]=>0
[4,1]=>24
[3,2]=>54
[3,1,1]=>60
[2,2,1]=>78
[2,1,1,1]=>90
[1,1,1,1,1]=>105
[6]=>0
[5,1]=>120
[4,2]=>336
[4,1,1]=>360
[3,3]=>450
[3,2,1]=>570
[3,1,1,1]=>630
[2,2,2]=>672
[2,2,1,1]=>750
[2,1,1,1,1]=>840
[1,1,1,1,1,1]=>945
[7]=>0
[6,1]=>720
[5,2]=>2400
[5,1,1]=>2520
[4,3]=>3960
[4,2,1]=>4680
[4,1,1,1]=>5040
[3,3,1]=>5670
[3,2,2]=>6360
[3,2,1,1]=>6930
[3,1,1,1,1]=>7560
[2,2,2,1]=>7860
[2,2,1,1,1]=>8610
[2,1,1,1,1,1]=>9450
[1,1,1,1,1,1,1]=>10395
[8]=>0
[7,1]=>5040
[6,2]=>19440
[6,1,1]=>20160
[5,3]=>37800
[5,2,1]=>42840
[5,1,1,1]=>45360
[4,4]=>46440
[4,3,1]=>60480
[4,2,2]=>65880
[4,2,1,1]=>70560
[4,1,1,1,1]=>75600
[3,3,2]=>75600
[3,3,1,1]=>81270
[3,2,2,1]=>89460
[3,2,1,1,1]=>96390
[3,1,1,1,1,1]=>103950
[2,2,2,2]=>98820
[2,2,2,1,1]=>106680
[2,2,1,1,1,1]=>115290
[2,1,1,1,1,1,1]=>124740
[1,1,1,1,1,1,1,1]=>135135
[9]=>0
[8,1]=>40320
[7,2]=>176400
[7,1,1]=>181440
[6,3]=>393120
[6,2,1]=>433440
[6,1,1,1]=>453600
[5,4]=>567000
[5,3,1]=>695520
[5,2,2]=>743400
[5,2,1,1]=>786240
[5,1,1,1,1]=>831600
[4,4,1]=>808920
[4,3,2]=>960120
[4,3,1,1]=>1020600
[4,2,2,1]=>1101240
[4,2,1,1,1]=>1171800
[4,1,1,1,1,1]=>1247400
[3,3,3]=>1065960
[3,3,2,1]=>1228500
[3,3,1,1,1]=>1309770
[3,2,2,2]=>1331820
[3,2,2,1,1]=>1421280
[3,2,1,1,1,1]=>1517670
[3,1,1,1,1,1,1]=>1621620
[2,2,2,2,1]=>1545180
[2,2,2,1,1,1]=>1651860
[2,2,1,1,1,1,1]=>1767150
[2,1,1,1,1,1,1,1]=>1891890
[1,1,1,1,1,1,1,1,1]=>2027025
[10]=>0
[9,1]=>362880
[8,2]=>1774080
[8,1,1]=>1814400
[7,3]=>4445280
[7,2,1]=>4808160
[7,1,1,1]=>4989600
[6,4]=>7318080
[6,3,1]=>8618400
[6,2,2]=>9092160
[6,2,1,1]=>9525600
[6,1,1,1,1]=>9979200
[5,5]=>8580600
[5,4,1]=>11362680
[5,3,2]=>13025880
[5,3,1,1]=>13721400
[5,2,2,1]=>14598360
[5,2,1,1,1]=>15384600
[5,1,1,1,1,1]=>16216200
[4,4,2]=>14636160
[4,4,1,1]=>15445080
[4,3,3]=>15898680
[4,3,2,1]=>17939880
[4,3,1,1,1]=>18960480
[4,2,2,2]=>19182240
[4,2,2,1,1]=>20283480
[4,2,1,1,1,1]=>21455280
[4,1,1,1,1,1,1]=>22702680
[3,3,3,1]=>19584180
[3,3,2,2]=>20975220
[3,3,2,1,1]=>22203720
[3,3,1,1,1,1]=>23513490
[3,2,2,2,1]=>23817780
[3,2,2,1,1,1]=>25239060
[3,2,1,1,1,1,1]=>26756730
[3,1,1,1,1,1,1,1]=>28378350
[2,2,2,2,2]=>25576320
[2,2,2,2,1,1]=>27121500
[2,2,2,1,1,1,1]=>28773360
[2,2,1,1,1,1,1,1]=>30540510
[2,1,1,1,1,1,1,1,1]=>32432400
[1,1,1,1,1,1,1,1,1,1]=>34459425
[11]=>0
[10,1]=>3628800
[9,2]=>19595520
[9,1,1]=>19958400
[8,3]=>54432000
[8,2,1]=>58060800
[8,1,1,1]=>59875200
[7,4]=>100336320
[7,3,1]=>114760800
[7,2,2]=>119931840
[7,2,1,1]=>124740000
[7,1,1,1,1]=>129729600
[6,5]=>134038800
[6,4,1]=>168512400
[6,3,2]=>188470800
[6,3,1,1]=>197089200
[6,2,2,1]=>207522000
[6,2,1,1,1]=>217047600
[6,1,1,1,1,1]=>227026800
[5,5,1]=>190852200
[5,4,2]=>234375120
[5,4,1,1]=>245737800
[5,3,3]=>250727400
[5,3,2,1]=>278170200
[5,3,1,1,1]=>291891600
[5,2,2,2]=>294341040
[5,2,2,1,1]=>308939400
[5,2,1,1,1,1]=>324324000
[5,1,1,1,1,1,1]=>340540200
[4,4,3]=>275244480
[4,4,2,1]=>306134640
[4,4,1,1,1]=>321579720
[4,3,3,1]=>329064120
[4,3,2,2]=>349045200
[4,3,2,1,1]=>366985080
[4,3,1,1,1,1]=>385945560
[4,2,2,2,1]=>389612160
[4,2,2,1,1,1]=>409895640
[4,2,1,1,1,1,1]=>431350920
[4,1,1,1,1,1,1,1]=>454053600
[3,3,3,2]=>376091100
[3,3,3,1,1]=>395675280
[3,3,2,2,1]=>420498540
[3,3,2,1,1,1]=>442702260
[3,3,1,1,1,1,1]=>466215750
[3,2,2,2,2]=>447158880
[3,2,2,2,1,1]=>470976660
[3,2,2,1,1,1,1]=>496215720
[3,2,1,1,1,1,1,1]=>522972450
[3,1,1,1,1,1,1,1,1]=>551350800
[2,2,2,2,2,1]=>501401880
[2,2,2,2,1,1,1]=>528523380
[2,2,2,1,1,1,1,1]=>557296740
[2,2,1,1,1,1,1,1,1]=>587837250
[2,1,1,1,1,1,1,1,1,1]=>620269650
[1,1,1,1,1,1,1,1,1,1,1]=>654729075
[12]=>0
[11,1]=>39916800
[10,2]=>235872000
[10,1,1]=>239500800
[9,3]=>718502400
[9,2,1]=>758419200
[9,1,1,1]=>778377600
[8,4]=>1462406400
[8,3,1]=>1636588800
[8,2,2]=>1698278400
[8,2,1,1]=>1756339200
[8,1,1,1,1]=>1816214400
                    

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$F_{2} = 1 + q$

$F_{3} = 1 + q^{2} + q^{3}$

$F_{4} = 1 + q^{6} + q^{10} + q^{12} + q^{15}$

$F_{5} = 1 + q^{24} + q^{54} + q^{60} + q^{78} + q^{90} + q^{105}$

$F_{6} = 1 + q^{120} + q^{336} + q^{360} + q^{450} + q^{570} + q^{630} + q^{672} + q^{750} + q^{840} + q^{945}$

$F_{7} = 1 + q^{720} + q^{2400} + q^{2520} + q^{3960} + q^{4680} + q^{5040} + q^{5670} + q^{6360} + q^{6930} + q^{7560} + q^{7860} + q^{8610} + q^{9450} + q^{10395}$

$F_{8} = 1 + q^{5040} + q^{19440} + q^{20160} + q^{37800} + q^{42840} + q^{45360} + q^{46440} + q^{60480} + q^{65880} + q^{70560} + 2\ q^{75600} + q^{81270} + q^{89460} + q^{96390} + q^{98820} + q^{103950} + q^{106680} + q^{115290} + q^{124740} + q^{135135}$

$F_{9} = 1 + q^{40320} + q^{176400} + q^{181440} + q^{393120} + q^{433440} + q^{453600} + q^{567000} + q^{695520} + q^{743400} + q^{786240} + q^{808920} + q^{831600} + q^{960120} + q^{1020600} + q^{1065960} + q^{1101240} + q^{1171800} + q^{1228500} + q^{1247400} + q^{1309770} + q^{1331820} + q^{1421280} + q^{1517670} + q^{1545180} + q^{1621620} + q^{1651860} + q^{1767150} + q^{1891890} + q^{2027025}$

$F_{10} = 1 + q^{362880} + q^{1774080} + q^{1814400} + q^{4445280} + q^{4808160} + q^{4989600} + q^{7318080} + q^{8580600} + q^{8618400} + q^{9092160} + q^{9525600} + q^{9979200} + q^{11362680} + q^{13025880} + q^{13721400} + q^{14598360} + q^{14636160} + q^{15384600} + q^{15445080} + q^{15898680} + q^{16216200} + q^{17939880} + q^{18960480} + q^{19182240} + q^{19584180} + q^{20283480} + q^{20975220} + q^{21455280} + q^{22203720} + q^{22702680} + q^{23513490} + q^{23817780} + q^{25239060} + q^{25576320} + q^{26756730} + q^{27121500} + q^{28378350} + q^{28773360} + q^{30540510} + q^{32432400} + q^{34459425}$

$F_{11} = 1 + q^{3628800} + q^{19595520} + q^{19958400} + q^{54432000} + q^{58060800} + q^{59875200} + q^{100336320} + q^{114760800} + q^{119931840} + q^{124740000} + q^{129729600} + q^{134038800} + q^{168512400} + q^{188470800} + q^{190852200} + q^{197089200} + q^{207522000} + q^{217047600} + q^{227026800} + q^{234375120} + q^{245737800} + q^{250727400} + q^{275244480} + q^{278170200} + q^{291891600} + q^{294341040} + q^{306134640} + q^{308939400} + q^{321579720} + q^{324324000} + q^{329064120} + q^{340540200} + q^{349045200} + q^{366985080} + q^{376091100} + q^{385945560} + q^{389612160} + q^{395675280} + q^{409895640} + q^{420498540} + q^{431350920} + q^{442702260} + q^{447158880} + q^{454053600} + q^{466215750} + q^{470976660} + q^{496215720} + q^{501401880} + q^{522972450} + q^{528523380} + q^{551350800} + q^{557296740} + q^{587837250} + q^{620269650} + q^{654729075}$

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