- FindStat

Identifier

St001101: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

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

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{2} = 1 + q$

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

$F_{4} = 1 + q + q^{3} + q^{4} + q^{6}$

$F_{5} = 1 + q + q^{7} + q^{8} + q^{14} + q^{18} + q^{24}$

$F_{6} = 1 + q + q^{15} + q^{16} + q^{31} + q^{46} + q^{54} + q^{64} + q^{78} + q^{96} + q^{120}$

$F_{7} = 1 + q + q^{31} + q^{32} + q^{115} + q^{146} + q^{162} + q^{230} + q^{284} + q^{330} + q^{384} + q^{426} + q^{504} + q^{600} + q^{720}$

$F_{8} = 1 + q + q^{63} + q^{64} + q^{391} + q^{454} + q^{486} + q^{675} + q^{1066} + q^{1228} + q^{1374} + q^{1536} + q^{1672} + q^{1902} + q^{2286} + q^{2616} + q^{2790} + q^{3000} + q^{3216} + q^{3720} + q^{4320} + q^{5040}$

$F_{9} = 1 + q + q^{127} + q^{128} + q^{1267} + q^{1394} + q^{1458} + q^{3451} + q^{4718} + q^{5204} + q^{5658} + q^{6144} + q^{6902} + q^{9488} + q^{10554} + q^{11828} + q^{12090} + q^{13464} + q^{15000} + q^{15402} + q^{17304} + q^{18018} + q^{20304} + q^{22920} + q^{24024} + q^{25920} + q^{27240} + q^{30960} + q^{35280} + q^{40320}$

$F_{10} = 1 + q + q^{255} + q^{256} + q^{3991} + q^{4246} + q^{4374} + q^{16275} + q^{20266} + q^{21724} + q^{23118} + q^{24576} + q^{25231} + q^{41506} + q^{52336} + q^{57054} + q^{63198} + q^{68856} + q^{69208} + q^{75000} + q^{76110} + q^{81460} + q^{101502} + q^{112056} + q^{114966} + q^{122190} + q^{127056} + q^{139494} + q^{140520} + q^{154896} + q^{155520} + q^{172200} + q^{177816} + q^{198120} + q^{205056} + q^{221040} + q^{229080} + q^{246960} + q^{256320} + q^{287280} + q^{322560} + q^{362880}$

$F_{11} = 1 + q + q^{511} + q^{512} + q^{12355} + q^{12866} + q^{13122} + q^{72955} + q^{85310} + q^{89684} + q^{93930} + q^{98304} + q^{164731} + q^{237686} + q^{282464} + q^{302730} + q^{327306} + q^{329462} + q^{350424} + q^{375000} + q^{484136} + q^{525642} + q^{547820} + q^{657210} + q^{677636} + q^{714264} + q^{726066} + q^{789264} + q^{822954} + q^{858120} + q^{899064} + q^{933120} + q^{951546} + q^{1063602} + q^{1165104} + q^{1244034} + q^{1277160} + q^{1305624} + q^{1366224} + q^{1432680} + q^{1538424} + q^{1573200} + q^{1693320} + q^{1728720} + q^{1736544} + q^{1865520} + q^{1914360} + q^{2112480} + q^{2170680} + q^{2333520} + q^{2399760} + q^{2580480} + q^{2656080} + q^{2943360} + q^{3265920} + q^{3628800}$

$F_{12} = 1 + q + q^{1023} + q^{1024} + q^{37831} + q^{38854} + q^{39366} + q^{316275} + q^{354106} + q^{367228} + q^{380094} + q^{393216} + q^{999391} + q^{1315666} + q^{1441923} + q^{1499152} + q^{1584462} + q^{1682766} + q^{1776696} + q^{1875000} + q^{2441314} + q^{3281608} + q^{3519294} + q^{3610372} + q^{4195566} + q^{4223704} + q^{4498296} + q^{4545990} + q^{4553166} + q^{4873296} + q^{5223720} + q^{5485756} + q^{5598720} + q^{6476644} + q^{6495534} + q^{7021176} + q^{7290942} + q^{8005206} + q^{8662416} + q^{8724078} + q^{9376680} + q^{9520536} + q^{9623142} + q^{10309800} + q^{10446096} + q^{10942230} + q^{11167920} + q^{11345160} + q^{11893776} + q^{12101040} + q^{12497958} + q^{13170936} + q^{14336040} + q^{14603616} + q^{15108216} + q^{15613200} + q^{15909240} + q^{16474440} + q^{16801536} + q^{17341920} + q^{18339960} + q^{18915120} + q^{20033280} + q^{20452440} + q^{20643840} + q^{21898800} + q^{22366800} + q^{22852200} + q^{24479280} + q^{25022880} + q^{26812800} + q^{27422640} + q^{29393280} + q^{30078720} + q^{33022080} + q^{36288000} + q^{39916800}$

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