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Identifier

St001601: Integer partitions ⟶ ℤ

Values

[1] => 1

[2] => 1

[1,1] => 0

[3] => 1

[2,1] => 1

[1,1,1] => 0

[4] => 2

[3,1] => 2

[2,2] => 2

[2,1,1] => 1

[1,1,1,1] => 1

[5] => 3

[4,1] => 6

[3,2] => 6

[3,1,1] => 5

[2,2,1] => 5

[2,1,1,1] => 3

[1,1,1,1,1] => 1

[6] => 6

[5,1] => 14

[4,2] => 23

[4,1,1] => 18

[3,3] => 8

[3,2,1] => 28

[3,1,1,1] => 16

[2,2,2] => 10

[2,2,1,1] => 11

[2,1,1,1,1] => 7

[1,1,1,1,1,1] => 1

[7] => 11

[6,1] => 37

[5,2] => 68

[5,1,1] => 61

[4,3] => 59

[4,2,1] => 128

[4,1,1,1] => 64

[3,3,1] => 68

[3,2,2] => 69

[3,2,1,1] => 101

[3,1,1,1,1] => 41

[2,2,2,1] => 39

[2,2,1,1,1] => 33

[2,1,1,1,1,1] => 14

[1,1,1,1,1,1,1] => 2

[8] => 23

[7,1] => 92

[6,2] => 214

[6,1,1] => 191

[5,3] => 244

[5,2,1] => 507

[5,1,1,1] => 251

[4,4] => 125

[4,3,1] => 499

[4,2,2] => 393

[4,2,1,1] => 562

[4,1,1,1,1] => 204

[3,3,2] => 260

[3,3,1,1] => 342

[3,2,2,1] => 400

[3,2,1,1,1] => 337

[3,1,1,1,1,1] => 99

[2,2,2,2] => 81

[2,2,2,1,1] => 134

[2,2,1,1,1,1] => 95

[2,1,1,1,1,1,1] => 29

[1,1,1,1,1,1,1,1] => 5

[9] => 47

[8,1] => 239

[7,2] => 632

[7,1,1] => 590

[6,3] => 948

[6,2,1] => 1877

[6,1,1,1] => 906

[5,4] => 760

[5,3,1] => 2568

[5,2,2] => 1831

[5,2,1,1] => 2689

[5,1,1,1,1] => 913

[4,4,1] => 1273

[4,3,2] => 2346

[4,3,1,1] => 2848

[4,2,2,1] => 2738

[4,2,1,1,1] => 2220

[4,1,1,1,1,1] => 602

[3,3,3] => 533

[3,3,2,1] => 2022

[3,3,1,1,1] => 1358

[3,2,2,2] => 962

[3,2,2,1,1] => 1746

[3,2,1,1,1,1] => 1055

[3,1,1,1,1,1,1] => 254

[2,2,2,2,1] => 427

[2,2,2,1,1,1] => 454

[2,2,1,1,1,1,1] => 242

[2,1,1,1,1,1,1,1] => 66

[1,1,1,1,1,1,1,1,1] => 9

[10] => 106

[9,1] => 613

[8,2] => 1890

[8,1,1] => 1773

[7,3] => 3369

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Description

The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees.

Code

def statistic(mu):
    s = SymmetricFunctions(QQ).s()
    A = CombinatorialSpecies()
    X = species.SingletonSpecies()
    E = species.SetSpecies()
    A.define(X*E(A))
    V = (X + species.CharacteristicSpecies(2)).cycle_index_series() - (X^2).cycle_index_series()
    F = V(A.cycle_index_series())
    return F.coefficient(mu.size()).scalar(s(mu))

Created

Sep 27, 2020 at 00:18 by Martin Rubey

Updated

Sep 27, 2020 at 00:18 by Martin Rubey