- FindStat

Identifier

Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤ (values match St000159The number of distinct parts of the integer partition., St000318The number of addable cells of the Ferrers diagram of an integer partition.)

Values

[1,1] => [1,1] => 0

[2] => [2] => 0

[1,1,1] => [1,1,1] => 0

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

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

[3] => [3] => 0

[1,1,1,1] => [1,1,1,1] => 0

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

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

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

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

[2,2] => [2,2] => 0

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

[4] => [4] => 0

[1,1,1,1,1] => [1,1,1,1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[5] => [5] => 0

[1,1,1,1,1,1] => [1,1,1,1,1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[2,2,2] => [2,2,2] => 0

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

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

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

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

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

[3,3] => [3,3] => 0

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

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

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

[6] => [6] => 0

[1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1191 entries. <<<

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$F_{2} = 2$

$F_{3} = 2 + 2\ q$

$F_{4} = 3 + 5\ q$

$F_{5} = 2 + 14\ q$

$F_{6} = 4 + 22\ q + 6\ q^{2}$

$F_{7} = 2 + 44\ q + 18\ q^{2}$

$F_{8} = 4 + 68\ q + 56\ q^{2}$

$F_{9} = 3 + 107\ q + 146\ q^{2}$

$F_{10} = 4 + 172\ q + 312\ q^{2} + 24\ q^{3}$

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$
This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than St000159The number of distinct parts of the integer partition., the number of distinct parts of the partition.

Map

to partition

Description

Sends a composition to the partition obtained by sorting the entries.