- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ 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

[1,1,0,0,1,0,1,0] => [2,1,1] => [[2,2,2],[1,1]] => [1,1] => 0

[1,1,0,1,0,0,1,0] => [3,1] => [[3,3],[2]] => [2] => 0

[1,1,1,0,0,0,1,0] => [3,1] => [[3,3],[2]] => [2] => 0

[1,0,1,1,0,0,1,0,1,0] => [1,2,1,1] => [[2,2,2,1],[1,1]] => [1,1] => 0

[1,0,1,1,0,1,0,0,1,0] => [1,3,1] => [[3,3,1],[2]] => [2] => 0

[1,0,1,1,1,0,0,0,1,0] => [1,3,1] => [[3,3,1],[2]] => [2] => 0

[1,1,0,0,1,0,1,0,1,0] => [2,1,1,1] => [[2,2,2,2],[1,1,1]] => [1,1,1] => 0

[1,1,0,0,1,0,1,1,0,0] => [2,1,2] => [[3,2,2],[1,1]] => [1,1] => 0

[1,1,0,0,1,1,0,0,1,0] => [2,2,1] => [[3,3,2],[2,1]] => [2,1] => 1

[1,1,0,1,0,0,1,0,1,0] => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 0

[1,1,0,1,0,0,1,1,0,0] => [3,2] => [[4,3],[2]] => [2] => 0

[1,1,0,1,0,1,0,0,1,0] => [4,1] => [[4,4],[3]] => [3] => 0

[1,1,0,1,1,0,0,0,1,0] => [4,1] => [[4,4],[3]] => [3] => 0

[1,1,1,0,0,0,1,0,1,0] => [3,1,1] => [[3,3,3],[2,2]] => [2,2] => 0

[1,1,1,0,0,0,1,1,0,0] => [3,2] => [[4,3],[2]] => [2] => 0

[1,1,1,0,0,1,0,0,1,0] => [4,1] => [[4,4],[3]] => [3] => 0

[1,1,1,0,1,0,0,0,1,0] => [4,1] => [[4,4],[3]] => [3] => 0

[1,1,1,1,0,0,0,0,1,0] => [4,1] => [[4,4],[3]] => [3] => 0

[1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]] => [1,1] => 0

[1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 0

[1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,3,1] => [[3,3,1,1],[2]] => [2] => 0

[1,0,1,1,0,0,1,0,1,0,1,0] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]] => [1,1,1] => 0

[1,0,1,1,0,0,1,0,1,1,0,0] => [1,2,1,2] => [[3,2,2,1],[1,1]] => [1,1] => 0

[1,0,1,1,0,0,1,1,0,0,1,0] => [1,2,2,1] => [[3,3,2,1],[2,1]] => [2,1] => 1

[1,0,1,1,0,1,0,0,1,0,1,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 0

[1,0,1,1,0,1,0,0,1,1,0,0] => [1,3,2] => [[4,3,1],[2]] => [2] => 0

[1,0,1,1,0,1,0,1,0,0,1,0] => [1,4,1] => [[4,4,1],[3]] => [3] => 0

[1,0,1,1,0,1,1,0,0,0,1,0] => [1,4,1] => [[4,4,1],[3]] => [3] => 0

[1,0,1,1,1,0,0,0,1,0,1,0] => [1,3,1,1] => [[3,3,3,1],[2,2]] => [2,2] => 0

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,3,2] => [[4,3,1],[2]] => [2] => 0

[1,0,1,1,1,0,0,1,0,0,1,0] => [1,4,1] => [[4,4,1],[3]] => [3] => 0

[1,0,1,1,1,0,1,0,0,0,1,0] => [1,4,1] => [[4,4,1],[3]] => [3] => 0

[1,0,1,1,1,1,0,0,0,0,1,0] => [1,4,1] => [[4,4,1],[3]] => [3] => 0

[1,1,0,0,1,0,1,0,1,0,1,0] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]] => [1,1,1,1] => 0

[1,1,0,0,1,0,1,0,1,1,0,0] => [2,1,1,2] => [[3,2,2,2],[1,1,1]] => [1,1,1] => 0

[1,1,0,0,1,0,1,1,0,0,1,0] => [2,1,2,1] => [[3,3,2,2],[2,1,1]] => [2,1,1] => 1

[1,1,0,0,1,0,1,1,0,1,0,0] => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 0

[1,1,0,0,1,0,1,1,1,0,0,0] => [2,1,3] => [[4,2,2],[1,1]] => [1,1] => 0

[1,1,0,0,1,1,0,0,1,0,1,0] => [2,2,1,1] => [[3,3,3,2],[2,2,1]] => [2,2,1] => 1

[1,1,0,0,1,1,0,0,1,1,0,0] => [2,2,2] => [[4,3,2],[2,1]] => [2,1] => 1

[1,1,0,0,1,1,0,1,0,0,1,0] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 1

[1,1,0,0,1,1,1,0,0,0,1,0] => [2,3,1] => [[4,4,2],[3,1]] => [3,1] => 1

[1,1,0,1,0,0,1,0,1,0,1,0] => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 0

[1,1,0,1,0,0,1,0,1,1,0,0] => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 0

[1,1,0,1,0,0,1,1,0,0,1,0] => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 1

[1,1,0,1,0,0,1,1,0,1,0,0] => [3,3] => [[5,3],[2]] => [2] => 0

[1,1,0,1,0,0,1,1,1,0,0,0] => [3,3] => [[5,3],[2]] => [2] => 0

[1,1,0,1,0,1,0,0,1,0,1,0] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0

[1,1,0,1,0,1,0,0,1,1,0,0] => [4,2] => [[5,4],[3]] => [3] => 0

[1,1,0,1,0,1,0,1,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,0,1,0,1,1,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,0,1,1,0,0,0,1,0,1,0] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0

[1,1,0,1,1,0,0,0,1,1,0,0] => [4,2] => [[5,4],[3]] => [3] => 0

[1,1,0,1,1,0,0,1,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,0,1,1,0,1,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,0,1,1,1,0,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,0,0,0,1,0,1,0,1,0] => [3,1,1,1] => [[3,3,3,3],[2,2,2]] => [2,2,2] => 0

[1,1,1,0,0,0,1,0,1,1,0,0] => [3,1,2] => [[4,3,3],[2,2]] => [2,2] => 0

[1,1,1,0,0,0,1,1,0,0,1,0] => [3,2,1] => [[4,4,3],[3,2]] => [3,2] => 1

[1,1,1,0,0,0,1,1,0,1,0,0] => [3,3] => [[5,3],[2]] => [2] => 0

[1,1,1,0,0,0,1,1,1,0,0,0] => [3,3] => [[5,3],[2]] => [2] => 0

[1,1,1,0,0,1,0,0,1,0,1,0] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0

[1,1,1,0,0,1,0,0,1,1,0,0] => [4,2] => [[5,4],[3]] => [3] => 0

[1,1,1,0,0,1,0,1,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,0,0,1,1,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,0,1,0,0,0,1,0,1,0] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0

[1,1,1,0,1,0,0,0,1,1,0,0] => [4,2] => [[5,4],[3]] => [3] => 0

[1,1,1,0,1,0,0,1,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,0,1,0,1,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,0,1,1,0,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,1,0,0,0,0,1,0,1,0] => [4,1,1] => [[4,4,4],[3,3]] => [3,3] => 0

[1,1,1,1,0,0,0,0,1,1,0,0] => [4,2] => [[5,4],[3]] => [3] => 0

[1,1,1,1,0,0,0,1,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,1,0,0,1,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,1,0,1,0,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,1,1,1,1,0,0,0,0,0,1,0] => [5,1] => [[5,5],[4]] => [4] => 0

[1,0,1,0,1,0,1,1,0,0,1,0,1,0] => [1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]] => [1,1] => 0

[1,0,1,0,1,0,1,1,0,1,0,0,1,0] => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 0

[1,0,1,0,1,0,1,1,1,0,0,0,1,0] => [1,1,1,3,1] => [[3,3,1,1,1],[2]] => [2] => 0

[1,0,1,0,1,1,0,0,1,0,1,0,1,0] => [1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]] => [1,1,1] => 0

[1,0,1,0,1,1,0,0,1,0,1,1,0,0] => [1,1,2,1,2] => [[3,2,2,1,1],[1,1]] => [1,1] => 0

[1,0,1,0,1,1,0,0,1,1,0,0,1,0] => [1,1,2,2,1] => [[3,3,2,1,1],[2,1]] => [2,1] => 1

[1,0,1,0,1,1,0,1,0,0,1,0,1,0] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 0

[1,0,1,0,1,1,0,1,0,0,1,1,0,0] => [1,1,3,2] => [[4,3,1,1],[2]] => [2] => 0

[1,0,1,0,1,1,0,1,0,1,0,0,1,0] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0

[1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0

[1,0,1,0,1,1,1,0,0,0,1,0,1,0] => [1,1,3,1,1] => [[3,3,3,1,1],[2,2]] => [2,2] => 0

[1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,1,3,2] => [[4,3,1,1],[2]] => [2] => 0

[1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0

[1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0

[1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,1,4,1] => [[4,4,1,1],[3]] => [3] => 0

[1,0,1,1,0,0,1,0,1,0,1,0,1,0] => [1,2,1,1,1,1] => [[2,2,2,2,2,1],[1,1,1,1]] => [1,1,1,1] => 0

[1,0,1,1,0,0,1,0,1,0,1,1,0,0] => [1,2,1,1,2] => [[3,2,2,2,1],[1,1,1]] => [1,1,1] => 0

[1,0,1,1,0,0,1,0,1,1,0,0,1,0] => [1,2,1,2,1] => [[3,3,2,2,1],[2,1,1]] => [2,1,1] => 1

[1,0,1,1,0,0,1,0,1,1,0,1,0,0] => [1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 0

[1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,2,1,3] => [[4,2,2,1],[1,1]] => [1,1] => 0

[1,0,1,1,0,0,1,1,0,0,1,0,1,0] => [1,2,2,1,1] => [[3,3,3,2,1],[2,2,1]] => [2,2,1] => 1

[1,0,1,1,0,0,1,1,0,0,1,1,0,0] => [1,2,2,2] => [[4,3,2,1],[2,1]] => [2,1] => 1

[1,0,1,1,0,0,1,1,0,1,0,0,1,0] => [1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 1

[1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,2,3,1] => [[4,4,2,1],[3,1]] => [3,1] => 1

[1,0,1,1,0,1,0,0,1,0,1,0,1,0] => [1,3,1,1,1] => [[3,3,3,3,1],[2,2,2]] => [2,2,2] => 0

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search for individual values

searching the database for the individual values of this statistic

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 ribbon

Description

The ribbon shape corresponding to an integer composition.
For an integer composition

$(a_1, \dots, a_n)$ , this is the ribbon shape whose

$i$ th row from the bottom has

$a_i$ cells.

Map

touch composition

Description

Sends a Dyck path to its touch composition given by the composition of lengths of its touch points.

Map

inner shape

Description

The inner shape of a skew partition.