- FindStat

Identifier

Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,1] => [6] => [6] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 3520 entries. <<<

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

touch composition

Description

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

Map

delta morphism

Description

Apply the delta morphism to an integer composition.
The delta morphism of a composition $C$ is the compositions composed of the lengths of consecutive runs of the same integer in $C$.

Map

to partition

Description

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