Your data matches 182 different statistics following compositions of up to 3 maps.
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Mp00248: Permutations DEX compositionInteger compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => 1 = -1 + 2
[2,1] => [2] => 1 = -1 + 2
[1,2,3] => [3] => 1 = -1 + 2
[1,3,2] => [1,2] => 2 = 0 + 2
[2,1,3] => [3] => 1 = -1 + 2
[2,3,1] => [3] => 1 = -1 + 2
[3,1,2] => [3] => 1 = -1 + 2
[3,2,1] => [2,1] => 2 = 0 + 2
[1,2,3,4] => [4] => 1 = -1 + 2
[1,3,2,4] => [1,3] => 2 = 0 + 2
[1,3,4,2] => [1,3] => 2 = 0 + 2
[1,4,2,3] => [1,3] => 2 = 0 + 2
[1,4,3,2] => [1,2,1] => 2 = 0 + 2
[2,1,3,4] => [4] => 1 = -1 + 2
[2,3,1,4] => [4] => 1 = -1 + 2
[2,3,4,1] => [4] => 1 = -1 + 2
[2,4,1,3] => [4] => 1 = -1 + 2
[2,4,3,1] => [3,1] => 2 = 0 + 2
[3,1,2,4] => [4] => 1 = -1 + 2
[3,4,1,2] => [4] => 1 = -1 + 2
[3,4,2,1] => [3,1] => 2 = 0 + 2
[4,1,2,3] => [4] => 1 = -1 + 2
[4,1,3,2] => [3,1] => 2 = 0 + 2
[4,2,3,1] => [3,1] => 2 = 0 + 2
[4,3,1,2] => [1,3] => 2 = 0 + 2
[4,3,2,1] => [1,2,1] => 2 = 0 + 2
[1,2,3,4,5] => [5] => 1 = -1 + 2
[1,3,2,4,5] => [1,4] => 2 = 0 + 2
[1,3,4,2,5] => [1,4] => 2 = 0 + 2
[1,3,4,5,2] => [1,4] => 2 = 0 + 2
[1,3,5,2,4] => [1,4] => 2 = 0 + 2
[1,3,5,4,2] => [1,3,1] => 2 = 0 + 2
[1,4,2,3,5] => [1,4] => 2 = 0 + 2
[1,4,5,2,3] => [1,4] => 2 = 0 + 2
[1,4,5,3,2] => [1,3,1] => 2 = 0 + 2
[1,5,2,3,4] => [1,4] => 2 = 0 + 2
[1,5,2,4,3] => [1,3,1] => 2 = 0 + 2
[1,5,3,4,2] => [1,3,1] => 2 = 0 + 2
[1,5,4,2,3] => [1,1,3] => 2 = 0 + 2
[1,5,4,3,2] => [1,1,2,1] => 2 = 0 + 2
[2,1,3,4,5] => [5] => 1 = -1 + 2
[2,3,1,4,5] => [5] => 1 = -1 + 2
[2,3,4,1,5] => [5] => 1 = -1 + 2
[2,3,4,5,1] => [5] => 1 = -1 + 2
[2,3,5,1,4] => [5] => 1 = -1 + 2
[2,3,5,4,1] => [4,1] => 2 = 0 + 2
[2,4,1,3,5] => [5] => 1 = -1 + 2
[2,4,5,1,3] => [5] => 1 = -1 + 2
[2,4,5,3,1] => [4,1] => 2 = 0 + 2
[2,5,1,3,4] => [5] => 1 = -1 + 2
Description
The number of different parts of an integer composition.
Matching statistic: St000481
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000481: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 0 = -1 + 1
[2,1] => [2] => [2]
=> 0 = -1 + 1
[1,2,3] => [3] => [3]
=> 0 = -1 + 1
[1,3,2] => [1,2] => [2,1]
=> 1 = 0 + 1
[2,1,3] => [3] => [3]
=> 0 = -1 + 1
[2,3,1] => [3] => [3]
=> 0 = -1 + 1
[3,1,2] => [3] => [3]
=> 0 = -1 + 1
[3,2,1] => [2,1] => [2,1]
=> 1 = 0 + 1
[1,2,3,4] => [4] => [4]
=> 0 = -1 + 1
[1,3,2,4] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,3,4,2] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,4,2,3] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [4] => [4]
=> 0 = -1 + 1
[2,3,1,4] => [4] => [4]
=> 0 = -1 + 1
[2,3,4,1] => [4] => [4]
=> 0 = -1 + 1
[2,4,1,3] => [4] => [4]
=> 0 = -1 + 1
[2,4,3,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[3,1,2,4] => [4] => [4]
=> 0 = -1 + 1
[3,4,1,2] => [4] => [4]
=> 0 = -1 + 1
[3,4,2,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,1,2,3] => [4] => [4]
=> 0 = -1 + 1
[4,1,3,2] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,2,3,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,3,1,2] => [1,3] => [3,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => [5]
=> 0 = -1 + 1
[1,3,2,4,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,1,4,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,4,1,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,4,5,1] => [5] => [5]
=> 0 = -1 + 1
[2,3,5,1,4] => [5] => [5]
=> 0 = -1 + 1
[2,3,5,4,1] => [4,1] => [4,1]
=> 1 = 0 + 1
[2,4,1,3,5] => [5] => [5]
=> 0 = -1 + 1
[2,4,5,1,3] => [5] => [5]
=> 0 = -1 + 1
[2,4,5,3,1] => [4,1] => [4,1]
=> 1 = 0 + 1
[2,5,1,3,4] => [5] => [5]
=> 0 = -1 + 1
Description
The number of upper covers of a partition in dominance order.
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 0 = -1 + 1
[2,1] => [2] => [2]
=> 0 = -1 + 1
[1,2,3] => [3] => [3]
=> 0 = -1 + 1
[1,3,2] => [1,2] => [2,1]
=> 1 = 0 + 1
[2,1,3] => [3] => [3]
=> 0 = -1 + 1
[2,3,1] => [3] => [3]
=> 0 = -1 + 1
[3,1,2] => [3] => [3]
=> 0 = -1 + 1
[3,2,1] => [2,1] => [2,1]
=> 1 = 0 + 1
[1,2,3,4] => [4] => [4]
=> 0 = -1 + 1
[1,3,2,4] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,3,4,2] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,4,2,3] => [1,3] => [3,1]
=> 1 = 0 + 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 1 = 0 + 1
[2,1,3,4] => [4] => [4]
=> 0 = -1 + 1
[2,3,1,4] => [4] => [4]
=> 0 = -1 + 1
[2,3,4,1] => [4] => [4]
=> 0 = -1 + 1
[2,4,1,3] => [4] => [4]
=> 0 = -1 + 1
[2,4,3,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[3,1,2,4] => [4] => [4]
=> 0 = -1 + 1
[3,4,1,2] => [4] => [4]
=> 0 = -1 + 1
[3,4,2,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,1,2,3] => [4] => [4]
=> 0 = -1 + 1
[4,1,3,2] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,2,3,1] => [3,1] => [3,1]
=> 1 = 0 + 1
[4,3,1,2] => [1,3] => [3,1]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => [5]
=> 0 = -1 + 1
[1,3,2,4,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,4] => [4,1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,1,4,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,4,1,5] => [5] => [5]
=> 0 = -1 + 1
[2,3,4,5,1] => [5] => [5]
=> 0 = -1 + 1
[2,3,5,1,4] => [5] => [5]
=> 0 = -1 + 1
[2,3,5,4,1] => [4,1] => [4,1]
=> 1 = 0 + 1
[2,4,1,3,5] => [5] => [5]
=> 0 = -1 + 1
[2,4,5,1,3] => [5] => [5]
=> 0 = -1 + 1
[2,4,5,3,1] => [4,1] => [4,1]
=> 1 = 0 + 1
[2,5,1,3,4] => [5] => [5]
=> 0 = -1 + 1
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 [[St000159]], the number of distinct parts of the partition.
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 1 = -1 + 2
[2,1] => [2] => [2]
=> 1 = -1 + 2
[1,2,3] => [3] => [3]
=> 1 = -1 + 2
[1,3,2] => [1,2] => [2,1]
=> 2 = 0 + 2
[2,1,3] => [3] => [3]
=> 1 = -1 + 2
[2,3,1] => [3] => [3]
=> 1 = -1 + 2
[3,1,2] => [3] => [3]
=> 1 = -1 + 2
[3,2,1] => [2,1] => [2,1]
=> 2 = 0 + 2
[1,2,3,4] => [4] => [4]
=> 1 = -1 + 2
[1,3,2,4] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,3,4,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,2,3] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[2,1,3,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,1,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,4,1] => [4] => [4]
=> 1 = -1 + 2
[2,4,1,3] => [4] => [4]
=> 1 = -1 + 2
[2,4,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[3,1,2,4] => [4] => [4]
=> 1 = -1 + 2
[3,4,1,2] => [4] => [4]
=> 1 = -1 + 2
[3,4,2,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,1,2,3] => [4] => [4]
=> 1 = -1 + 2
[4,1,3,2] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,2,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,3,1,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[1,3,2,4,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,2,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 2 = 0 + 2
[2,1,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,1,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,1,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,5,1] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,1,4] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,4,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,4,1,3,5] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,1,3] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,3,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,5,1,3,4] => [5] => [5]
=> 1 = -1 + 2
Description
The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000783: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 1 = -1 + 2
[2,1] => [2] => [2]
=> 1 = -1 + 2
[1,2,3] => [3] => [3]
=> 1 = -1 + 2
[1,3,2] => [1,2] => [2,1]
=> 2 = 0 + 2
[2,1,3] => [3] => [3]
=> 1 = -1 + 2
[2,3,1] => [3] => [3]
=> 1 = -1 + 2
[3,1,2] => [3] => [3]
=> 1 = -1 + 2
[3,2,1] => [2,1] => [2,1]
=> 2 = 0 + 2
[1,2,3,4] => [4] => [4]
=> 1 = -1 + 2
[1,3,2,4] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,3,4,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,2,3] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[2,1,3,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,1,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,4,1] => [4] => [4]
=> 1 = -1 + 2
[2,4,1,3] => [4] => [4]
=> 1 = -1 + 2
[2,4,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[3,1,2,4] => [4] => [4]
=> 1 = -1 + 2
[3,4,1,2] => [4] => [4]
=> 1 = -1 + 2
[3,4,2,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,1,2,3] => [4] => [4]
=> 1 = -1 + 2
[4,1,3,2] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,2,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,3,1,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[1,3,2,4,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,2,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 2 = 0 + 2
[2,1,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,1,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,1,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,5,1] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,1,4] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,4,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,4,1,3,5] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,1,3] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,3,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,5,1,3,4] => [5] => [5]
=> 1 = -1 + 2
Description
The side length of the largest staircase partition fitting into a partition. For an integer partition $(\lambda_1\geq \lambda_2\geq\dots)$ this is the largest integer $k$ such that $\lambda_i > k-i$ for $i\in\{1,\dots,k\}$. In other words, this is the length of a longest (strict) north-east chain of cells in the Ferrers diagram of the partition, using the English convention. Equivalently, this is the maximal number of non-attacking rooks that can be placed on the Ferrers diagram. This is also the maximal number of occurrences of a colour in a proper colouring of a Ferrers diagram. A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1]. This statistic records the largest part occurring in any of these partitions.
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St001432: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 1 = -1 + 2
[2,1] => [2] => [2]
=> 1 = -1 + 2
[1,2,3] => [3] => [3]
=> 1 = -1 + 2
[1,3,2] => [1,2] => [2,1]
=> 2 = 0 + 2
[2,1,3] => [3] => [3]
=> 1 = -1 + 2
[2,3,1] => [3] => [3]
=> 1 = -1 + 2
[3,1,2] => [3] => [3]
=> 1 = -1 + 2
[3,2,1] => [2,1] => [2,1]
=> 2 = 0 + 2
[1,2,3,4] => [4] => [4]
=> 1 = -1 + 2
[1,3,2,4] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,3,4,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,2,3] => [1,3] => [3,1]
=> 2 = 0 + 2
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[2,1,3,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,1,4] => [4] => [4]
=> 1 = -1 + 2
[2,3,4,1] => [4] => [4]
=> 1 = -1 + 2
[2,4,1,3] => [4] => [4]
=> 1 = -1 + 2
[2,4,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[3,1,2,4] => [4] => [4]
=> 1 = -1 + 2
[3,4,1,2] => [4] => [4]
=> 1 = -1 + 2
[3,4,2,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,1,2,3] => [4] => [4]
=> 1 = -1 + 2
[4,1,3,2] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,2,3,1] => [3,1] => [3,1]
=> 2 = 0 + 2
[4,3,1,2] => [1,3] => [3,1]
=> 2 = 0 + 2
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 2 = 0 + 2
[1,2,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[1,3,2,4,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,2,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,4,5,2] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,2,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,4,2,3,5] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,2,3] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,2,3,4] => [1,4] => [4,1]
=> 2 = 0 + 2
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 2 = 0 + 2
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 2 = 0 + 2
[2,1,3,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,1,4,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,1,5] => [5] => [5]
=> 1 = -1 + 2
[2,3,4,5,1] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,1,4] => [5] => [5]
=> 1 = -1 + 2
[2,3,5,4,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,4,1,3,5] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,1,3] => [5] => [5]
=> 1 = -1 + 2
[2,4,5,3,1] => [4,1] => [4,1]
=> 2 = 0 + 2
[2,5,1,3,4] => [5] => [5]
=> 1 = -1 + 2
Description
The order dimension of the partition. Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> 2 = -1 + 3
[2,1] => [2] => [2]
=> 2 = -1 + 3
[1,2,3] => [3] => [3]
=> 2 = -1 + 3
[1,3,2] => [1,2] => [2,1]
=> 3 = 0 + 3
[2,1,3] => [3] => [3]
=> 2 = -1 + 3
[2,3,1] => [3] => [3]
=> 2 = -1 + 3
[3,1,2] => [3] => [3]
=> 2 = -1 + 3
[3,2,1] => [2,1] => [2,1]
=> 3 = 0 + 3
[1,2,3,4] => [4] => [4]
=> 2 = -1 + 3
[1,3,2,4] => [1,3] => [3,1]
=> 3 = 0 + 3
[1,3,4,2] => [1,3] => [3,1]
=> 3 = 0 + 3
[1,4,2,3] => [1,3] => [3,1]
=> 3 = 0 + 3
[1,4,3,2] => [1,2,1] => [2,1,1]
=> 3 = 0 + 3
[2,1,3,4] => [4] => [4]
=> 2 = -1 + 3
[2,3,1,4] => [4] => [4]
=> 2 = -1 + 3
[2,3,4,1] => [4] => [4]
=> 2 = -1 + 3
[2,4,1,3] => [4] => [4]
=> 2 = -1 + 3
[2,4,3,1] => [3,1] => [3,1]
=> 3 = 0 + 3
[3,1,2,4] => [4] => [4]
=> 2 = -1 + 3
[3,4,1,2] => [4] => [4]
=> 2 = -1 + 3
[3,4,2,1] => [3,1] => [3,1]
=> 3 = 0 + 3
[4,1,2,3] => [4] => [4]
=> 2 = -1 + 3
[4,1,3,2] => [3,1] => [3,1]
=> 3 = 0 + 3
[4,2,3,1] => [3,1] => [3,1]
=> 3 = 0 + 3
[4,3,1,2] => [1,3] => [3,1]
=> 3 = 0 + 3
[4,3,2,1] => [1,2,1] => [2,1,1]
=> 3 = 0 + 3
[1,2,3,4,5] => [5] => [5]
=> 2 = -1 + 3
[1,3,2,4,5] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,3,4,2,5] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,3,4,5,2] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,3,5,2,4] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> 3 = 0 + 3
[1,4,2,3,5] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,4,5,2,3] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> 3 = 0 + 3
[1,5,2,3,4] => [1,4] => [4,1]
=> 3 = 0 + 3
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> 3 = 0 + 3
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> 3 = 0 + 3
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> 3 = 0 + 3
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> 3 = 0 + 3
[2,1,3,4,5] => [5] => [5]
=> 2 = -1 + 3
[2,3,1,4,5] => [5] => [5]
=> 2 = -1 + 3
[2,3,4,1,5] => [5] => [5]
=> 2 = -1 + 3
[2,3,4,5,1] => [5] => [5]
=> 2 = -1 + 3
[2,3,5,1,4] => [5] => [5]
=> 2 = -1 + 3
[2,3,5,4,1] => [4,1] => [4,1]
=> 3 = 0 + 3
[2,4,1,3,5] => [5] => [5]
=> 2 = -1 + 3
[2,4,5,1,3] => [5] => [5]
=> 2 = -1 + 3
[2,4,5,3,1] => [4,1] => [4,1]
=> 3 = 0 + 3
[2,5,1,3,4] => [5] => [5]
=> 2 = -1 + 3
Description
The number of addable cells of the Ferrers diagram of an integer partition.
Mp00248: Permutations DEX compositionInteger compositions
Mp00041: Integer compositions conjugateInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000455: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1] => ([(0,1)],2)
=> -1
[2,1] => [2] => [1,1] => ([(0,1)],2)
=> -1
[1,2,3] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1
[1,3,2] => [1,2] => [1,2] => ([(1,2)],3)
=> 0
[2,1,3] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1
[2,3,1] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1
[3,1,2] => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> -1
[3,2,1] => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[1,2,3,4] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[1,3,2,4] => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,3,4,2] => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[1,4,3,2] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[2,1,3,4] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[2,3,1,4] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[2,3,4,1] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[2,4,1,3] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[2,4,3,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[3,4,1,2] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[3,4,2,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,1,2,3] => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> -1
[4,1,3,2] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,2,3,1] => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 0
[4,3,1,2] => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[4,3,2,1] => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[1,2,3,4,5] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[1,3,2,4,5] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,4,2,5] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,4,5,2] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,3,5,4,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,5,2,3] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,4,5,3,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,2,3,4] => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,2,4,3] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,3,4,2] => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[1,5,4,2,3] => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,5,4,3,2] => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[2,1,3,4,5] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,3,1,4,5] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,3,4,1,5] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,3,4,5,1] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,3,5,1,4] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,3,5,4,1] => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,4,5,1,3] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
[2,4,5,3,1] => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,5,1,3,4] => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> -1
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.
Matching statistic: St000024
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
Mp00230: Integer partitions parallelogram polyominoDyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> [1,0,1,0]
=> 0 = -1 + 1
[2,1] => [2] => [2]
=> [1,0,1,0]
=> 0 = -1 + 1
[1,2,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = -1 + 1
[1,3,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,3] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = -1 + 1
[3,1,2] => [3] => [3]
=> [1,0,1,0,1,0]
=> 0 = -1 + 1
[3,2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[1,3,2,4] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,2,3] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,1,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,4,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,4,1,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,4,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[3,1,2,4] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[3,4,1,2] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[3,4,2,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,1,2,3] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[4,1,3,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,2,3,1] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,1,2] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[1,3,2,4,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,3,4,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,1,4,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,4,1,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,4,5,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,5,1,4] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,3,5,4,1] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,4,1,3,5] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,4,5,1,3] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
[2,4,5,3,1] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,5,1,3,4] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = -1 + 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000147
Mp00248: Permutations DEX compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [2]
=> []
=> 0 = -1 + 1
[2,1] => [2] => [2]
=> []
=> 0 = -1 + 1
[1,2,3] => [3] => [3]
=> []
=> 0 = -1 + 1
[1,3,2] => [1,2] => [2,1]
=> [1]
=> 1 = 0 + 1
[2,1,3] => [3] => [3]
=> []
=> 0 = -1 + 1
[2,3,1] => [3] => [3]
=> []
=> 0 = -1 + 1
[3,1,2] => [3] => [3]
=> []
=> 0 = -1 + 1
[3,2,1] => [2,1] => [2,1]
=> [1]
=> 1 = 0 + 1
[1,2,3,4] => [4] => [4]
=> []
=> 0 = -1 + 1
[1,3,2,4] => [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,2] => [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,2,3] => [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[1,4,3,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[2,1,3,4] => [4] => [4]
=> []
=> 0 = -1 + 1
[2,3,1,4] => [4] => [4]
=> []
=> 0 = -1 + 1
[2,3,4,1] => [4] => [4]
=> []
=> 0 = -1 + 1
[2,4,1,3] => [4] => [4]
=> []
=> 0 = -1 + 1
[2,4,3,1] => [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[3,1,2,4] => [4] => [4]
=> []
=> 0 = -1 + 1
[3,4,1,2] => [4] => [4]
=> []
=> 0 = -1 + 1
[3,4,2,1] => [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,1,2,3] => [4] => [4]
=> []
=> 0 = -1 + 1
[4,1,3,2] => [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,2,3,1] => [3,1] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,3,1,2] => [1,3] => [3,1]
=> [1]
=> 1 = 0 + 1
[4,3,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4,5] => [5] => [5]
=> []
=> 0 = -1 + 1
[1,3,2,4,5] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,2,5] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,4,5,2] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,2,4] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,3,5,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,4,2,3,5] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,2,3] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,4,5,3,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,2,3,4] => [1,4] => [4,1]
=> [1]
=> 1 = 0 + 1
[1,5,2,4,3] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,3,4,2] => [1,3,1] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,4,2,3] => [1,1,3] => [3,1,1]
=> [1,1]
=> 1 = 0 + 1
[1,5,4,3,2] => [1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,3,4,5] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,3,1,4,5] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,3,4,1,5] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,3,4,5,1] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,3,5,1,4] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,3,5,4,1] => [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[2,4,1,3,5] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,4,5,1,3] => [5] => [5]
=> []
=> 0 = -1 + 1
[2,4,5,3,1] => [4,1] => [4,1]
=> [1]
=> 1 = 0 + 1
[2,5,1,3,4] => [5] => [5]
=> []
=> 0 = -1 + 1
Description
The largest part of an integer partition.
The following 172 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000183The side length of the Durfee square of an integer partition. St000292The number of ascents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000378The diagonal inversion number of an integer partition. St000442The maximal area to the right of an up step of a Dyck path. St000480The number of lower covers of a partition in dominance order. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000535The rank-width of a graph. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001712The number of natural descents of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000013The height of a Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000444The length of the maximal rise of a Dyck path. St000759The smallest missing part in an integer partition. St000905The number of different multiplicities of parts of an integer composition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001261The Castelnuovo-Mumford regularity of a graph. St001732The number of peaks visible from the left. St001814The number of partitions interlacing the given partition. St001884The number of borders of a binary word. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000260The radius of a connected graph. St000058The order of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000298The order dimension or Dushnik-Miller dimension of a poset. St000264The girth of a graph, which is not a tree. St000478Another weight of a partition according to Alladi. St000485The length of the longest cycle of a permutation. St000862The number of parts of the shifted shape of a permutation. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000259The diameter of a connected graph. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001734The lettericity of a graph. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001271The competition number of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001512The minimum rank of a graph. St001638The book thickness of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000443The number of long tunnels of a Dyck path. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000918The 2-limited packing number of a graph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001093The detour number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001644The dimension of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001962The proper pathwidth of a graph. St001112The 3-weak dynamic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000640The rank of the largest boolean interval in a poset. St001060The distinguishing index of a graph. St001488The number of corners of a skew partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000454The largest eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1.