Your data matches 391 different statistics following compositions of up to 3 maps.
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St001122: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 0
[3,1,1]
=> 1
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 0
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 0
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
Description
The multiplicity of the sign 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 $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00119: Dyck paths to 321-avoiding permutation (Krattenthaler)Permutations
Mp00223: Permutations runsortPermutations
St000215: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => [1,2] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,2,3] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [1,2,3] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3,4,2] => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,4,5,3] => 0
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,4,2,3] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,3,4,5,2] => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,5,6,4] => 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [1,2,3,4] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,2,5,3] => 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,4,5,6,3] => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,3,4] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [1,4,5,2,3] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,3,4,5,6,2] => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0
Description
The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended. $(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00222: Dyck paths peaks-to-valleysDyck paths
Mp00024: Dyck paths to 321-avoiding permutationPermutations
St001850: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,2,4,5,3] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,2,5,3,4] => 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,2,3,5,6,4] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,2,5,3] => 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,2,3,6,4,5] => 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,2,3,4,6,7,5] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,2,4,6,3,5] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 0
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 0
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 0
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,2,5,3,6,4] => 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,2,3,4,7,5,6] => 0
Description
The number of Hecke atoms of a permutation. For a permutation $z\in\mathfrak S_n$, this is the cardinality of the set $$ \{ w\in\mathfrak S_n | w^{-1} \star w = z\}, $$ where $\star$ denotes the Demazure product. Note that $w\mapsto w^{-1}\star w$ is a surjection onto the set of involutions.
Matching statistic: St001939
Mp00042: Integer partitions initial tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00204: Permutations LLPSInteger partitions
St001939: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => [1]
=> 1
[2]
=> [[1,2]]
=> [1,2] => [1,1]
=> 0
[1,1]
=> [[1],[2]]
=> [2,1] => [2]
=> 0
[3]
=> [[1,2,3]]
=> [1,2,3] => [1,1,1]
=> 0
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,1]
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3]
=> 0
[4]
=> [[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> 0
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,1,1]
=> 0
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,1,1]
=> 0
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [3,1]
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4]
=> 0
[5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,1,1,1]
=> 0
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,1,1]
=> 0
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,1,1]
=> 0
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,1,1]
=> 0
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5]
=> 0
[6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 0
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 0
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 0
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [3,1,1,1]
=> 0
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 0
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [3,1,1,1]
=> 0
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [4,1,1]
=> 0
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [3,1,1,1]
=> 0
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [4,1,1]
=> 0
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [5,1]
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6]
=> 0
Description
The number of parts that are equal to their multiplicity in the integer partition.
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00189: Skew partitions rotateSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
St001940: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1],[]]
=> [[1],[]]
=> [1]
=> 1
[2]
=> [[2],[]]
=> [[2],[]]
=> [2]
=> 0
[1,1]
=> [[1,1],[]]
=> [[1,1],[]]
=> [1,1]
=> 0
[3]
=> [[3],[]]
=> [[3],[]]
=> [3]
=> 0
[2,1]
=> [[2,1],[]]
=> [[2,2],[1]]
=> [2,2]
=> 1
[1,1,1]
=> [[1,1,1],[]]
=> [[1,1,1],[]]
=> [1,1,1]
=> 0
[4]
=> [[4],[]]
=> [[4],[]]
=> [4]
=> 0
[3,1]
=> [[3,1],[]]
=> [[3,3],[2]]
=> [3,3]
=> 0
[2,2]
=> [[2,2],[]]
=> [[2,2],[]]
=> [2,2]
=> 1
[2,1,1]
=> [[2,1,1],[]]
=> [[2,2,2],[1,1]]
=> [2,2,2]
=> 0
[1,1,1,1]
=> [[1,1,1,1],[]]
=> [[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[5]
=> [[5],[]]
=> [[5],[]]
=> [5]
=> 0
[4,1]
=> [[4,1],[]]
=> [[4,4],[3]]
=> [4,4]
=> 0
[3,2]
=> [[3,2],[]]
=> [[3,3],[1]]
=> [3,3]
=> 0
[3,1,1]
=> [[3,1,1],[]]
=> [[3,3,3],[2,2]]
=> [3,3,3]
=> 1
[2,2,1]
=> [[2,2,1],[]]
=> [[2,2,2],[1]]
=> [2,2,2]
=> 0
[2,1,1,1]
=> [[2,1,1,1],[]]
=> [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> 0
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> 0
[6]
=> [[6],[]]
=> [[6],[]]
=> [6]
=> 0
[5,1]
=> [[5,1],[]]
=> [[5,5],[4]]
=> [5,5]
=> 0
[4,2]
=> [[4,2],[]]
=> [[4,4],[2]]
=> [4,4]
=> 0
[4,1,1]
=> [[4,1,1],[]]
=> [[4,4,4],[3,3]]
=> [4,4,4]
=> 0
[3,3]
=> [[3,3],[]]
=> [[3,3],[]]
=> [3,3]
=> 0
[3,2,1]
=> [[3,2,1],[]]
=> [[3,3,3],[2,1]]
=> [3,3,3]
=> 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> 0
[2,2,2]
=> [[2,2,2],[]]
=> [[2,2,2],[]]
=> [2,2,2]
=> 0
[2,2,1,1]
=> [[2,2,1,1],[]]
=> [[2,2,2,2],[1,1]]
=> [2,2,2,2]
=> 0
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> 0
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> 0
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
Matching statistic: St000655
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00132: Dyck paths switch returns and last double riseDyck paths
Mp00143: Dyck paths inverse promotionDyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1 = 0 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 0 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1 = 0 + 1
Description
The length of the minimal rise of a Dyck path. For the length of a maximal rise, see [[St000444]].
Matching statistic: St000657
Mp00095: Integer partitions to binary wordBinary words
Mp00272: Binary words Gray nextBinary words
Mp00097: Binary words delta morphismInteger compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 11 => [2] => 2 = 1 + 1
[2]
=> 100 => 110 => [2,1] => 1 = 0 + 1
[1,1]
=> 110 => 010 => [1,1,1] => 1 = 0 + 1
[3]
=> 1000 => 1100 => [2,2] => 2 = 1 + 1
[2,1]
=> 1010 => 0010 => [2,1,1] => 1 = 0 + 1
[1,1,1]
=> 1110 => 1010 => [1,1,1,1] => 1 = 0 + 1
[4]
=> 10000 => 11000 => [2,3] => 2 = 1 + 1
[3,1]
=> 10010 => 00010 => [3,1,1] => 1 = 0 + 1
[2,2]
=> 1100 => 0100 => [1,1,2] => 1 = 0 + 1
[2,1,1]
=> 10110 => 11110 => [4,1] => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01110 => [1,3,1] => 1 = 0 + 1
[5]
=> 100000 => 110000 => [2,4] => 2 = 1 + 1
[4,1]
=> 100010 => 000010 => [4,1,1] => 1 = 0 + 1
[3,2]
=> 10100 => 00100 => [2,1,2] => 1 = 0 + 1
[3,1,1]
=> 100110 => 110110 => [2,1,2,1] => 1 = 0 + 1
[2,2,1]
=> 11010 => 10010 => [1,2,1,1] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001110 => [2,3,1] => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 101110 => [1,1,3,1] => 1 = 0 + 1
[6]
=> 1000000 => 1100000 => [2,5] => 2 = 1 + 1
[5,1]
=> 1000010 => 0000010 => [5,1,1] => 1 = 0 + 1
[4,2]
=> 100100 => 000100 => [3,1,2] => 1 = 0 + 1
[4,1,1]
=> 1000110 => 1100110 => [2,2,2,1] => 1 = 0 + 1
[3,3]
=> 11000 => 01000 => [1,1,3] => 1 = 0 + 1
[3,2,1]
=> 101010 => 111010 => [3,1,1,1] => 1 = 0 + 1
[3,1,1,1]
=> 1001110 => 0001110 => [3,3,1] => 1 = 0 + 1
[2,2,2]
=> 11100 => 10100 => [1,1,1,2] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 010110 => [1,1,1,2,1] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 1111110 => [6,1] => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111110 => [1,5,1] => 1 = 0 + 1
Description
The smallest part of an integer composition.
Matching statistic: St000899
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00097: Binary words delta morphismInteger compositions
St000899: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => [1,1] => 2 = 1 + 1
[2]
=> 100 => 001 => [2,1] => 1 = 0 + 1
[1,1]
=> 110 => 011 => [1,2] => 1 = 0 + 1
[3]
=> 1000 => 0001 => [3,1] => 1 = 0 + 1
[2,1]
=> 1010 => 0011 => [2,2] => 2 = 1 + 1
[1,1,1]
=> 1110 => 0111 => [1,3] => 1 = 0 + 1
[4]
=> 10000 => 00001 => [4,1] => 1 = 0 + 1
[3,1]
=> 10010 => 00011 => [3,2] => 1 = 0 + 1
[2,2]
=> 1100 => 0011 => [2,2] => 2 = 1 + 1
[2,1,1]
=> 10110 => 00111 => [2,3] => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1 = 0 + 1
[5]
=> 100000 => 000001 => [5,1] => 1 = 0 + 1
[4,1]
=> 100010 => 000011 => [4,2] => 1 = 0 + 1
[3,2]
=> 10100 => 00011 => [3,2] => 1 = 0 + 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2 = 1 + 1
[2,2,1]
=> 11010 => 00111 => [2,3] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => [6,1] => 1 = 0 + 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1 = 0 + 1
[4,2]
=> 100100 => 000011 => [4,2] => 1 = 0 + 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1 = 0 + 1
[3,3]
=> 11000 => 00011 => [3,2] => 1 = 0 + 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1 = 0 + 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1 = 0 + 1
Description
The maximal number of repetitions of an integer composition. This is the maximal part of the composition obtained by applying the delta morphism.
Matching statistic: St000902
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00097: Binary words delta morphismInteger compositions
St000902: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => [1,1] => 2 = 1 + 1
[2]
=> 100 => 001 => [2,1] => 1 = 0 + 1
[1,1]
=> 110 => 011 => [1,2] => 1 = 0 + 1
[3]
=> 1000 => 0001 => [3,1] => 1 = 0 + 1
[2,1]
=> 1010 => 0011 => [2,2] => 2 = 1 + 1
[1,1,1]
=> 1110 => 0111 => [1,3] => 1 = 0 + 1
[4]
=> 10000 => 00001 => [4,1] => 1 = 0 + 1
[3,1]
=> 10010 => 00011 => [3,2] => 1 = 0 + 1
[2,2]
=> 1100 => 0011 => [2,2] => 2 = 1 + 1
[2,1,1]
=> 10110 => 00111 => [2,3] => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1 = 0 + 1
[5]
=> 100000 => 000001 => [5,1] => 1 = 0 + 1
[4,1]
=> 100010 => 000011 => [4,2] => 1 = 0 + 1
[3,2]
=> 10100 => 00011 => [3,2] => 1 = 0 + 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2 = 1 + 1
[2,2,1]
=> 11010 => 00111 => [2,3] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => [6,1] => 1 = 0 + 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1 = 0 + 1
[4,2]
=> 100100 => 000011 => [4,2] => 1 = 0 + 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1 = 0 + 1
[3,3]
=> 11000 => 00011 => [3,2] => 1 = 0 + 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1 = 0 + 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1 = 0 + 1
Description
The minimal number of repetitions of an integer composition.
Matching statistic: St000904
Mp00095: Integer partitions to binary wordBinary words
Mp00224: Binary words runsortBinary words
Mp00097: Binary words delta morphismInteger compositions
St000904: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => [1,1] => 2 = 1 + 1
[2]
=> 100 => 001 => [2,1] => 1 = 0 + 1
[1,1]
=> 110 => 011 => [1,2] => 1 = 0 + 1
[3]
=> 1000 => 0001 => [3,1] => 1 = 0 + 1
[2,1]
=> 1010 => 0011 => [2,2] => 2 = 1 + 1
[1,1,1]
=> 1110 => 0111 => [1,3] => 1 = 0 + 1
[4]
=> 10000 => 00001 => [4,1] => 1 = 0 + 1
[3,1]
=> 10010 => 00011 => [3,2] => 1 = 0 + 1
[2,2]
=> 1100 => 0011 => [2,2] => 2 = 1 + 1
[2,1,1]
=> 10110 => 00111 => [2,3] => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01111 => [1,4] => 1 = 0 + 1
[5]
=> 100000 => 000001 => [5,1] => 1 = 0 + 1
[4,1]
=> 100010 => 000011 => [4,2] => 1 = 0 + 1
[3,2]
=> 10100 => 00011 => [3,2] => 1 = 0 + 1
[3,1,1]
=> 100110 => 000111 => [3,3] => 2 = 1 + 1
[2,2,1]
=> 11010 => 00111 => [2,3] => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 001111 => [2,4] => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 011111 => [1,5] => 1 = 0 + 1
[6]
=> 1000000 => 0000001 => [6,1] => 1 = 0 + 1
[5,1]
=> 1000010 => 0000011 => [5,2] => 1 = 0 + 1
[4,2]
=> 100100 => 000011 => [4,2] => 1 = 0 + 1
[4,1,1]
=> 1000110 => 0000111 => [4,3] => 1 = 0 + 1
[3,3]
=> 11000 => 00011 => [3,2] => 1 = 0 + 1
[3,2,1]
=> 101010 => 001011 => [2,1,1,2] => 2 = 1 + 1
[3,1,1,1]
=> 1001110 => 0001111 => [3,4] => 1 = 0 + 1
[2,2,2]
=> 11100 => 00111 => [2,3] => 1 = 0 + 1
[2,2,1,1]
=> 110110 => 001111 => [2,4] => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0011111 => [2,5] => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => [1,6] => 1 = 0 + 1
Description
The maximal number of repetitions of an integer composition.
The following 381 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001141The number of occurrences of hills of size 3 in a Dyck path. St000633The size of the automorphism group of a poset. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001399The distinguishing number of a poset. St001481The minimal height of a peak of a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. 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 $eA$. St000214The number of adjacencies of a permutation. St000929The constant term of the character polynomial of an integer partition. 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. St001175The size of a partition minus the hook length of the base cell. St000017The number of inversions of a standard tableau. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000256The number of parts from which one can substract 2 and still get an integer partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000347The inversion sum of a binary word. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001092The number of distinct even parts of a partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001438The number of missing boxes of a skew partition. St001485The modular major index of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001596The number of two-by-two squares inside a skew partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St000009The charge of a standard tableau. St000042The number of crossings of a perfect matching. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000065The number of entries equal to -1 in an alternating sign matrix. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000117The number of centered tunnels of a Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000119The number of occurrences of the pattern 321 in a permutation. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000143The largest repeated part of a partition. St000150The floored half-sum of the multiplicities of a partition. St000185The weighted size of a partition. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000210Minimum over maximum difference of elements in cycles. St000221The number of strong fixed points of a permutation. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000252The number of nodes of degree 3 of a binary tree. St000257The number of distinct parts of a partition that occur at least twice. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000369The dinv deficit of a Dyck path. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000386The number of factors DDU in a Dyck path. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000462The major index minus the number of excedences of a permutation. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000497The lcb statistic of a set partition. St000516The number of stretching pairs of a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000674The number of hills of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000691The number of changes of a binary word. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000761The number of ascents in an integer composition. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000769The major index of a composition regarded as a word. St000779The tier of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000872The number of very big descents of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000885The number of critical steps in the Catalan decomposition of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000974The length of the trunk of an ordered tree. St000995The largest even part of an integer partition. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area 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. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001204Call 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 special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001301The first Betti number of the order complex associated with the poset. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001377The major index minus the number of inversions of a permutation. St001381The fertility of a permutation. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001403The number of vertical separators in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001470The cyclic holeyness of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001513The number of nested exceedences of a permutation. St001535The number of cyclic alignments of a permutation. St001536The number of cyclic misalignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001673The degree of asymmetry of an integer composition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001696The natural major index of a standard Young tableau. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001715The number of non-records in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001777The number of weak descents in an integer composition. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001856The number of edges in the reduced word graph of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001931The weak major index of an integer composition regarded as a word. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001964The interval resolution global dimension of a poset. St000488The number of cycles of a permutation of length at most 2. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000478Another weight of a partition according to Alladi. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000219The number of occurrences of the pattern 231 in a permutation. St000353The number of inner valleys of a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000565The major index of a set partition. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000748The major index of the permutation obtained by flattening the set partition. St000837The number of ascents of distance 2 of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001082The number of boxed occurrences of 123 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001557The number of inversions of the second entry of a permutation. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St001615The number of join prime elements of a lattice. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of 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. St000016The number of attacking pairs of a standard tableau. St000455The second largest eigenvalue of a graph if it is integral. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001721The degree of a binary word. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001910The height of the middle non-run of a Dyck path. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001851The number of Hecke atoms of a signed permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St000741The Colin de Verdière graph invariant. St000124The cardinality of the preimage of the Simion-Schmidt map. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. 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. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001651The Frankl number of a lattice. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000729The minimal arc length of a set partition. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000850The number of 1/2-balanced pairs in a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001754The number of tolerances of a finite lattice.