- FindStat

Your data matches 366 different statistics following compositions of up to 3 maps.
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Matching statistic: St000897

Mapping

St000897: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 1
[2]
 => 1
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 1
[1,1,1]
 => 1
[4]
 => 1
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 1
[5]
 => 1
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 2
[2,2,1]
 => 2
[2,1,1,1]
 => 2
[1,1,1,1,1]
 => 1
[6]
 => 1
[5,1]
 => 1
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 1
[3,2,1]
 => 1
[3,1,1,1]
 => 2
[2,2,2]
 => 1
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 1
[7]
 => 1
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 2
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 2
[3,3,1]
 => 2
[3,2,2]
 => 2
[3,2,1,1]
 => 2
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 1
[8]
 => 1
[7,1]
 => 1
[6,2]
 => 1
[6,1,1]
 => 2
[5,3]
 => 1
[5,2,1]
 => 1

Description

The number of different multiplicities of parts of an integer partition.

Matching statistic: St000905
(load all 9 compositions to match this statistic)

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000905: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => [[1]]
 => [1] => 1
[2]
 => [[1,2]]
 => [2] => 1
[1,1]
 => [[1],[2]]
 => [1,1] => 1
[3]
 => [[1,2,3]]
 => [3] => 1
[2,1]
 => [[1,2],[3]]
 => [2,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => 1
[4]
 => [[1,2,3,4]]
 => [4] => 1
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => 1
[5]
 => [[1,2,3,4,5]]
 => [5] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => 2
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => 2
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => 2
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => 2
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => 2
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => 2
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => 2
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [7,1] => 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [6,2] => 1
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [6,1,1] => 2
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [5,3] => 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [5,2,1] => 1

Description

The number of different multiplicities of parts of an integer composition.

Matching statistic: St000903
(load all 8 compositions to match this statistic)

Mapping

Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => [[1]]
 => [1] => [1] => 1
[2]
 => [[1,2]]
 => [2] => [1] => 1
[1,1]
 => [[1],[2]]
 => [1,1] => [2] => 1
[3]
 => [[1,2,3]]
 => [3] => [1] => 1
[2,1]
 => [[1,2],[3]]
 => [2,1] => [1,1] => 1
[1,1,1]
 => [[1],[2],[3]]
 => [1,1,1] => [3] => 1
[4]
 => [[1,2,3,4]]
 => [4] => [1] => 1
[3,1]
 => [[1,2,3],[4]]
 => [3,1] => [1,1] => 1
[2,2]
 => [[1,2],[3,4]]
 => [2,2] => [2] => 1
[2,1,1]
 => [[1,2],[3],[4]]
 => [2,1,1] => [1,2] => 2
[1,1,1,1]
 => [[1],[2],[3],[4]]
 => [1,1,1,1] => [4] => 1
[5]
 => [[1,2,3,4,5]]
 => [5] => [1] => 1
[4,1]
 => [[1,2,3,4],[5]]
 => [4,1] => [1,1] => 1
[3,2]
 => [[1,2,3],[4,5]]
 => [3,2] => [1,1] => 1
[3,1,1]
 => [[1,2,3],[4],[5]]
 => [3,1,1] => [1,2] => 2
[2,2,1]
 => [[1,2],[3,4],[5]]
 => [2,2,1] => [2,1] => 2
[2,1,1,1]
 => [[1,2],[3],[4],[5]]
 => [2,1,1,1] => [1,3] => 2
[1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [1,1,1,1,1] => [5] => 1
[6]
 => [[1,2,3,4,5,6]]
 => [6] => [1] => 1
[5,1]
 => [[1,2,3,4,5],[6]]
 => [5,1] => [1,1] => 1
[4,2]
 => [[1,2,3,4],[5,6]]
 => [4,2] => [1,1] => 1
[4,1,1]
 => [[1,2,3,4],[5],[6]]
 => [4,1,1] => [1,2] => 2
[3,3]
 => [[1,2,3],[4,5,6]]
 => [3,3] => [2] => 1
[3,2,1]
 => [[1,2,3],[4,5],[6]]
 => [3,2,1] => [1,1,1] => 1
[3,1,1,1]
 => [[1,2,3],[4],[5],[6]]
 => [3,1,1,1] => [1,3] => 2
[2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [2,2,2] => [3] => 1
[2,2,1,1]
 => [[1,2],[3,4],[5],[6]]
 => [2,2,1,1] => [2,2] => 1
[2,1,1,1,1]
 => [[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1] => [1,4] => 2
[1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1] => [6] => 1
[7]
 => [[1,2,3,4,5,6,7]]
 => [7] => [1] => 1
[6,1]
 => [[1,2,3,4,5,6],[7]]
 => [6,1] => [1,1] => 1
[5,2]
 => [[1,2,3,4,5],[6,7]]
 => [5,2] => [1,1] => 1
[5,1,1]
 => [[1,2,3,4,5],[6],[7]]
 => [5,1,1] => [1,2] => 2
[4,3]
 => [[1,2,3,4],[5,6,7]]
 => [4,3] => [1,1] => 1
[4,2,1]
 => [[1,2,3,4],[5,6],[7]]
 => [4,2,1] => [1,1,1] => 1
[4,1,1,1]
 => [[1,2,3,4],[5],[6],[7]]
 => [4,1,1,1] => [1,3] => 2
[3,3,1]
 => [[1,2,3],[4,5,6],[7]]
 => [3,3,1] => [2,1] => 2
[3,2,2]
 => [[1,2,3],[4,5],[6,7]]
 => [3,2,2] => [1,2] => 2
[3,2,1,1]
 => [[1,2,3],[4,5],[6],[7]]
 => [3,2,1,1] => [1,1,2] => 2
[3,1,1,1,1]
 => [[1,2,3],[4],[5],[6],[7]]
 => [3,1,1,1,1] => [1,4] => 2
[2,2,2,1]
 => [[1,2],[3,4],[5,6],[7]]
 => [2,2,2,1] => [3,1] => 2
[2,2,1,1,1]
 => [[1,2],[3,4],[5],[6],[7]]
 => [2,2,1,1,1] => [2,3] => 2
[2,1,1,1,1,1]
 => [[1,2],[3],[4],[5],[6],[7]]
 => [2,1,1,1,1,1] => [1,5] => 2
[1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => [1,1,1,1,1,1,1] => [7] => 1
[8]
 => [[1,2,3,4,5,6,7,8]]
 => [8] => [1] => 1
[7,1]
 => [[1,2,3,4,5,6,7],[8]]
 => [7,1] => [1,1] => 1
[6,2]
 => [[1,2,3,4,5,6],[7,8]]
 => [6,2] => [1,1] => 1
[6,1,1]
 => [[1,2,3,4,5,6],[7],[8]]
 => [6,1,1] => [1,2] => 2
[5,3]
 => [[1,2,3,4,5],[6,7,8]]
 => [5,3] => [1,1] => 1
[5,2,1]
 => [[1,2,3,4,5],[6,7],[8]]
 => [5,2,1] => [1,1,1] => 1

Description

The number of different parts of an integer composition.

Matching statistic: St001124

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 10 => [1,2] => [2,1]
 => 1
[2]
 => 100 => [1,3] => [3,1]
 => 1
[1,1]
 => 110 => [1,1,2] => [2,1,1]
 => 1
[3]
 => 1000 => [1,4] => [4,1]
 => 1
[2,1]
 => 1010 => [1,2,2] => [2,2,1]
 => 1
[1,1,1]
 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 1
[4]
 => 10000 => [1,5] => [5,1]
 => 1
[3,1]
 => 10010 => [1,3,2] => [3,2,1]
 => 2
[2,2]
 => 1100 => [1,1,3] => [3,1,1]
 => 1
[2,1,1]
 => 10110 => [1,2,1,2] => [2,2,1,1]
 => 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
 => 1
[5]
 => 100000 => [1,6] => [6,1]
 => 1
[4,1]
 => 100010 => [1,4,2] => [4,2,1]
 => 2
[3,2]
 => 10100 => [1,2,3] => [3,2,1]
 => 2
[3,1,1]
 => 100110 => [1,3,1,2] => [3,2,1,1]
 => 2
[2,2,1]
 => 11010 => [1,1,2,2] => [2,2,1,1]
 => 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [2,2,1,1,1]
 => 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
 => 1
[6]
 => 1000000 => [1,7] => [7,1]
 => 1
[5,1]
 => 1000010 => [1,5,2] => [5,2,1]
 => 2
[4,2]
 => 100100 => [1,3,3] => [3,3,1]
 => 1
[4,1,1]
 => 1000110 => [1,4,1,2] => [4,2,1,1]
 => 2
[3,3]
 => 11000 => [1,1,4] => [4,1,1]
 => 1
[3,2,1]
 => 101010 => [1,2,2,2] => [2,2,2,1]
 => 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
 => 2
[2,2,2]
 => 11100 => [1,1,1,3] => [3,1,1,1]
 => 1
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [2,2,1,1,1]
 => 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
 => 1
[7]
 => 10000000 => [1,8] => [8,1]
 => 1
[6,1]
 => 10000010 => [1,6,2] => [6,2,1]
 => 2
[5,2]
 => 1000100 => [1,4,3] => [4,3,1]
 => 2
[5,1,1]
 => 10000110 => [1,5,1,2] => [5,2,1,1]
 => 2
[4,3]
 => 101000 => [1,2,4] => [4,2,1]
 => 2
[4,2,1]
 => 1001010 => [1,3,2,2] => [3,2,2,1]
 => 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
 => 2
[3,3,1]
 => 110010 => [1,1,3,2] => [3,2,1,1]
 => 2
[3,2,2]
 => 101100 => [1,2,1,3] => [3,2,1,1]
 => 2
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
 => 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
 => 2
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [2,2,1,1,1]
 => 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
 => 1
[8]
 => 100000000 => [1,9] => [9,1]
 => 1
[7,1]
 => 100000010 => [1,7,2] => [7,2,1]
 => 2
[6,2]
 => 10000100 => [1,5,3] => [5,3,1]
 => 2
[6,1,1]
 => 100000110 => [1,6,1,2] => [6,2,1,1]
 => 2
[5,3]
 => 1001000 => [1,3,4] => [4,3,1]
 => 2
[5,2,1]
 => 10001010 => [1,4,2,2] => [4,2,2,1]
 => 2

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.

Matching statistic: St000159

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 10 => [1,2] => [2,1]
 => 2 = 1 + 1
[2]
 => 100 => [1,3] => [3,1]
 => 2 = 1 + 1
[1,1]
 => 110 => [1,1,2] => [2,1,1]
 => 2 = 1 + 1
[3]
 => 1000 => [1,4] => [4,1]
 => 2 = 1 + 1
[2,1]
 => 1010 => [1,2,2] => [2,2,1]
 => 2 = 1 + 1
[1,1,1]
 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 2 = 1 + 1
[4]
 => 10000 => [1,5] => [5,1]
 => 2 = 1 + 1
[3,1]
 => 10010 => [1,3,2] => [3,2,1]
 => 3 = 2 + 1
[2,2]
 => 1100 => [1,1,3] => [3,1,1]
 => 2 = 1 + 1
[2,1,1]
 => 10110 => [1,2,1,2] => [2,2,1,1]
 => 2 = 1 + 1
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
 => 2 = 1 + 1
[5]
 => 100000 => [1,6] => [6,1]
 => 2 = 1 + 1
[4,1]
 => 100010 => [1,4,2] => [4,2,1]
 => 3 = 2 + 1
[3,2]
 => 10100 => [1,2,3] => [3,2,1]
 => 3 = 2 + 1
[3,1,1]
 => 100110 => [1,3,1,2] => [3,2,1,1]
 => 3 = 2 + 1
[2,2,1]
 => 11010 => [1,1,2,2] => [2,2,1,1]
 => 2 = 1 + 1
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [2,2,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[6]
 => 1000000 => [1,7] => [7,1]
 => 2 = 1 + 1
[5,1]
 => 1000010 => [1,5,2] => [5,2,1]
 => 3 = 2 + 1
[4,2]
 => 100100 => [1,3,3] => [3,3,1]
 => 2 = 1 + 1
[4,1,1]
 => 1000110 => [1,4,1,2] => [4,2,1,1]
 => 3 = 2 + 1
[3,3]
 => 11000 => [1,1,4] => [4,1,1]
 => 2 = 1 + 1
[3,2,1]
 => 101010 => [1,2,2,2] => [2,2,2,1]
 => 2 = 1 + 1
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
 => 3 = 2 + 1
[2,2,2]
 => 11100 => [1,1,1,3] => [3,1,1,1]
 => 2 = 1 + 1
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [2,2,1,1,1]
 => 2 = 1 + 1
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
 => 2 = 1 + 1
[7]
 => 10000000 => [1,8] => [8,1]
 => 2 = 1 + 1
[6,1]
 => 10000010 => [1,6,2] => [6,2,1]
 => 3 = 2 + 1
[5,2]
 => 1000100 => [1,4,3] => [4,3,1]
 => 3 = 2 + 1
[5,1,1]
 => 10000110 => [1,5,1,2] => [5,2,1,1]
 => 3 = 2 + 1
[4,3]
 => 101000 => [1,2,4] => [4,2,1]
 => 3 = 2 + 1
[4,2,1]
 => 1001010 => [1,3,2,2] => [3,2,2,1]
 => 3 = 2 + 1
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
 => 3 = 2 + 1
[3,3,1]
 => 110010 => [1,1,3,2] => [3,2,1,1]
 => 3 = 2 + 1
[3,2,2]
 => 101100 => [1,2,1,3] => [3,2,1,1]
 => 3 = 2 + 1
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
 => 2 = 1 + 1
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
 => 3 = 2 + 1
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [2,2,1,1,1]
 => 2 = 1 + 1
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
 => 2 = 1 + 1
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
 => 2 = 1 + 1
[8]
 => 100000000 => [1,9] => [9,1]
 => 2 = 1 + 1
[7,1]
 => 100000010 => [1,7,2] => [7,2,1]
 => 3 = 2 + 1
[6,2]
 => 10000100 => [1,5,3] => [5,3,1]
 => 3 = 2 + 1
[6,1,1]
 => 100000110 => [1,6,1,2] => [6,2,1,1]
 => 3 = 2 + 1
[5,3]
 => 1001000 => [1,3,4] => [4,3,1]
 => 3 = 2 + 1
[5,2,1]
 => 10001010 => [1,4,2,2] => [4,2,2,1]
 => 3 = 2 + 1

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.

Matching statistic: St000318

Mapping

Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1]
 => 10 => [1,2] => [2,1]
 => 3 = 1 + 2
[2]
 => 100 => [1,3] => [3,1]
 => 3 = 1 + 2
[1,1]
 => 110 => [1,1,2] => [2,1,1]
 => 3 = 1 + 2
[3]
 => 1000 => [1,4] => [4,1]
 => 3 = 1 + 2
[2,1]
 => 1010 => [1,2,2] => [2,2,1]
 => 3 = 1 + 2
[1,1,1]
 => 1110 => [1,1,1,2] => [2,1,1,1]
 => 3 = 1 + 2
[4]
 => 10000 => [1,5] => [5,1]
 => 3 = 1 + 2
[3,1]
 => 10010 => [1,3,2] => [3,2,1]
 => 4 = 2 + 2
[2,2]
 => 1100 => [1,1,3] => [3,1,1]
 => 3 = 1 + 2
[2,1,1]
 => 10110 => [1,2,1,2] => [2,2,1,1]
 => 3 = 1 + 2
[1,1,1,1]
 => 11110 => [1,1,1,1,2] => [2,1,1,1,1]
 => 3 = 1 + 2
[5]
 => 100000 => [1,6] => [6,1]
 => 3 = 1 + 2
[4,1]
 => 100010 => [1,4,2] => [4,2,1]
 => 4 = 2 + 2
[3,2]
 => 10100 => [1,2,3] => [3,2,1]
 => 4 = 2 + 2
[3,1,1]
 => 100110 => [1,3,1,2] => [3,2,1,1]
 => 4 = 2 + 2
[2,2,1]
 => 11010 => [1,1,2,2] => [2,2,1,1]
 => 3 = 1 + 2
[2,1,1,1]
 => 101110 => [1,2,1,1,2] => [2,2,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1]
 => 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
 => 3 = 1 + 2
[6]
 => 1000000 => [1,7] => [7,1]
 => 3 = 1 + 2
[5,1]
 => 1000010 => [1,5,2] => [5,2,1]
 => 4 = 2 + 2
[4,2]
 => 100100 => [1,3,3] => [3,3,1]
 => 3 = 1 + 2
[4,1,1]
 => 1000110 => [1,4,1,2] => [4,2,1,1]
 => 4 = 2 + 2
[3,3]
 => 11000 => [1,1,4] => [4,1,1]
 => 3 = 1 + 2
[3,2,1]
 => 101010 => [1,2,2,2] => [2,2,2,1]
 => 3 = 1 + 2
[3,1,1,1]
 => 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
 => 4 = 2 + 2
[2,2,2]
 => 11100 => [1,1,1,3] => [3,1,1,1]
 => 3 = 1 + 2
[2,2,1,1]
 => 110110 => [1,1,2,1,2] => [2,2,1,1,1]
 => 3 = 1 + 2
[2,1,1,1,1]
 => 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1,1]
 => 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
 => 3 = 1 + 2
[7]
 => 10000000 => [1,8] => [8,1]
 => 3 = 1 + 2
[6,1]
 => 10000010 => [1,6,2] => [6,2,1]
 => 4 = 2 + 2
[5,2]
 => 1000100 => [1,4,3] => [4,3,1]
 => 4 = 2 + 2
[5,1,1]
 => 10000110 => [1,5,1,2] => [5,2,1,1]
 => 4 = 2 + 2
[4,3]
 => 101000 => [1,2,4] => [4,2,1]
 => 4 = 2 + 2
[4,2,1]
 => 1001010 => [1,3,2,2] => [3,2,2,1]
 => 4 = 2 + 2
[4,1,1,1]
 => 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
 => 4 = 2 + 2
[3,3,1]
 => 110010 => [1,1,3,2] => [3,2,1,1]
 => 4 = 2 + 2
[3,2,2]
 => 101100 => [1,2,1,3] => [3,2,1,1]
 => 4 = 2 + 2
[3,2,1,1]
 => 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
 => 3 = 1 + 2
[3,1,1,1,1]
 => 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
 => 4 = 2 + 2
[2,2,2,1]
 => 111010 => [1,1,1,2,2] => [2,2,1,1,1]
 => 3 = 1 + 2
[2,2,1,1,1]
 => 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
 => 3 = 1 + 2
[2,1,1,1,1,1]
 => 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
 => 3 = 1 + 2
[1,1,1,1,1,1,1]
 => 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
 => 3 = 1 + 2
[8]
 => 100000000 => [1,9] => [9,1]
 => 3 = 1 + 2
[7,1]
 => 100000010 => [1,7,2] => [7,2,1]
 => 4 = 2 + 2
[6,2]
 => 10000100 => [1,5,3] => [5,3,1]
 => 4 = 2 + 2
[6,1,1]
 => 100000110 => [1,6,1,2] => [6,2,1,1]
 => 4 = 2 + 2
[5,3]
 => 1001000 => [1,3,4] => [4,3,1]
 => 4 = 2 + 2
[5,2,1]
 => 10001010 => [1,4,2,2] => [4,2,2,1]
 => 4 = 2 + 2

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000919

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000919: Binary trees ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [[.,.],.]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [[.,[.,.]],.]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [.,[[.,.],.]]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [[.,[.,[.,.]]],.]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [[[.,.],.],.]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => [.,[.,[[.,.],.]]]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[.,[.,[.,[.,.]]]],.]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [[[.,.],[.,.]],.]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => [.,[[.,[.,.]],.]]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [[.,.],[[.,.],.]]
 => 2
[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
[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
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[.,[[.,.],[.,.]]],.]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [[[.,[.,.]],.],.]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => [[.,[[.,.],.]],.]
 => 2
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [.,[[[.,.],.],.]]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [.,[[.,.],[[.,.],.]]]
 => 2
[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
[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
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[[.,[.,.]],[.,.]],.]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[[.,.],[.,[.,.]]],.]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [.,[[.,[.,[.,.]]],.]]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [[[[.,.],.],.],.]
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [.,[[[.,.],[.,.]],.]]
 => 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
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[.,.],[.,[[.,.],.]]]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [.,[.,[[.,.],[[.,.],.]]]]
 => 2
[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
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,[.,[.,.]]]]]]],.]
 => ? = 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[.,[[.,.],[.,.]]]]],.]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0,1,0]
 => [[.,[[.,[.,.]],[.,.]]],.]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[[.,[.,[.,.]]],.],.]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[[[.,.],.],[.,.]],.]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[.,[.,[[.,.],.]]],.]
 => 2
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[.,.],[[.,[.,.]],.]]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[.,[.,.]],[[.,.],.]]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[[.,.],.],[[.,.],.]]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [.,[.,[[[.,.],[.,.]],.]]]
 => 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [.,[.,[[[.,.],.],.]]]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [.,[[.,.],[.,[[.,.],.]]]]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [.,[.,[.,[[.,.],[[.,.],.]]]]]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]
 => 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]],.]
 => ? ∊ {1,1,1}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => [[.,[.,[.,[.,[[.,.],[.,.]]]]]],.]
 => ? ∊ {1,1,1}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
 => [[.,[.,[[.,[.,.]],[.,.]]]],.]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
 => [[.,[.,[[.,.],[.,[.,.]]]]],.]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [[[.,[.,[.,.]]],[.,.]],.]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => [[.,[[[.,.],.],[.,.]]],.]
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => [.,[[.,[.,[.,[.,.]]]],.]]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[[[.,.],[.,.]],.],.]
 => 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
 => [.,[.,[.,[.,[.,[.,[.,[[.,.],.]]]]]]]]
 => ? ∊ {1,1,1}

Description

The number of maximal left branches of a binary tree. A maximal left branch of a binary tree is an inclusion wise maximal path which consists of left edges only. This statistic records the number of distinct maximal left branches in the tree.

Matching statistic: St001732

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00327: Dyck paths —inverse Kreweras complement⟶ Dyck paths
St001732: Dyck paths ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 2
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 2
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
 => ? ∊ {1,1}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,1,0,0]
 => 2
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ?
 => ? ∊ {1,1}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
 => 2
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,1,0,0,0]
 => 2
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
 => 2
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => 2
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,0,0,0,1,0,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 2
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ?
 => ? ∊ {1,1}

Description

The number of peaks visible from the left. This is, the number of left-to-right maxima of the heights of the peaks of a Dyck path.

Matching statistic: St001037

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00118: Dyck paths —swap returns and last descent⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 92% ●values known / values provided: 92%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0 = 1 - 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 0 = 1 - 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 0 = 1 - 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 0 = 1 - 1
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 0 = 1 - 1
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 0 = 1 - 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 1 = 2 - 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 0 = 1 - 1
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0 = 1 - 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]
 => [1,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => 0 = 1 - 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {1,1} - 1
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 0 = 1 - 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,0,1,0,0,0]
 => 1 = 2 - 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1 = 2 - 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1 = 2 - 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1 = 2 - 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 1 = 2 - 1
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0,1,0]
 => 0 = 1 - 1
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => 1 = 2 - 1
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => 0 = 1 - 1
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {1,1} - 1
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {1,1,1} - 1
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 1 = 2 - 1
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => 0 = 1 - 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
 => 1 = 2 - 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 2 - 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,1,0,1,0,0]
 => 1 = 2 - 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => 0 = 1 - 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1 = 2 - 1
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
 => ? ∊ {1,1,1} - 1
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
 => ? ∊ {1,1,1} - 1

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St001199
(load all 14 compositions to match this statistic)

Mapping

Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1]
 => [1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => ? = 1
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => ? = 1
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => ? = 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 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]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 1
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => 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]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 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]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 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
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 1
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => 1
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2}
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 1
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 1
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,0,0,0]
 => 2
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? ∊ {2,2}
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2}
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2}
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
 => 1
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,1,0,0,0]
 => 1
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => 1
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => 1
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 2
[4,2,2]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => 1
[4,2,1,1]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 1
[4,1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 2
[3,3,2]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 2
[3,3,1,1]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,2,2,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 1
[3,2,1,1,1]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,1,0,0]
 => 2
[2,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? ∊ {2,2,2,2}
[1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? ∊ {2,2,2,2}

Description

The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.

The following 356 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001737The number of descents of type 2 in a permutation. St000872The number of very big descents of a permutation. St001114The number of odd descents of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001487The number of inner corners of a skew partition. St001712The number of natural descents of a standard Young tableau. St001955The number of natural descents for set-valued two row standard Young tableaux. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St001469The holeyness of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000834The number of right outer peaks of a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000007The number of saliances of the permutation. St001964The interval resolution global dimension of a poset. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000092The number of outer peaks of a permutation. St000542The number of left-to-right-minima of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001728The number of invisible descents of a permutation. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000035The number of left outer peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001162The minimum jump of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001202Call 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. 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 $eA$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001344The neighbouring number of a permutation. St001394The genus of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001569The maximal modular displacement of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000061The number of nodes on the left branch of a binary tree. St000099The number of valleys of a permutation, including the boundary. St000252The number of nodes of degree 3 of a binary tree. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000664The number of right ropes of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000991The number of right-to-left minima of a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. 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. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001715The number of non-records in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000871The number of very big ascents of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001568The smallest positive integer that does not appear twice in the partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000256The number of parts from which one can substract 2 and still get an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000668The least common multiple of the parts of the partition. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001128The exponens consonantiae of a partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001532The leading coefficient of the Poincare polynomial of the poset cone. 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 $eA$. 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. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000260The radius of a connected graph. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001095The number of non-isomorphic posets with precisely one further covering relation. St001490The number of connected components of a skew partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001720The minimal length of a chain of small intervals in a lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001846The number of elements which do not have a complement in the lattice. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000284The Plancherel distribution on integer partitions. St000454The largest eigenvalue of a graph if it is integral. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. 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. St000937The number of positive values of the symmetric group character corresponding to the partition. St000741The Colin de Verdière graph invariant. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001896The number of right descents of a signed permutations. St001330The hat guessing number of a graph. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000629The defect of a binary word. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001616The number of neutral elements in a lattice. St000326The position of the first one in a binary word after appending a 1 at the end. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001811The Castelnuovo-Mumford regularity of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000782The indicator function of whether a given perfect matching is an L & P matching. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001578The minimal number of edges to add or remove to make a graph a line graph. St000527The width of the poset. St000627The exponent of a binary word. St000455The second largest eigenvalue of a graph if it is integral. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St000878The number of ones minus the number of zeros of a binary word. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000056The decomposition (or block) number of a permutation. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000317The cycle descent number of a permutation. St000657The smallest part of an integer composition. St000694The number of affine bounded permutations that project to a given permutation. St000761The number of ascents in an integer composition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001413Half the length of the longest even length palindromic prefix of a binary word. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001590The crossing number of a perfect matching. St001768The number of reduced words of a signed permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001884The number of borders of a binary word. St001889The size of the connectivity set of a signed permutation. St001946The number of descents in a parking function. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000297The number of leading ones in a binary word. St000392The length of the longest run of ones in a binary word. St000405The number of occurrences of the pattern 1324 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000758The length of the longest staircase fitting into an integer composition. St000787The number of flips required to make a perfect matching noncrossing. St000842The breadth of a permutation. St000876The number of factors in the Catalan decomposition of a binary word. St000877The depth of the binary word interpreted as a path. St000885The number of critical steps in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000943The number of spots the most unlucky car had to go further in a parking function. St000982The length of the longest constant subword. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. 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 $eA$. 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. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001524The degree of symmetry of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001850The number of Hecke atoms of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St000767The number of runs in an integer composition. St000891The number of distinct diagonal sums of a permutation matrix. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001481The minimal height of a peak of a Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000417The size of the automorphism group of the ordered tree. St001058The breadth of the ordered tree. St000068The number of minimal elements in a poset. St000731The number of double exceedences of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000679The pruning number of an ordered tree. St001624The breadth of a lattice. 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. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000570The Edelman-Greene number of a permutation. St000729The minimal arc length of a set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001806The upper middle entry of a permutation. St000039The number of crossings of a permutation. St000084The number of subtrees. St000091The descent variation of a composition. St000105The number of blocks in the set partition. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. 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. St000406The number of occurrences of the pattern 3241 in a permutation. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000732The number of double deficiencies of a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000823The number of unsplittable factors of the set partition. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001537The number of cyclic crossings of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001781The interlacing number of a set partition. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000451The length of the longest pattern of the form k 1 2. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000862The number of parts of the shifted shape of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation.