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Your data matches 187 different statistics following compositions of up to 3 maps.
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Matching statistic: St000758
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St000758: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 0 + 1
[1,1] => 1 = 0 + 1
[2] => 1 = 0 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 2 = 1 + 1
[2,1] => 1 = 0 + 1
[3] => 1 = 0 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 2 = 1 + 1
[1,2,1] => 2 = 1 + 1
[1,3] => 2 = 1 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 2 = 1 + 1
[3,1] => 1 = 0 + 1
[4] => 1 = 0 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 2 = 1 + 1
[1,1,2,1] => 2 = 1 + 1
[1,1,3] => 2 = 1 + 1
[1,2,1,1] => 2 = 1 + 1
[1,2,2] => 2 = 1 + 1
[1,3,1] => 2 = 1 + 1
[1,4] => 2 = 1 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 2 = 1 + 1
[2,2,1] => 2 = 1 + 1
[2,3] => 2 = 1 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 2 = 1 + 1
[4,1] => 1 = 0 + 1
[5] => 1 = 0 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 2 = 1 + 1
[1,1,1,2,1] => 2 = 1 + 1
[1,1,1,3] => 2 = 1 + 1
[1,1,2,1,1] => 2 = 1 + 1
[1,1,2,2] => 2 = 1 + 1
[1,1,3,1] => 2 = 1 + 1
[1,1,4] => 2 = 1 + 1
[1,2,1,1,1] => 2 = 1 + 1
[1,2,1,2] => 2 = 1 + 1
[1,2,2,1] => 2 = 1 + 1
[1,2,3] => 3 = 2 + 1
[1,3,1,1] => 2 = 1 + 1
[1,3,2] => 2 = 1 + 1
[1,4,1] => 2 = 1 + 1
[1,5] => 2 = 1 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 2 = 1 + 1
[2,1,2,1] => 2 = 1 + 1
Description
The length of the longest staircase fitting into an integer composition.
For a given composition $c_1,\dots,c_n$, this is the maximal number $\ell$ such that there are indices $i_1 < \dots < i_\ell$ with $c_{i_k} \geq k$, see [def.3.1, 1]
Matching statistic: St001221
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001221: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001221: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
[1,4,1] => [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
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
Description
The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module.
Matching statistic: St000862
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [1,2] => 1 = 0 + 1
[2] => [1,1,0,0]
=> [2,1] => 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2 = 1 + 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2 = 1 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2 = 1 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 1 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2 = 1 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2 = 1 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 2 = 1 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 2 = 1 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 3 = 2 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2 = 1 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 2 = 1 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2 = 1 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2 = 1 + 1
Description
The number of parts of the shifted shape of a permutation.
The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing.
The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled.
This statistic records the number of parts of the shifted shape.
Matching statistic: St000024
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,1] => [1,1]
=> [1]
=> [1,0]
=> 0
[2] => [2]
=> []
=> []
=> 0
[1,1,1] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2] => [2,1]
=> [1]
=> [1,0]
=> 0
[2,1] => [2,1]
=> [1]
=> [1,0]
=> 0
[3] => [3]
=> []
=> []
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,3] => [3,1]
=> [1]
=> [1,0]
=> 0
[2,1,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,1] => [3,1]
=> [1]
=> [1,0]
=> 0
[4] => [4]
=> []
=> []
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,4] => [4,1]
=> [1]
=> [1,0]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,1,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [4,1]
=> [1]
=> [1,0]
=> 0
[5] => [5]
=> []
=> []
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,5] => [5,1]
=> [1]
=> [1,0]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
Description
The number of double up and double down steps of a Dyck path.
In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
Matching statistic: St000052
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1,0]
=> 0
[1,1] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000358
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000358: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[1,1] => [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2] => [2]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[3] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 1
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => 1
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 1
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [2,3,5,6,1,4] => 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => 1
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
Description
The number of occurrences of the pattern 31-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $31\!\!-\!\!2$.
Matching statistic: St000473
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000473: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> []
=> []
=> 0
[1,1] => [1,1]
=> [1]
=> [1]
=> 0
[2] => [2]
=> []
=> []
=> 0
[1,1,1] => [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,2] => [2,1]
=> [1]
=> [1]
=> 0
[2,1] => [2,1]
=> [1]
=> [1]
=> 0
[3] => [3]
=> []
=> []
=> 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,3] => [3,1]
=> [1]
=> [1]
=> 0
[2,1,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1
[2,2] => [2,2]
=> [2]
=> [1,1]
=> 0
[3,1] => [3,1]
=> [1]
=> [1]
=> 0
[4] => [4]
=> []
=> []
=> 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,4] => [4,1]
=> [1]
=> [1]
=> 0
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,3] => [3,2]
=> [2]
=> [1,1]
=> 0
[3,1,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1
[3,2] => [3,2]
=> [2]
=> [1,1]
=> 0
[4,1] => [4,1]
=> [1]
=> [1]
=> 0
[5] => [5]
=> []
=> []
=> 0
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [2]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [2]
=> 1
[1,5] => [5,1]
=> [1]
=> [1]
=> 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
Description
The number of parts of a partition that are strictly bigger than the number of ones.
This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000647
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000647: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0]
=> [1] => 0
[1,1] => [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[2] => [2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[4] => [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,1,1,1] => [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
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1
[1,1,4] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[1,4,1] => [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1
Description
The number of big descents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i)-\pi(i+1) > 1$.
The generating functions of big descents is equal to the generating function of (normal) descents after sending a permutation from cycle to one-line notation [[Mp00090]], see [Theorem 2.5, 1].
For the number of small descents, see [[St000214]].
Matching statistic: St000660
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000660: 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]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1] => [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
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
Description
The number of rises of length at least 3 of a Dyck path.
The number of Dyck paths without such rises are counted by the Motzkin numbers [1].
Matching statistic: St000697
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00154: Graphs —core⟶ Graphs
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
St000697: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => ([],1)
=> ([],1)
=> [1]
=> 0
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> [2]
=> 0
[2] => ([],2)
=> ([],1)
=> [1]
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [2]
=> 0
[3] => ([],3)
=> ([],1)
=> [1]
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3] => ([(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,2] => ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [2]
=> 0
[4] => ([],4)
=> ([],1)
=> [1]
=> 0
[1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4] => ([(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> 0
[2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[2,3] => ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> 0
[3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> 0
[4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [2]
=> 0
[5] => ([],5)
=> ([],1)
=> [1]
=> 0
[1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [6]
=> 2
[1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1
[1,5] => ([(4,5)],6)
=> ([(0,1)],2)
=> [2]
=> 0
[2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1
[2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
[2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [4]
=> 1
Description
The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $3$-rim hooks that are removed in this process to obtain a $3$-core.
The following 177 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000944The 3-degree of an integer partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001727The number of invisible inversions of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000486The number of cycles of length at least 3 of a permutation. St000661The number of rises of length 3 of a Dyck path. St000710The number of big deficiencies of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001335The cardinality of a minimal cycle-isolating set of a graph. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001871The number of triconnected components of a graph. St000619The number of cyclic descents of a permutation. St001093The detour number of a graph. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000147The largest part of an integer partition. St000378The diagonal inversion number of an integer partition. St001280The number of parts of an integer partition that are at least two. St000256The number of parts from which one can substract 2 and still get an integer partition. St000143The largest repeated part of a partition. St001330The hat guessing number of a graph. St001556The number of inversions of the third entry of a permutation. St001469The holeyness of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000183The side length of the Durfee square of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000897The number of different multiplicities of parts of an integer partition. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000662The staircase size of the code of a permutation. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000470The number of runs in a permutation. St001394The genus of a permutation. St001597The Frobenius rank of a skew partition. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000259The diameter of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000659The number of rises of length at least 2 of a Dyck path. St000914The sum of the values of the Möbius function of a poset. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001890The maximum magnitude of the Möbius function of a poset. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000021The number of descents of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St000325The width of the tree associated to a permutation. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001845The number of join irreducibles minus the rank of a lattice. St000137The Grundy value of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000035The number of left outer peaks of a permutation. St000782The indicator function of whether a given perfect matching is an L & P matching. St001651The Frankl number of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000456The monochromatic index of a connected graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000807The sum of the heights of the valleys of the associated bargraph. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001432The order dimension of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000761The number of ascents in an integer composition. St000805The number of peaks of the associated bargraph. 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. St001729The number of visible descents of a permutation. St001928The number of non-overlapping descents in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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. St001095The number of non-isomorphic posets with precisely one further covering relation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St000091The descent variation of a composition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000562The number of internal points of a set partition. St000709The number of occurrences of 14-2-3 or 14-3-2. St000872The number of very big descents of a permutation. 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. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001130The number of two successive successions in a permutation. St001470The cyclic holeyness of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001857The number of edges in the reduced word graph of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001344The neighbouring number of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001896The number of right descents of a signed permutations. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001964The interval resolution global dimension of a poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000879The number of long braid edges in the graph of braid moves of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph.
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