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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
[1,1] => 1
[2] => 1
[1,1,1] => 1
[1,2] => 2
[2,1] => 1
[3] => 1
[1,1,1,1] => 1
[1,1,2] => 2
[1,2,1] => 2
[1,3] => 2
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 1
[1,1,1,2] => 2
[1,1,2,1] => 2
[1,1,3] => 2
[1,2,1,1] => 2
[1,2,2] => 2
[1,3,1] => 2
[1,4] => 2
[2,1,1,1] => 1
[2,1,2] => 2
[2,2,1] => 2
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 2
[1,1,1,2,1] => 2
[1,1,1,3] => 2
[1,1,2,1,1] => 2
[1,1,2,2] => 2
[1,1,3,1] => 2
[1,1,4] => 2
[1,2,1,1,1] => 2
[1,2,1,2] => 2
[1,2,2,1] => 2
[1,2,3] => 3
[1,3,1,1] => 2
[1,3,2] => 2
[1,4,1] => 2
[1,5] => 2
[2,1,1,1,1] => 1
[2,1,1,2] => 2
[2,1,2,1] => 2
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: 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
[1,1] => [1,0,1,0]
=> [1,2] => 1
[2] => [1,1,0,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 1
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 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
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 2
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 2
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => 2
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 2
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 3
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
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: 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
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[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
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 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 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 3 - 1
[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 = 2 - 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 = 2 - 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 = 1 - 1
[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
[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
[2,1,2,1] => [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 = 2 - 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: St000760
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000760: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => [1] => 1
[1,1] => 11 => 11 => [2] => 1
[2] => 10 => 10 => [1,1] => 1
[1,1,1] => 111 => 111 => [3] => 1
[1,2] => 110 => 110 => [2,1] => 2
[2,1] => 101 => 101 => [1,1,1] => 1
[3] => 100 => 010 => [1,1,1] => 1
[1,1,1,1] => 1111 => 1111 => [4] => 1
[1,1,2] => 1110 => 1110 => [3,1] => 2
[1,2,1] => 1101 => 1101 => [2,1,1] => 2
[1,3] => 1100 => 1010 => [1,1,1,1] => 1
[2,1,1] => 1011 => 1011 => [1,1,2] => 1
[2,2] => 1010 => 0110 => [1,2,1] => 2
[3,1] => 1001 => 0101 => [1,1,1,1] => 1
[4] => 1000 => 0010 => [2,1,1] => 2
[1,1,1,1,1] => 11111 => 11111 => [5] => 1
[1,1,1,2] => 11110 => 11110 => [4,1] => 2
[1,1,2,1] => 11101 => 11101 => [3,1,1] => 2
[1,1,3] => 11100 => 11010 => [2,1,1,1] => 2
[1,2,1,1] => 11011 => 11011 => [2,1,2] => 2
[1,2,2] => 11010 => 10110 => [1,1,2,1] => 2
[1,3,1] => 11001 => 10101 => [1,1,1,1,1] => 1
[1,4] => 11000 => 01010 => [1,1,1,1,1] => 1
[2,1,1,1] => 10111 => 10111 => [1,1,3] => 1
[2,1,2] => 10110 => 01110 => [1,3,1] => 2
[2,2,1] => 10101 => 01101 => [1,2,1,1] => 2
[2,3] => 10100 => 10010 => [1,2,1,1] => 2
[3,1,1] => 10011 => 01011 => [1,1,1,2] => 1
[3,2] => 10010 => 00110 => [2,2,1] => 2
[4,1] => 10001 => 00101 => [2,1,1,1] => 2
[5] => 10000 => 00010 => [3,1,1] => 2
[1,1,1,1,1,1] => 111111 => 111111 => [6] => 1
[1,1,1,1,2] => 111110 => 111110 => [5,1] => 2
[1,1,1,2,1] => 111101 => 111101 => [4,1,1] => 2
[1,1,1,3] => 111100 => 111010 => [3,1,1,1] => 2
[1,1,2,1,1] => 111011 => 111011 => [3,1,2] => 2
[1,1,2,2] => 111010 => 110110 => [2,1,2,1] => 2
[1,1,3,1] => 111001 => 110101 => [2,1,1,1,1] => 2
[1,1,4] => 111000 => 101010 => [1,1,1,1,1,1] => 1
[1,2,1,1,1] => 110111 => 110111 => [2,1,3] => 2
[1,2,1,2] => 110110 => 101110 => [1,1,3,1] => 2
[1,2,2,1] => 110101 => 101101 => [1,1,2,1,1] => 2
[1,2,3] => 110100 => 011010 => [1,2,1,1,1] => 2
[1,3,1,1] => 110011 => 101011 => [1,1,1,1,2] => 1
[1,3,2] => 110010 => 010110 => [1,1,1,2,1] => 2
[1,4,1] => 110001 => 010101 => [1,1,1,1,1,1] => 1
[1,5] => 110000 => 001010 => [2,1,1,1,1] => 2
[2,1,1,1,1] => 101111 => 101111 => [1,1,4] => 1
[2,1,1,2] => 101110 => 011110 => [1,4,1] => 2
[2,1,2,1] => 101101 => 011101 => [1,3,1,1] => 2
Description
The length of the longest strictly decreasing subsequence of parts of an integer composition.
By the Greene-Kleitman theorem, regarding the composition as a word, this is the length of the partition associated by the Robinson-Schensted-Knuth correspondence.
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]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[2] => [2]
=> []
=> []
=> 0 = 1 - 1
[1,1,1] => [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2] => [2,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[2,1] => [2,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[3] => [3]
=> []
=> []
=> 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,3] => [3,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[2,2] => [2,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,1] => [3,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[4] => [4]
=> []
=> []
=> 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,4] => [4,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[2,3] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[3,2] => [3,2]
=> [2]
=> [1,0,1,0]
=> 0 = 1 - 1
[4,1] => [4,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[5] => [5]
=> []
=> []
=> 0 = 1 - 1
[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 = 2 - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1 = 2 - 1
[1,5] => [5,1]
=> [1]
=> [1,0]
=> 0 = 1 - 1
[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
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 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]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2] => [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 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 = 2 - 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 = 1 - 1
[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
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 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 - 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[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 = 2 - 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[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 = 1 - 1
[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]
=> [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 = 2 - 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 = 2 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 1 - 1
[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
[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
[2,1,2,1] => [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 = 2 - 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]
=> [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[2] => [2]
=> [1,0,1,0]
=> [2,1] => 0 = 1 - 1
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1 = 2 - 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[3] => [3]
=> [1,0,1,0,1,0]
=> [2,3,1] => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1 = 2 - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1 = 2 - 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 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 = 2 - 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 = 2 - 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 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 = 2 - 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0 = 1 - 1
[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
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1 = 2 - 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1 = 2 - 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [3,4,5,6,1,2] => 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 2 - 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 = 2 - 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 1 = 2 - 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 = 2 - 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0 = 1 - 1
[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
[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
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1 = 2 - 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]
=> [1]
=> [1]
=> 0 = 1 - 1
[2] => [2]
=> []
=> []
=> 0 = 1 - 1
[1,1,1] => [1,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,2] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1] => [2,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[3] => [3]
=> []
=> []
=> 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,1,2] => [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,2,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,3] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[2,2] => [2,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[3,1] => [3,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[4] => [4]
=> []
=> []
=> 0 = 1 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,1,1,2] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,1,2,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,1,3] => [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,2,1,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,2,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,4] => [4,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1,1,1] => [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[2,1,2] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,2,1] => [2,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,3] => [3,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[3,1,1] => [3,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[3,2] => [3,2]
=> [2]
=> [1,1]
=> 0 = 1 - 1
[4,1] => [4,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[5] => [5]
=> []
=> []
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 1 = 2 - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,1,1,3] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,1,2,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,1,3,1] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,1,4] => [4,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[1,2,1,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,2,2,1] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[1,2,3] => [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,3,1,1] => [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1 = 2 - 1
[1,3,2] => [3,2,1]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[1,4,1] => [4,1,1]
=> [1,1]
=> [2]
=> 1 = 2 - 1
[1,5] => [5,1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 1 = 2 - 1
[2,1,1,2] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[2,1,2,1] => [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 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]
=> [1,1,0,0]
=> [2,1] => 0 = 1 - 1
[2] => [2]
=> [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[1,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3] => [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1 = 2 - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1 = 2 - 1
[2,2] => [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1 = 2 - 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 = 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 = 2 - 1
[1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 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 = 2 - 1
[1,2,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[1,3,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[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
[2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,2,1] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1 = 2 - 1
[2,3] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1 = 2 - 1
[3,2] => [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 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]
=> [1,3,4,5,6,2] => 1 = 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 = 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 = 2 - 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 = 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 = 2 - 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 = 2 - 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 = 2 - 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 = 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 = 2 - 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 = 2 - 1
[1,2,3] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 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 = 2 - 1
[1,3,2] => [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1 = 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 = 2 - 1
[1,5] => [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0 = 1 - 1
[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
[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
[2,1,2,1] => [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 1 = 2 - 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
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 2 - 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1] => [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
[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 - 1
[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 = 2 - 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 = 1 - 1
[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 - 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 = 2 - 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 = 2 - 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
[1,1,1,1,1] => [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
[1,1,1,2] => [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
[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 = 2 - 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 - 1
[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 = 2 - 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 = 2 - 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 = 2 - 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 = 1 - 1
[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
[2,1,2] => [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 - 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 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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
[1,1,1,1,1,1] => [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
[1,1,1,1,2] => [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
[1,1,1,2,1] => [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 = 2 - 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
[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 = 2 - 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 = 2 - 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 = 2 - 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 - 1
[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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 2 - 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 = 1 - 1
[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
[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
[2,1,2,1] => [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 = 2 - 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].
The following 117 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. 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. St000619The number of cyclic descents of a permutation. 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. St001093The detour number of a graph. 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. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St000443The number of long tunnels of a Dyck path. St000759The smallest missing part in an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001814The number of partitions interlacing the given partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St001330The hat guessing number of a graph. St000143The largest repeated part of a partition. St001469The holeyness of a permutation. St001556The number of inversions of the third entry of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000444The length of the maximal rise of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000662The staircase size of the code of a permutation. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St000470The number of runs in a permutation. St001597The Frobenius rank of a skew partition. St001394The genus of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000260The radius of a connected graph. St001642The Prague dimension of a graph. St000822The Hadwiger number of the graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000021The number of descents of a permutation. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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 $eA$. St001845The number of join irreducibles minus the rank of a lattice. St000035The number of left outer peaks of a permutation. 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. St001960The number of descents of a permutation minus one if its first entry is not one. St000460The hook length of the last cell along the main diagonal of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000761The number of ascents in an integer composition. St000805The number of peaks of the associated bargraph. St001487The number of inner corners of a skew partition. 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001928The number of non-overlapping descents in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000668The least common multiple of the parts of the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St001128The exponens consonantiae of a partition. St001344The neighbouring number of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001896The number of right descents of a signed permutations. St000091The descent variation of a composition. 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. 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. 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. 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. St001964The interval resolution global dimension of a poset. St001645The pebbling number of a connected graph. 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.
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