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Your data matches 210 different statistics following compositions of up to 3 maps.
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Matching statistic: St000142
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 1
Description
The number of even parts of a partition.
Matching statistic: St000157
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [[1,3],[2]]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [[1,3,4],[2]]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [[1,3,4,5,6],[2]]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [[1,3,4],[2]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [[1,2],[3,4]]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [[1,3,4,5,6,7],[2]]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [[1,3,4,5,6],[2]]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [[1,3,4,5,6,7,8],[2]]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [[1,3,5],[2,4]]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [[1,3,4,5,6,7,8,9],[2]]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [[1,2,5,6],[3,4]]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [[1,2,4],[3,5]]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [[1,2,3,4],[5,6]]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [[1,2,3,4],[5,6]]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9,10],[2]]
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [[1,2,3],[4,5,6]]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [[1,2,3,5],[4,6]]
=> 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [[1,2,3,4],[5,6]]
=> 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [[1,2,5,6,7],[3,4]]
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [[1,2,5,6],[3,4]]
=> 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [[1,2,4],[3,5,6]]
=> 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [[1,2,3],[4,5,6]]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [[1,2,3,6],[4,5]]
=> 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [[1,2,3,4,5],[6,7]]
=> 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [[1,2,5],[3,4,6]]
=> 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [[1,2,3],[4,5,6]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000183
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000660
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000660: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[2]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 2
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> 2
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,1,0,0,0,0]
=> 2
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,1,0,0,0,0]
=> 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> 1
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> 2
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0]
=> 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> 1
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> 2
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0]
=> 2
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> 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: St001251
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 1
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001252
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 1
Description
Half the sum of the even parts of a partition.
Matching statistic: St000321
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 3 = 2 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The number of integer partitions of n that are dominated by an integer partition.
A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000345
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 3 = 2 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The number of refinements of a partition.
A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000935
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000935: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [2,1] => [2]
=> 2 = 1 + 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> 2 = 1 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> 2 = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> 2 = 1 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => [2,1]
=> 2 = 1 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> 2 = 1 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,1,1]
=> 2 = 1 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> 2 = 1 + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [2,2]
=> 3 = 2 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> 2 = 1 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [2,1,1,1,1]
=> 2 = 1 + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => [2,2,1]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,2]
=> 3 = 2 + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [2,1,1,1]
=> 2 = 1 + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => [2,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [2,2,1]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [2,2,1]
=> 3 = 2 + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [2,1,1,1]
=> 2 = 1 + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [2,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [2,2,1]
=> 3 = 2 + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => [2,1,1,1,1,1,1,1,1]
=> 2 = 1 + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [2,1,1,1]
=> 2 = 1 + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => [2,2,1,1]
=> 3 = 2 + 1
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => [2,1,1,1,1]
=> 2 = 1 + 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => [2,1,1,1,1]
=> 2 = 1 + 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => [2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => [2,1,1,1,1,1]
=> 2 = 1 + 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => [2,2,1,1]
=> 3 = 2 + 1
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => [2,1,1,1,1]
=> 2 = 1 + 1
Description
The number of ordered refinements of an integer partition.
This is, for an integer partition $\mu = (\mu_1,\ldots,\mu_n)$ the number of integer partition $\lambda = (\lambda_1,\ldots,\lambda_m)$ such that there are indices $1 = a_0 < \ldots < a_n = m$ with $\mu_j = \lambda_{a_{j-1}} + \ldots + \lambda_{a_j-1}$.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 98%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0,1,0]
=> [2,1] => 1 => 1
[2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 01 => 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 01 => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 001 => 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,1,3] => 10 => 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 001 => 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0001 => 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 010 => 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 001 => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 010 => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0001 => 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5] => 00001 => 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 0010 => 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 011 => 2
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 100 => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 001 => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 0010 => 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => 00001 => 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6] => 000001 => 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,4,6] => 00010 => 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 0011 => 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0001 => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 101 => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0001 => 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 0001 => 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7] => 0000001 => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 0011 => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 0001 => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 0011 => 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 0101 => 2
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0001 => 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => 00000001 => 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [5,6,1,2,3,4] => 00001 => 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0010 => 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 0011 => 2
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,1,2] => 00001 => 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,3,5,6,1,4] => 00001 => 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9] => 000000001 => 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,5,6,1,2,3] => 00001 => 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0010 => 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [3,4,5,1,6,2] => 00011 => 2
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,4,5,6,1,3] => 00001 => 1
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [6,7,1,2,3,4,5] => 000001 => 1
[4,4,2]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> [4,5,1,2,3,6] => 00010 => 1
[4,3,3]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> [4,5,1,6,2,3] => 00011 => 2
[3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,5,6,1,2,4] => 00001 => 1
[3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> [3,4,5,1,2,6] => 00010 => 1
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [3,4,5,6,7,1,2] => 000001 => 1
[4,4,3]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,5,1,2,6,3] => 00011 => 2
[3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,4,6,1,2,5] => 00001 => 1
[]
=> []
=> [] => => ? = 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
The following 200 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000993The multiplicity of the largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St001657The number of twos in an integer partition. St001175The size of a partition minus the hook length of the base cell. St001389The number of partitions of the same length below the given integer partition. St000862The number of parts of the shifted shape of a permutation. St000884The number of isolated descents of a permutation. St000035The number of left outer peaks of a permutation. St000201The number of leaf nodes in a binary tree. St000758The length of the longest staircase fitting into an integer composition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000396The register function (or Horton-Strahler number) of a binary tree. St000662The staircase size of the code of a permutation. St001732The number of peaks visible from the left. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000919The number of maximal left branches of a binary tree. St000254The nesting number of a set partition. St000872The number of very big descents of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000664The number of right ropes of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000092The number of outer peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St001427The number of descents of a signed permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000021The number of descents of a permutation. St001665The number of pure excedances of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St000355The number of occurrences of the pattern 21-3. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000619The number of cyclic descents of a permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St000023The number of inner peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001960The number of descents of a permutation minus one if its first entry is not one. St001715The number of non-records in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001811The Castelnuovo-Mumford regularity of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St001520The number of strict 3-descents. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001621The number of atoms of a lattice. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001946The number of descents in a parking function. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001771The number of occurrences of the signed pattern 1-2 in 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. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St000068The number of minimal elements in a poset. St001410The minimal entry of a semistandard tableau. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. 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 $eA$. St001720The minimal length of a chain of small intervals in a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000056The decomposition (or block) number of a permutation. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000729The minimal arc length of a set partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001806The upper middle entry of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000039The number of crossings of a permutation. St000084The number of subtrees. St000091The descent variation of a composition. St000105The number of blocks in the set partition. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000317The cycle descent number of a permutation. St000328The maximum number of child nodes in a tree. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000462The major index minus the number of excedences of a permutation. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000516The number of stretching pairs of a permutation. St000542The number of left-to-right-minima of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000823The number of unsplittable factors of the set partition. St000962The 3-shifted major index of a permutation. St000989The number of final rises of a permutation. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000879The number of long braid edges in the graph of braid moves of a permutation. St001624The breadth of a lattice.
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