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Your data matches 129 different statistics following compositions of up to 3 maps.
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Matching statistic: St000023
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(load all 12 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000023: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => 0
[[1,2],[3]]
=> [3,1,2] => 0
[[1],[2],[3]]
=> [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 1
Description
The number of inner peaks of a permutation.
The number of peaks including the boundary is [[St000092]].
Matching statistic: St000099
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(load all 12 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => 1 = 0 + 1
[[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => 1 = 0 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => 1 = 0 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1 = 0 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 2 = 1 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 1 = 0 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 1 = 0 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 1 = 0 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1 = 0 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 1 = 0 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 1 = 0 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 2 = 1 + 1
Description
The number of valleys of a permutation, including the boundary.
The number of valleys excluding the boundary is [[St000353]].
Matching statistic: St001086
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00064: Permutations —reverse⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 0
[[1,2]]
=> [1,2] => [2,1] => 0
[[1],[2]]
=> [2,1] => [1,2] => 0
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 0
[[1,3],[2]]
=> [2,1,3] => [3,1,2] => 0
[[1,2],[3]]
=> [3,1,2] => [2,1,3] => 0
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [4,3,1,2] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [4,2,1,3] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [3,2,1,4] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [4,1,2,3] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,2,4] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [2,1,3,4] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5,4,3,1,2] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5,4,2,1,3] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5,3,2,1,4] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [4,3,2,1,5] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5,3,1,4,2] => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5,2,1,4,3] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,3,1,5,2] => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [4,2,1,5,3] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,2,1,5,4] => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [5,4,1,2,3] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [5,3,1,2,4] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [5,2,1,3,4] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,3,1,2,5] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [4,2,1,3,5] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2,1,4,5] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1,5,2,3] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [3,1,5,2,4] => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,1,5,3,4] => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [3,1,4,2,5] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,1,4,3,5] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [5,1,2,3,4] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,1,2,3,5] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [3,1,2,4,5] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,1,3,4,5] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6,5,4,3,1,2] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6,5,4,2,1,3] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6,5,3,2,1,4] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6,4,3,2,1,5] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [5,4,3,2,1,6] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6,5,3,1,4,2] => 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001251
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001251: 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
St001251: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> []
=> 0
[[1,2]]
=> [2]
=> []
=> 0
[[1],[2]]
=> [1,1]
=> [1]
=> 0
[[1,2,3]]
=> [3]
=> []
=> 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3,4]]
=> [4]
=> []
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5]]
=> [5]
=> []
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> 1
Description
The number of parts of a partition that are not congruent 1 modulo 3.
Matching statistic: St001840
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00171: Set partitions —intertwining number to dual major index⟶ Set partitions
St001840: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> {{1}}
=> {{1}}
=> 0
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 0
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 0
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 0
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 0
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 0
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,4},{2,3}}
=> 1
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 0
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1},{2,3,4}}
=> 0
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 0
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> 0
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 0
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,4,5},{2,3}}
=> 1
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,5},{3,4}}
=> 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 0
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1},{2,3,4,5}}
=> 0
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,2,4},{3,5}}
=> 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,2},{3,4,5}}
=> 0
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,5},{2},{3,4}}
=> 1
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1},{2,3,5},{4}}
=> 1
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 0
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1,4},{2,3},{5}}
=> 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> 0
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1},{2},{3,4,5}}
=> 0
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1},{2,3},{4,5}}
=> 0
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,3,4},{5}}
=> 0
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 0
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2},{3,4},{5}}
=> 0
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> 0
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 0
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,4,5,6},{2,3}}
=> 1
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,5,6},{3,4}}
=> 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,3,6},{4,5}}
=> 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> 0
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 0
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,6},{2,3,4,5}}
=> 1
Description
The number of descents of a set partition.
The Mahonian representation of a set partition $\{B_1,\dots,B_k\}$ of $\{1,\dots,n\}$ is the restricted growth word $w_1\dots w_n\}$ obtained by sorting the blocks of the set partition according to their maximal element, and setting $w_i$ to the index of the block containing $i$.
The word $w$ has a descent at position $i$ if $w_i > w_{i+1}$.
Matching statistic: St000092
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 = 0 + 1
[[1,2]]
=> [1,2] => [1,2] => 1 = 0 + 1
[[1],[2]]
=> [2,1] => [2,1] => 1 = 0 + 1
[[1,2,3]]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 1 = 0 + 1
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1 = 0 + 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => 1 = 0 + 1
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[[1,3,4],[2]]
=> [2,1,3,4] => [2,3,4,1] => 1 = 0 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3,4,2] => 1 = 0 + 1
[[1,2,3],[4]]
=> [4,1,2,3] => [1,2,4,3] => 1 = 0 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,4,2,1] => 1 = 0 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => 1 = 0 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,4,3,2] => 1 = 0 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,3,4,5,1] => 1 = 0 + 1
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,4,5,2] => 1 = 0 + 1
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,2,4,5,3] => 1 = 0 + 1
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,2,3,5,4] => 1 = 0 + 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,4,5,1,3] => 2 = 1 + 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,4,5,1,2] => 2 = 1 + 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,3,5,1,4] => 2 = 1 + 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [1,4,5,2,3] => 2 = 1 + 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => 1 = 0 + 1
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => 1 = 0 + 1
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4,5,3,2] => 1 = 0 + 1
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,3,5,4,1] => 1 = 0 + 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1 = 0 + 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 0 + 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,5,2,4,1] => 2 = 1 + 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [4,5,2,3,1] => 2 = 1 + 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,5,1,3,2] => 2 = 1 + 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,5,1,4,3] => 2 = 1 + 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [3,5,1,4,2] => 2 = 1 + 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,5,3,2,1] => 1 = 0 + 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,4,2,1] => 1 = 0 + 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,3,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,5,4,3,2] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1 = 0 + 1
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,3,4,5,6,1] => 1 = 0 + 1
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,4,5,6,2] => 1 = 0 + 1
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,2,4,5,6,3] => 1 = 0 + 1
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,2,3,5,6,4] => 1 = 0 + 1
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,2,3,4,6,5] => 1 = 0 + 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4,5,6,1,3] => 2 = 1 + 1
Description
The number of outer peaks of a permutation.
An outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$ or $n$ if $w_{n} > w_{n-1}$.
In other words, it is a peak in the word $[0,w_1,..., w_n,0]$.
Matching statistic: St000052
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ 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
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ 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
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[[1,2,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[[1,2,3,4,5]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 0
[[1,3,4,5,6],[2]]
=> [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
[[1,2,4,5,6],[3]]
=> [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
[[1,2,3,5,6],[4]]
=> [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
[[1,2,3,4,6],[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
[[1,2,3,4,5],[6]]
=> [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
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
Description
The number of valleys of a Dyck path not on the x-axis.
That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Matching statistic: St000196
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00018: Binary trees —left border symmetry⟶ Binary trees
St000196: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [.,.]
=> [.,.]
=> 0
[[1,2]]
=> [1,2] => [.,[.,.]]
=> [.,[.,.]]
=> 0
[[1],[2]]
=> [2,1] => [[.,.],.]
=> [[.,.],.]
=> 0
[[1,2,3]]
=> [1,2,3] => [.,[.,[.,.]]]
=> [.,[.,[.,.]]]
=> 0
[[1,3],[2]]
=> [2,1,3] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[[1,2],[3]]
=> [3,1,2] => [[.,.],[.,.]]
=> [[.,[.,.]],.]
=> 0
[[1],[2],[3]]
=> [3,2,1] => [[[.,.],.],.]
=> [[[.,.],.],.]
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [.,[.,[.,[.,.]]]]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[1,2,4],[3]]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[1,2,3],[4]]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[.,[.,[.,.]]],.]
=> 0
[[1,3],[2,4]]
=> [2,4,1,3] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[.,[.,.]],[.,.]]
=> 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[[.,[.,.]],.],.]
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[[[.,.],.],.],.]
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [.,[.,[.,[.,[.,.]]]]]
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [[.,.],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],.]
=> 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [[.,[.,.]],[.,[.,.]]]
=> [[.,[.,[.,.]]],[.,.]]
=> 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [[[.,.],.],[.,[.,.]]]
=> [[[.,[.,[.,.]]],.],.]
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [[[.,.],[.,.]],[.,.]]
=> [[[.,[.,.]],[.,.]],.]
=> 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [[[[.,.],.],.],[.,.]]
=> [[[[.,[.,.]],.],.],.]
=> 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
=> [[[[[.,.],.],.],.],.]
=> 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> [.,[.,[.,[.,[.,[.,.]]]]]]
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [[.,.],[.,[.,[.,[.,.]]]]]
=> [[.,[.,[.,[.,[.,.]]]]],.]
=> 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
=> [[.,[.,[.,[.,.]]]],[.,.]]
=> 1
Description
The number of occurrences of the contiguous pattern {{{[[.,.],[.,.]]}}} in a binary tree.
Equivalently, this is the number of branches in the tree, i.e. the number of nodes with two children. Binary trees avoiding this pattern are counted by $2^{n-2}$.
Matching statistic: St000223
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000223: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1]
=> [1,0]
=> [1] => 0
[[1,2]]
=> [2]
=> [1,0,1,0]
=> [1,2] => 0
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,1] => 0
[[1,2,3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1,2],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[[1,2,3,4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,2,3],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[[1,2,3,4,5]]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[[1,3,5],[2,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[1,2,5],[3,4]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[1,3,4],[2,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[1,2,4],[3,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[1,2,3],[4,5]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 0
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[[1,2,3,4,5,6]]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
Description
The number of nestings in the permutation.
Matching statistic: St000292
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => 1 => 0
[[1,2]]
=> [1,2] => [2] => 10 => 0
[[1],[2]]
=> [2,1] => [1,1] => 11 => 0
[[1,2,3]]
=> [1,2,3] => [3] => 100 => 0
[[1,3],[2]]
=> [2,1,3] => [1,2] => 110 => 0
[[1,2],[3]]
=> [3,1,2] => [1,2] => 110 => 0
[[1],[2],[3]]
=> [3,2,1] => [1,1,1] => 111 => 0
[[1,2,3,4]]
=> [1,2,3,4] => [4] => 1000 => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [1,3] => 1100 => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [1,3] => 1100 => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [1,3] => 1100 => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [2,2] => 1010 => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,2] => 1010 => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,2] => 1110 => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,2] => 1110 => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,2] => 1110 => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => 1111 => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5] => 10000 => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,4] => 11000 => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,4] => 11000 => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [1,4] => 11000 => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,4] => 11000 => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [2,3] => 10100 => 1
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,3] => 10100 => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [2,3] => 10100 => 1
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [2,3] => 10100 => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,3] => 10100 => 1
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,1,3] => 11100 => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [1,1,3] => 11100 => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,1,3] => 11100 => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,1,3] => 11100 => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,1,3] => 11100 => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,1,3] => 11100 => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [1,2,2] => 11010 => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [1,2,2] => 11010 => 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,2,2] => 11010 => 1
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,2,2] => 11010 => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,2,2] => 11010 => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,1,1,2] => 11110 => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,1,1,2] => 11110 => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,1,1,2] => 11110 => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,1,1,2] => 11110 => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,1,1,1,1] => 11111 => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6] => 100000 => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,5] => 110000 => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,5] => 110000 => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [1,5] => 110000 => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [1,5] => 110000 => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,5] => 110000 => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [2,4] => 101000 => 1
Description
The number of ascents of a binary word.
The following 119 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000682The Grundy value of Welter's game on a binary word. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001689The number of celebrities in a graph. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001777The number of weak descents in an integer composition. St001839The number of excedances of a set partition. St000201The number of leaf nodes in a binary tree. St000346The number of coarsenings of a partition. St000390The number of runs of ones in a binary word. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000353The number of inner valleys of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000568The hook number of a binary tree. St000291The number of descents of a binary word. St000486The number of cycles of length at least 3 of a permutation. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000872The number of very big descents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001520The number of strict 3-descents. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000934The 2-degree of an integer partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001498The normalised height of a Nakayama algebra with magnitude 1. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000929The constant term of the character polynomial of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000455The second largest eigenvalue of a graph if it is integral. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. 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 $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001427The number of descents of a signed permutation. St001487The number of inner corners of a skew partition. St001960The number of descents of a permutation minus one if its first entry is not one. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000699The toughness times the least common multiple of 1,. St001862The number of crossings of a signed permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001964The interval resolution global dimension of a poset. St001868The number of alignments of type NE of a signed permutation. St000068The number of minimal elements in a poset. St001845The number of join irreducibles minus the rank of a lattice. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000914The sum of the values of the Möbius function of a poset. St001768The number of reduced words of a signed permutation. St001408The number of maximal entries in a semistandard tableau. St001857The number of edges in the reduced word graph of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001407The number of minimal entries in a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
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