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Your data matches 7 different statistics following compositions of up to 3 maps.
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Matching statistic: St000152
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Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000152: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000152: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [[2,1],[2]]
=> 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [[2,1],[1]]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [[3,2,1],[3,2],[3]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [[3,2,1],[3,2],[2]]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[3,2,1],[3,1],[3]]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [[3,2,1],[3,1],[2]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [[3,2,1],[2,1],[2]]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [[3,2,1],[3,1],[1]]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [[3,2,1],[2,1],[1]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[4]]
=> 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[3]]
=> 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[4]]
=> 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[3]]
=> 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[3]]
=> 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[2]]
=> 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[2]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[4]]
=> 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[3]]
=> 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[4]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[3]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,2],[3]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[2]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,2],[2]]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[4]]
=> 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[3]]
=> 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,2],[3]]
=> 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[3]]
=> 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[2]]
=> 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,2],[2]]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[2]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[4]]
=> 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[3]]
=> 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[3]]
=> 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[2]]
=> 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[2]]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[4]]
=> 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[3]]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[3]]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[3]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[2]]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[2]]
=> 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[2]]
=> 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[2,1],[2]]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[2]]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[1]]
=> 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[1]]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[1]]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[1]]
=> 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[1]]
=> 5
Description
The number of boxed plus the number of special entries.
This is the sum of statistics [[St000073]] and [[St000074]].
Matching statistic: St000173
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(load all 4 compositions to match this statistic)
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000173: Semistandard tableaux ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000173: Semistandard tableaux ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 0
[[1,0],[0,1]]
=> [[0,1],[1,0]]
=> [[1,2],[2]]
=> 1
[[0,1],[1,0]]
=> [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ? = 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> ? = 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> ? = 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> ? = 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> ? = 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> ? = 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> ? = 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> ? = 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> ? = 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> ? = 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> ? = 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 6
Description
The segment statistic of a semistandard tableau.
Let ''T'' be a tableau. A ''k''-segment of ''T'' (in the ''i''th row) is defined to be a maximal consecutive sequence of ''k''-boxes in the ith row. Note that the possible ''i''-boxes in the ''i''th row are not considered to be ''i''-segments. Then seg(''T'') is the total number of ''k''-segments in ''T'' as ''k'' varies over all possible values.
Matching statistic: St000072
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000072: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00078: Gelfand-Tsetlin patterns —Schuetzenberger involution⟶ Gelfand-Tsetlin patterns
St000072: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [[1]]
=> [[1]]
=> [[1]]
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [[2,1],[2]]
=> [[2,1],[1]]
=> 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [[2,1],[1]]
=> [[2,1],[2]]
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [[3,2,1],[3,2],[3]]
=> [[3,2,1],[2,1],[1]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [[3,2,1],[3,2],[2]]
=> [[3,2,1],[3,1],[1]]
=> 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[3,2,1],[3,1],[3]]
=> [[3,2,1],[2,1],[2]]
=> 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [[3,2,1],[3,1],[2]]
=> [[3,2,1],[2,2],[2]]
=> 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [[3,2,1],[2,1],[2]]
=> [[3,2,1],[3,1],[3]]
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [[3,2,1],[3,1],[1]]
=> [[3,2,1],[3,2],[2]]
=> 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [[3,2,1],[2,1],[1]]
=> [[3,2,1],[3,2],[3]]
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[1]]
=> ? = 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,3],[3]]
=> [[4,3,2,1],[4,2,1],[2,1],[1]]
=> ? = 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[1]]
=> ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[3]]
=> [[4,3,2,1],[3,3,1],[3,1],[1]]
=> ? = 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[3]]
=> [[4,3,2,1],[4,2,1],[4,1],[1]]
=> ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[4,2],[2]]
=> [[4,3,2,1],[4,3,1],[3,1],[1]]
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [[4,3,2,1],[4,3,2],[3,2],[2]]
=> [[4,3,2,1],[4,3,1],[4,1],[1]]
=> ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[2]]
=> ? = 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,3],[3]]
=> [[4,3,2,1],[4,2,1],[2,1],[2]]
=> ? = 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[4]]
=> [[4,3,2,1],[3,2,1],[2,2],[2]]
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[3]]
=> [[4,3,2,1],[3,2,2],[2,2],[2]]
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,2],[3]]
=> [[4,3,2,1],[4,2,1],[3,2],[2]]
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,2],[2]]
=> [[4,3,2,1],[4,2,2],[2,2],[2]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,2],[2]]
=> [[4,3,2,1],[4,2,2],[3,2],[2]]
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[3]]
=> ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[3]]
=> [[4,3,2,1],[3,3,1],[3,1],[3]]
=> ? = 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,2],[3]]
=> [[4,3,2,1],[3,3,1],[3,2],[3]]
=> ? = 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[3]]
=> [[4,3,2,1],[4,2,1],[4,1],[4]]
=> ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,2],[2]]
=> [[4,3,2,1],[4,3,1],[3,1],[3]]
=> ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,2],[2]]
=> [[4,3,2,1],[4,3,1],[3,2],[3]]
=> ? = 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[2]]
=> [[4,3,2,1],[4,3,1],[4,1],[4]]
=> ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[2]]
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[3]]
=> [[4,3,2,1],[3,2,2],[3,2],[2]]
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[3]]
=> [[4,3,2,1],[4,2,1],[4,2],[2]]
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[2]]
=> [[4,3,2,1],[3,3,2],[3,2],[2]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[2]]
=> [[4,3,2,1],[4,2,2],[4,2],[2]]
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[4]]
=> [[4,3,2,1],[3,2,1],[3,2],[3]]
=> ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[3]]
=> [[4,3,2,1],[3,2,2],[3,2],[3]]
=> ? = 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[3]]
=> [[4,3,2,1],[3,3,1],[3,3],[3]]
=> ? = 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[3]]
=> [[4,3,2,1],[4,2,1],[4,2],[4]]
=> ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[2]]
=> [[4,3,2,1],[3,3,2],[3,2],[3]]
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[2]]
=> [[4,3,2,1],[3,3,2],[3,3],[3]]
=> ? = 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[2]]
=> [[4,3,2,1],[4,2,2],[4,2],[4]]
=> ? = 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[2,1],[2]]
=> [[4,3,2,1],[4,3,1],[4,3],[3]]
=> ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[2]]
=> [[4,3,2,1],[4,3,1],[4,3],[4]]
=> ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[4,1],[1]]
=> [[4,3,2,1],[4,3,2],[3,2],[2]]
=> ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [[4,3,2,1],[4,3,1],[3,1],[1]]
=> [[4,3,2,1],[4,3,2],[4,2],[2]]
=> ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[4,1],[1]]
=> [[4,3,2,1],[4,3,2],[3,2],[3]]
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[3,1],[1]]
=> [[4,3,2,1],[4,3,2],[3,3],[3]]
=> ? = 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[3,1],[1]]
=> [[4,3,2,1],[4,3,2],[4,2],[4]]
=> ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[4,2,1],[2,1],[1]]
=> [[4,3,2,1],[4,3,2],[4,3],[3]]
=> ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [[4,3,2,1],[3,2,1],[2,1],[1]]
=> [[4,3,2,1],[4,3,2],[4,3],[4]]
=> ? = 6
Description
The number of circled entries.
asdasda
An entry of a Gelfand-Tsetlin pattern is circled if $a_{i,j} = a_{i-1,j}$ (the northeast neighbor is the same).
Matching statistic: St001083
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 19% ●values known / values provided: 19%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [[1]]
=> [1] => [1] => 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => [1,3,2] => 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => [2,3,1] => 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [1,4,6,2,5,3] => 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [3,4,6,1,5,2] => 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [1,5,6,2,4,3] => 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => [1,5,6,3,4,2] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => [4,5,6,1,3,2] => 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => [2,5,6,3,4,1] => 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => [4,5,6,2,3,1] => 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => [1,5,8,10,2,6,9,3,7,4] => ? = 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => [4,5,8,10,1,6,9,2,7,3] => ? = 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => [1,7,8,10,2,5,9,3,6,4] => ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => [1,7,8,10,4,5,9,2,6,3] => ? = 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => [6,7,8,10,1,4,9,2,5,3] => ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => [3,7,8,10,4,5,9,1,6,2] => ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => [6,7,8,10,3,4,9,1,5,2] => ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => [1,5,9,10,2,6,8,3,7,4] => ? = 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => [4,5,9,10,1,6,8,2,7,3] => ? = 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => [1,5,9,10,2,7,8,3,6,4] => ? = 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => [1,4,9,10,2,7,8,5,6,3] => ? = 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => [4,6,9,10,1,7,8,2,5,3] => ? = 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => [3,4,9,10,1,7,8,5,6,2] => ? = 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => [3,6,9,10,1,7,8,4,5,2] => ? = 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => [1,8,9,10,2,5,7,3,6,4] => ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => [1,8,9,10,4,5,7,2,6,3] => ? = 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => [1,8,9,10,4,6,7,2,5,3] => ? = 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => [7,8,9,10,1,4,6,2,5,3] => ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => [3,8,9,10,4,5,7,1,6,2] => ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => [3,8,9,10,4,6,7,1,5,2] => ? = 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => [7,8,9,10,3,4,6,1,5,2] => ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => [1,6,9,10,2,7,8,3,5,4] => ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => [1,6,9,10,2,7,8,4,5,3] => ? = 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => [5,6,9,10,1,7,8,2,4,3] => ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => [1,6,9,10,3,7,8,4,5,2] => ? = 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => [5,6,9,10,1,7,8,3,4,2] => ? = 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => [1,8,9,10,2,6,7,3,5,4] => ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => [1,8,9,10,2,6,7,4,5,3] => ? = 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => [1,8,9,10,5,6,7,2,4,3] => ? = 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => [7,8,9,10,1,5,6,2,4,3] => ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => [1,8,9,10,3,6,7,4,5,2] => ? = 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => [1,8,9,10,5,6,7,3,4,2] => ? = 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => [7,8,9,10,1,5,6,3,4,2] => ? = 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => [4,8,9,10,5,6,7,1,3,2] => ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => [7,8,9,10,4,5,6,1,3,2] => ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => [2,6,9,10,3,7,8,4,5,1] => ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => [5,6,9,10,2,7,8,3,4,1] => ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => [2,8,9,10,3,6,7,4,5,1] => ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => [2,8,9,10,5,6,7,3,4,1] => ? = 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => [7,8,9,10,2,5,6,3,4,1] => ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => [4,8,9,10,5,6,7,2,3,1] => ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => [7,8,9,10,4,5,6,2,3,1] => ? = 6
Description
The number of boxed occurrences of 132 in a permutation.
This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St000646
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 57%
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
St000646: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 17%●distinct values known / distinct values provided: 57%
Values
[[1]]
=> [[1]]
=> [[1]]
=> [1] => ? = 0
[[1,0],[0,1]]
=> [[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => 1
[[0,1],[1,0]]
=> [[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => ? = 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => ? = 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => ? = 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => ? = 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => ? = 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => ? = 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => ? = 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => ? = 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => ? = 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => ? = 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => ? = 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => ? = 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => ? = 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => ? = 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => ? = 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => ? = 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => ? = 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => ? = 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => ? = 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => ? = 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => ? = 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => ? = 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => ? = 6
Description
The number of big ascents of a permutation.
For a permutation $\pi$, this is the number of indices $i$ such that $\pi(i+1)−\pi(i) > 1$.
For the number of small ascents, see [[St000441]].
Matching statistic: St001271
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 29%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001271: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 29%
Values
[[1]]
=> [[1]]
=> [1] => ([],1)
=> 0
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1,3] => ([(1,2)],3)
=> 0
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ? = 2
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [5,3,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [4,3,5,1,2,6] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [5,2,6,1,3,4] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ? = 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [4,2,5,1,3,6] => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> 0
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> [10,8,9,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 0
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> [10,8,9,4,5,6,1,2,3,7] => ([(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> [10,7,8,5,6,9,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,3],[3,3],[4]]
=> [10,7,8,4,5,9,1,2,3,6] => ([(0,1),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> [10,6,7,4,5,8,1,2,3,9] => ([(0,9),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,2],[2,2,3],[3,3],[4]]
=> [10,7,8,3,4,9,1,2,5,6] => ([(0,4),(0,7),(0,8),(0,9),(1,4),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> [10,6,7,3,4,8,1,2,5,9] => ([(0,9),(1,2),(1,7),(1,8),(1,9),(2,5),(2,6),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 3
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> [9,8,10,5,6,7,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(8,9)],10)
=> ? = 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> [9,8,10,4,5,6,1,2,3,7] => ([(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(8,9)],10)
=> ? = 2
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,3],[3,4],[4]]
=> [9,7,10,5,6,8,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,3],[3,4],[4]]
=> [9,7,10,4,5,8,1,2,3,6] => ([(0,6),(0,7),(0,8),(0,9),(1,3),(1,4),(1,5),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(8,9)],10)
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [[1,1,1,3],[2,2,3],[3,4],[4]]
=> [9,6,10,4,5,7,1,2,3,8] => ([(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,2],[2,2,3],[3,4],[4]]
=> [9,7,10,3,4,8,1,2,5,6] => ([(0,6),(0,7),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(8,9)],10)
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [[1,1,2,3],[2,2,3],[3,4],[4]]
=> [9,6,10,3,4,7,1,2,5,8] => ([(0,8),(0,9),(1,4),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9)],10)
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> [8,7,9,5,6,10,1,2,3,4] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,2],[2,2,4],[3,4],[4]]
=> [8,7,9,4,5,10,1,2,3,6] => ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,8),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,3],[2,2,4],[3,4],[4]]
=> [8,6,9,4,5,10,1,2,3,7] => ([(0,1),(0,8),(0,9),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 3
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> [7,6,8,4,5,9,1,2,3,10] => ([(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 3
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,2],[2,2,4],[3,4],[4]]
=> [8,7,9,3,4,10,1,2,5,6] => ([(0,2),(0,7),(0,8),(0,9),(1,2),(1,7),(1,8),(1,9),(2,5),(2,6),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(8,9)],10)
=> ? = 3
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,3],[2,2,4],[3,4],[4]]
=> [8,6,9,3,4,10,1,2,5,7] => ([(0,2),(0,8),(0,9),(1,2),(1,7),(1,8),(1,9),(2,5),(2,6),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9)],10)
=> ? = 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> [7,6,8,3,4,9,1,2,5,10] => ([(1,2),(1,8),(1,9),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,1],[2,3,3],[3,4],[4]]
=> [9,6,10,5,7,8,1,2,3,4] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(0,9),(1,3),(1,4),(1,5),(1,6),(1,7),(1,9),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(8,9)],10)
=> ? = 2
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,2],[2,3,3],[3,4],[4]]
=> [9,6,10,4,7,8,1,2,3,5] => ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 3
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[1,1,1,3],[2,3,3],[3,4],[4]]
=> [9,5,10,4,6,7,1,2,3,8] => ([(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 3
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,3,7,8,1,2,4,5] => ([(0,3),(0,4),(0,7),(0,8),(0,9),(1,5),(1,6),(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,9)],10)
=> ? = 3
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [[1,1,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,3,6,7,1,2,4,8] => ([(0,8),(0,9),(1,2),(1,4),(1,5),(1,8),(1,9),(2,3),(2,6),(2,7),(2,9),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> [8,6,9,5,7,10,1,2,3,4] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(1,9),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,2],[2,3,4],[3,4],[4]]
=> [8,6,9,4,7,10,1,2,3,5] => ([(0,1),(0,6),(0,7),(0,8),(1,3),(1,4),(1,5),(1,9),(2,3),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,3],[2,3,4],[3,4],[4]]
=> [8,5,9,4,6,10,1,2,3,7] => ([(0,1),(0,8),(0,9),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(2,9),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> [7,5,8,4,6,9,1,2,3,10] => ([(1,7),(1,8),(1,9),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9)],10)
=> ? = 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,3,7,10,1,2,4,5] => ([(0,2),(0,3),(0,7),(0,8),(1,4),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,9),(3,4),(3,5),(3,6),(3,9),(4,5),(4,7),(4,8),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9),(8,9)],10)
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,3,6,10,1,2,4,7] => ([(0,1),(0,8),(0,9),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [[1,1,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,3,6,9,1,2,4,10] => ([(1,6),(1,8),(1,9),(2,3),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(7,8),(7,9)],10)
=> ? = 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,3,5,10,1,2,6,7] => ([(0,4),(0,8),(0,9),(1,4),(1,8),(1,9),(2,6),(2,7),(2,8),(2,9),(3,5),(3,6),(3,7),(3,9),(4,6),(4,7),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,3,5,9,1,2,6,10] => ([(1,2),(1,8),(1,9),(2,6),(2,7),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,2],[2,3,3],[3,4],[4]]
=> [9,6,10,2,7,8,1,3,4,5] => ([(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,7),(1,8),(1,9),(2,4),(2,5),(2,7),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,9)],10)
=> ? = 3
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,2,2,3],[2,3,3],[3,4],[4]]
=> [9,5,10,2,6,7,1,3,4,8] => ([(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,4),(2,5),(2,6),(2,8),(2,9),(3,4),(3,5),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9)],10)
=> ? = 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,2],[2,3,4],[3,4],[4]]
=> [8,6,9,2,7,10,1,3,4,5] => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(2,5),(2,6),(2,7),(2,9),(3,5),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,8),(7,8),(7,9),(8,9)],10)
=> ? = 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,3],[2,3,4],[3,4],[4]]
=> [8,5,9,2,6,10,1,3,4,7] => ([(0,2),(0,8),(0,9),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,7),(3,5),(3,6),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9)],10)
=> ? = 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> [7,5,8,2,6,9,1,3,4,10] => ([(1,6),(1,7),(1,9),(2,4),(2,5),(2,8),(2,9),(3,4),(3,6),(3,7),(3,8),(3,9),(4,6),(4,7),(4,9),(5,6),(5,7),(5,8),(5,9),(6,8),(7,8),(8,9)],10)
=> ? = 5
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,3],[2,3,4],[3,4],[4]]
=> [8,4,9,2,5,10,1,3,6,7] => ([(0,5),(0,8),(0,9),(1,5),(1,8),(1,9),(2,3),(2,7),(2,8),(2,9),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 5
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> [7,4,8,2,5,9,1,3,6,10] => ([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> ? = 6
Description
The competition number of a graph.
The competition graph of a digraph $D$ is a (simple undirected) graph which has the same vertex set as $D$ and has an edge between $x$ and $y$ if and only if there exists a vertex $v$ in $D$ such that $(x, v)$ and $(y, v)$ are arcs of $D$. For any graph, $G$ together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number $k(G)$ is the smallest number of such isolated vertices.
Matching statistic: St000264
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 14%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 14%
Values
[[1]]
=> [1,0]
=> [1] => ([],1)
=> ? = 0 + 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> ? = 1 + 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> ? = 3 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? = 2 + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? = 2 + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 2 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 1 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? = 1 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> ? = 0 + 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 0 + 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? = 1 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 2 + 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? = 3 + 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? = 2 + 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 2 + 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 3 + 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 4 + 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 3 + 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? = 3 + 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 4 + 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 5 + 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 5 + 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 5 + 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4 = 3 + 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? = 4 + 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 5 + 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 5 + 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? = 5 + 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> ? = 6 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
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