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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000197
St000197: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 1
[[1,0],[0,1]]
=> 2
[[0,1],[1,0]]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> 5
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 6
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 5
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> 5
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> 5
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> 5
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> 5
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> 5
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> 6
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> 4
Description
The number of entries equal to positive one in the alternating sign matrix.
Matching statistic: St000176
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00076: Semistandard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
St000176: 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
St000176: Gelfand-Tsetlin patterns ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [[1]]
=> [[1]]
=> 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [[2,1],[2]]
=> 2
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [[2,1],[1]]
=> 2
[[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]]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [[3,2,1],[3,1],[3]]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [[3,2,1],[3,1],[2]]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [[3,2,1],[2,1],[2]]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [[3,2,1],[3,1],[1]]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [[3,2,1],[2,1],[1]]
=> 3
[[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
[[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
[[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
[[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]]
=> 5
[[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
[[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
[[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
[[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
[[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
[[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]]
=> 5
[[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]]
=> 6
[[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]]
=> 5
[[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]]
=> 5
[[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]]
=> 5
[[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
[[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]]
=> 5
[[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]]
=> 5
[[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
[[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
[[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]]
=> 5
[[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]]
=> 4
[[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]]
=> 5
[[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
[[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]]
=> 5
[[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]]
=> 5
[[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
[[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]]
=> 5
[[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]]
=> 5
[[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]]
=> 5
[[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]]
=> 6
[[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]]
=> 4
[[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
[[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]]
=> 4
Description
The total number of tiles in the Gelfand-Tsetlin pattern.
The tiling of a Gelfand-Tsetlin pattern is the finest partition of the entries in the pattern, such that adjacent (NW,NE,SW,SE) entries that are equal belong to the same part. These parts are called tiles, and each entry in a pattern belongs to exactly one tile.
Matching statistic: St001034
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1,0]
=> 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 4
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000395
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 6
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001018
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 6
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 6
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001348
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001348: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[0,1],[1,0]]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 4
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 6
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 6
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 5
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 5
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 5
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
Description
The bounce of the parallelogram polyomino associated with the Dyck path.
A bijection due to Delest and Viennot [1] associates a Dyck path with a parallelogram polyomino. The bounce statistic is defined in [2].
Matching statistic: St000245
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [2,4,3,1] => 1 = 2 - 1
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [2,3,6,4,5,1] => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,6,3,5,2,1] => 2 = 3 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,4,6,5,3] => 2 = 3 - 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,6,3,5,2,1] => 2 = 3 - 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [2,4,3,1,6,5] => 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => 5 = 6 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,4,5,8,6,7,3] => 4 = 5 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [2,3,6,4,5,1,8,7] => 4 = 5 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 3 = 4 - 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [2,4,3,1,6,8,7,5] => 3 = 4 - 1
Description
The number of ascents of a permutation.
Matching statistic: St000672
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00098: Alternating sign matrices —link pattern⟶ Perfect matchings
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 71% ●values known / values provided: 71%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [(1,2)]
=> [2,1] => [2,1] => 0 = 1 - 1
[[1,0],[0,1]]
=> [(1,4),(2,3)]
=> [3,4,2,1] => [2,4,3,1] => 1 = 2 - 1
[[0,1],[1,0]]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [(1,6),(2,5),(3,4)]
=> [4,5,6,3,2,1] => [2,3,6,4,5,1] => 3 = 4 - 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,6,3,5,2,1] => 2 = 3 - 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [(1,2),(3,6),(4,5)]
=> [2,1,5,6,4,3] => [2,1,4,6,5,3] => 2 = 3 - 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [(1,6),(2,3),(4,5)]
=> [3,5,2,6,4,1] => [4,6,3,5,2,1] => 2 = 3 - 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => 2 = 3 - 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [(1,4),(2,3),(5,6)]
=> [3,4,2,1,6,5] => [2,4,3,1,6,5] => 2 = 3 - 1
[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [5,6,7,8,4,3,2,1] => [2,3,4,8,5,6,7,1] => 5 = 6 - 1
[[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,6,7,8,5,4,3] => [2,1,4,5,8,6,7,3] => 4 = 5 - 1
[[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[0,0,1,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [4,6,7,3,8,5,2,1] => [2,5,8,4,7,6,3,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [3,6,2,7,8,5,4,1] => [4,5,3,8,6,2,7,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,0,0,0],[0,0,1,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[1,-1,1,0],[0,0,0,1],[0,1,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[0,0,1,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,5,6,4,3,8,7] => [2,1,4,6,5,3,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,0,0],[1,-1,0,1],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,1,0],[1,0,-1,1],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [4,5,6,3,2,1,8,7] => [2,3,6,4,5,1,8,7] => 4 = 5 - 1
[[0,0,0,1],[0,1,0,0],[1,-1,1,0],[0,1,0,0]]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [3,5,2,6,4,1,8,7] => [4,6,3,5,2,1,8,7] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [2,1,4,3,6,5,8,7] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [3,4,2,1,6,5,8,7] => [2,4,3,1,6,5,8,7] => 3 = 4 - 1
[[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,5,7,4,8,6,3] => [2,1,6,8,5,7,4,3] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,1,0,0],[0,0,0,1],[0,0,1,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,1,0],[0,1,-1,1],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [4,5,7,3,2,8,6,1] => [6,3,8,4,5,7,2,1] => ? ∊ {4,4,4,4,4,4,4,5,5,5,5,5,5,5,6} - 1
[[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [3,5,2,7,4,8,6,1] => [6,8,3,5,2,7,4,1] => 4 = 5 - 1
[[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,7,8,6,5] => [2,1,4,3,6,8,7,5] => 3 = 4 - 1
[[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [3,4,2,1,7,8,6,5] => [2,4,3,1,6,8,7,5] => 3 = 4 - 1
Description
The number of minimal elements in Bruhat order not less than the permutation.
The minimal elements in question are biGrassmannian, that is
$$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$$
for some $(r,a,b)$.
This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].
Matching statistic: St001515
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> [1,0,1,0]
=> 1
[[1,0],[0,1]]
=> [[1,1],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[0,1],[1,0]]
=> [[1,2],[2]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2
[[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[0,1,0],[1,-1,1],[0,1,0]]
=> [[1,1,2],[2,3],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 4
[[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[0,1,0],[0,0,1],[1,0,0]]
=> [[1,2,2],[2,3],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 3
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [3,3,2,2]
=> [1,1,0,0,1,1,0,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]]
=> [4,2,2,2]
=> [1,1,0,0,1,1,1,0,0,1,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 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]]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6}
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[[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]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000718
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00007: Alternating sign matrices —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000718: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[[1,0],[0,1]]
=> [1,0,1,0]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[[0,1],[1,0]]
=> [1,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[[1,0,0],[0,1,0],[0,0,1]]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0],[1,0,0],[0,0,1]]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {3,4} + 1
[[1,0,0],[0,0,1],[0,1,0]]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> ? ∊ {3,4} + 1
[[0,1,0],[1,-1,1],[0,1,0]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,0,1],[1,0,0],[0,1,0]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,1,0],[0,0,1],[1,0,0]]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[[0,0,1],[0,1,0],[1,0,0]]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4 = 3 + 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]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 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]
=> [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 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]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ? ∊ {5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6} + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 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]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5 = 4 + 1
Description
The largest Laplacian eigenvalue of a graph if it is integral.
This statistic is undefined if the largest Laplacian eigenvalue of the graph is not integral.
Various results are collected in Section 3.9 of [1]
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000670The reversal length of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001554The number of distinct nonempty subtrees of a binary tree. St001875The number of simple modules with projective dimension at most 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001645The pebbling number of a connected graph. St000703The number of deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St000308The height of the tree associated to a permutation. St000354The number of recoils of a permutation. St000451The length of the longest pattern of the form k 1 2. St000702The number of weak deficiencies of a permutation. St000730The maximal arc length of a set partition. St000829The Ulam distance of a permutation to the identity permutation. St001116The game chromatic number of a graph. St001286The annihilation number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001405The number of bonds in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001566The length of the longest arithmetic progression in a permutation. St001717The largest size of an interval in a poset. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000316The number of non-left-to-right-maxima of a permutation. St000624The normalized sum of the minimal distances to a greater element. St000663The number of right floats of a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000863The length of the first row of the shifted shape of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001403The number of vertical separators in a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001792The arboricity of a graph. St000619The number of cyclic descents of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000646The number of big ascents of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St001220The width of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000779The tier of a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000264The girth of a graph, which is not a tree. St001812The biclique partition number of a graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice.
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