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Your data matches 191 different statistics following compositions of up to 3 maps.
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Matching statistic: St000515
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 2
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
Description
The number of invariant set partitions when acting with a permutation of given cycle type.
Matching statistic: St001165
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 2
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 2
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 2
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 2
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[[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]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,0,1,1,1,0,0,1,0,0,0]
=> ? = 3
[[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]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> ? = 3
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 5
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001043
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001043: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 1
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 1
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 1
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
Description
The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
Matching statistic: St001045
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001045: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001045: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 1 = 2 - 1
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 1 = 2 - 1
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 1
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 1
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 1
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 1
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 1
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 1
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 1
Description
The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching.
The bijection between perfect matchings of $\{1,\dots,2n\}$ and trees with $n+1$ leaves is described in Example 5.2.6 of [1].
The number of trees having precisely $j$ leaves in the subtree not containing $1$, computed in [2], is
$$
\binom{n}{j}(2j-3)!!(2n-2j-1)!!
$$
Matching statistic: St001208
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1 = 2 - 1
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 2 - 1
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 1 = 2 - 1
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 2 - 1
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 5 - 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]
=> [3,5,6,1,2,4] => ? = 3 - 1
[[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]
=> [3,4,1,5,6,2] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,1,5,6,2] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,5,6,1,2,4] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,1,5,6,2] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,5,6,1,2,4] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,5,1,2,6] => ? = 3 - 1
[[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]
=> [3,4,1,5,6,2] => ? = 3 - 1
[[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]
=> [3,5,6,1,2,4] => ? = 3 - 1
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 5 - 1
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 5 - 1
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 3 - 1
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 3 - 1
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 2 - 1
Description
The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$.
Matching statistic: St000199
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 4 = 2 + 2
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5 + 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 5 + 2
[[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]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[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]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 3 + 2
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 5 + 2
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 5 + 2
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3 + 2
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 3 + 2
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [[1,0,0,0,0,0,0],[0,0,0,0,0,0,1],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0]]
=> ? = 2 + 2
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St001001
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001001: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0 = 2 - 2
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0 = 2 - 2
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 2 - 2
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[[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]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 3 - 2
[[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]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 - 2
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5 - 2
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 2
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 2
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 2
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001831
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001831: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001831: Perfect matchings ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 0 = 2 - 2
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 0 = 2 - 2
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> 0 = 2 - 2
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0 = 2 - 2
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> 0 = 2 - 2
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 2 - 2
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 2
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 5 - 2
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,7),(8,11),(9,10)]
=> ? = 3 - 2
[[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]
=> [(1,4),(2,3),(5,12),(6,9),(7,8),(10,11)]
=> ? = 3 - 2
[[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]
=> [(1,6),(2,3),(4,5),(7,12),(8,11),(9,10)]
=> ? = 3 - 2
[[1,2,3,6],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,4,6],[3,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,4],[3,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,4,6],[3],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,4],[3],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,5,6],[3],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,5],[3,6],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,5],[3],[4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,2,6],[3],[4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 2
[[1,2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 5 - 2
[[1,3,4,6],[2,5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,4],[2,5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,5,6],[2,4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,5],[2,4,6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,5],[2,4],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,6],[2,4],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 2
[[1,3],[2,4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 3 - 2
[[1,3,4,6],[2],[5]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,4],[2],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
[[1,3,5,6],[2],[4]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,2),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9)]
=> ? = 2 - 2
Description
The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching.
Given a perfect matching, we obtain a formal sum of non-crossing perfect matchings by replacing recursively every matching $M$ that has a crossing $(a, c), (b, d)$ with $a < b < c < d$ with the sum of the two matchings $(M\setminus \{(a,c), (b,d)\})\cup \{(a,b), (c,d)\}$ and $(M\setminus \{(a,c), (b,d)\})\cup \{(a,d), (b,c)\}$.
This statistic is the coefficient of the unique non-nesting perfect matching in the formal sum.
Computer experiments indicate that this statistic has the same distribution as [[StatisticsDatabase/St000383oMp00128oMp00215oMp00092]].
Matching statistic: St001200
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 57%●distinct values known / distinct values provided: 25%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 57%●distinct values known / distinct values provided: 25%
Values
[[1,4],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,5],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,5],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,5],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,1,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4,4],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,3,3],[2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[1,1,3],[2,2],[3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[[1,6],[2],[3]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[2],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[2],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,6],[3],[4]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,6],[3],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[3,6],[4],[5]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[3],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[2],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[3],[4],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[3],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[2],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[3],[4],[5],[6]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[[1,3,5],[2,4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,1,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,1,5],[3],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,2,5],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,3,5],[2],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,4,5],[2],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,5,5],[2],[3]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,4,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,5,5],[2],[4]]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3 = 2 + 1
[[1,3],[2,4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,4],[2,5],[3]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,3],[2],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,4],[2],[3],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2 + 1
[[1,5],[2],[3],[4]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5 + 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 5 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 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]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 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]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,1,1,1],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,1,1,2],[2,2,2],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,1,1],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,1,2,2],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,2,4],[2,2,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,1,1,1],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,1,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,1,1],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 1
[[1,1,1,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[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]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> ? = 3 + 1
[[1,1,3,3],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,1,3,4],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,2,3],[2,3,3],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[1,2,2,2],[2,3,4],[3,4],[4]]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 3 + 1
[[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]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 3 + 1
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000116
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000116: Semistandard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
St000116: Semistandard tableaux ⟶ ℤResult quality: 25% ●values known / values provided: 33%●distinct values known / distinct values provided: 25%
Values
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [[1,2,3,4]]
=> 0 = 2 - 2
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [[1,2,3],[5]]
=> [[1,2,3,5]]
=> ? = 2 - 2
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [[1,2,4],[5]]
=> [[1,2,4,5]]
=> ? = 2 - 2
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [[1,3,4],[5]]
=> [[1,3,4,5]]
=> ? = 2 - 2
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [[2,3,4],[5]]
=> [[2,3,4,5]]
=> ? = 2 - 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [[1,2,3],[5]]
=> [[1,2,3,5]]
=> ? = 2 - 2
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [[1,2,4],[5]]
=> [[1,2,4,5]]
=> ? = 2 - 2
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [[1,3,4],[5]]
=> [[1,3,4,5]]
=> ? = 2 - 2
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [[2,3,4],[5]]
=> [[2,3,4,5]]
=> ? = 2 - 2
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [[1,1,2,3],[4]]
=> [[1,1,2,3,4]]
=> 0 = 2 - 2
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [[1,2,2,3],[4]]
=> [[1,2,2,3,4]]
=> 0 = 2 - 2
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [[1,2,3,3],[4]]
=> [[1,2,3,3,4]]
=> 0 = 2 - 2
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [[1,2,3,4],[4]]
=> [[1,2,3,4,4]]
=> 0 = 2 - 2
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [[1,1,2,3],[4]]
=> [[1,1,2,3,4]]
=> 0 = 2 - 2
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [[1,2,2,3],[4]]
=> [[1,2,2,3,4]]
=> 0 = 2 - 2
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [[1,2,3,3],[4]]
=> [[1,2,3,3,4]]
=> 0 = 2 - 2
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [[1,2,3,4],[4]]
=> [[1,2,3,4,4]]
=> 0 = 2 - 2
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [[1,1,2,2,3,3]]
=> [[1,1,2,2,3,3]]
=> 0 = 2 - 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [[1,1,2,2,3,3]]
=> [[1,1,2,2,3,3]]
=> 0 = 2 - 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [[1,1,2,2,3,3]]
=> [[1,1,2,2,3,3]]
=> 0 = 2 - 2
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [[1,2,3],[6]]
=> [[1,2,3,6]]
=> ? = 2 - 2
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [[1,2,4],[6]]
=> [[1,2,4,6]]
=> ? = 2 - 2
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [[1,2,5],[6]]
=> [[1,2,5,6]]
=> ? = 2 - 2
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [[1,3,4],[6]]
=> [[1,3,4,6]]
=> ? = 2 - 2
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [[1,3,5],[6]]
=> [[1,3,5,6]]
=> ? = 2 - 2
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [[1,4,5],[6]]
=> [[1,4,5,6]]
=> ? = 2 - 2
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [[2,3,4],[6]]
=> [[2,3,4,6]]
=> ? = 2 - 2
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [[2,3,5],[6]]
=> [[2,3,5,6]]
=> ? = 2 - 2
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [[2,4,5],[6]]
=> [[2,4,5,6]]
=> ? = 2 - 2
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [[3,4,5],[6]]
=> [[3,4,5,6]]
=> ? = 2 - 2
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [[1,2,3],[6]]
=> [[1,2,3,6]]
=> ? = 2 - 2
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [[1,2,4],[6]]
=> [[1,2,4,6]]
=> ? = 2 - 2
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [[1,2,5],[6]]
=> [[1,2,5,6]]
=> ? = 2 - 2
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [[1,3,4],[6]]
=> [[1,3,4,6]]
=> ? = 2 - 2
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [[1,3,5],[6]]
=> [[1,3,5,6]]
=> ? = 2 - 2
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [[1,4,5],[6]]
=> [[1,4,5,6]]
=> ? = 2 - 2
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [[2,3,4],[6]]
=> [[2,3,4,6]]
=> ? = 2 - 2
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [[2,3,5],[6]]
=> [[2,3,5,6]]
=> ? = 2 - 2
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [[2,4,5],[6]]
=> [[2,4,5,6]]
=> ? = 2 - 2
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [[3,4,5],[6]]
=> [[3,4,5,6]]
=> ? = 2 - 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> ? = 2 - 2
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [[1,1,2,3],[5]]
=> [[1,1,2,3,5]]
=> ? = 2 - 2
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [[1,1,2,4],[5]]
=> [[1,1,2,4,5]]
=> ? = 2 - 2
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [[1,1,3,4],[5]]
=> [[1,1,3,4,5]]
=> ? = 2 - 2
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [[1,2,2,3],[5]]
=> [[1,2,2,3,5]]
=> ? = 2 - 2
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [[1,2,2,4],[5]]
=> [[1,2,2,4,5]]
=> ? = 2 - 2
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [[1,2,3,3],[5]]
=> [[1,2,3,3,5]]
=> ? = 2 - 2
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> ? = 2 - 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> ? = 2 - 2
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> ? = 2 - 2
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [[1,2,3,5],[5]]
=> [[1,2,3,5,5]]
=> ? = 2 - 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [[1,2,4,4],[5]]
=> [[1,2,4,4,5]]
=> ? = 2 - 2
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [[1,2,4,5],[5]]
=> [[1,2,4,5,5]]
=> ? = 2 - 2
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [[1,3,3,4],[5]]
=> [[1,3,3,4,5]]
=> ? = 2 - 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [[1,3,4,4],[5]]
=> [[1,3,4,4,5]]
=> ? = 2 - 2
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [[1,3,4,5],[5]]
=> [[1,3,4,5,5]]
=> ? = 2 - 2
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [[2,2,3,4],[5]]
=> [[2,2,3,4,5]]
=> ? = 2 - 2
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [[2,3,3,4],[5]]
=> [[2,3,3,4,5]]
=> ? = 2 - 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [[2,3,4,4],[5]]
=> [[2,3,4,4,5]]
=> ? = 2 - 2
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [[2,3,4,5],[5]]
=> [[2,3,4,5,5]]
=> ? = 2 - 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> ? = 2 - 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> ? = 2 - 2
[[1,1,3,4],[2,2]]
=> [[1,1,2,2],[3,4]]
=> [[1,1,2,2,3,4]]
=> [[1,1,2,2,3,4]]
=> 0 = 2 - 2
[[1,1,4,4],[2,2]]
=> [[1,1,2,2],[4,4]]
=> [[1,1,2,2,4,4]]
=> [[1,1,2,2,4,4]]
=> 0 = 2 - 2
[[1,1,3,4],[2,3]]
=> [[1,1,2,3],[3,4]]
=> [[1,1,2,3,3,4]]
=> [[1,1,2,3,3,4]]
=> 0 = 2 - 2
[[1,1,4,4],[2,3]]
=> [[1,1,2,3],[4,4]]
=> [[1,1,2,3,4,4]]
=> [[1,1,2,3,4,4]]
=> 0 = 2 - 2
[[1,1,4,4],[3,3]]
=> [[1,1,3,3],[4,4]]
=> [[1,1,3,3,4,4]]
=> [[1,1,3,3,4,4]]
=> 0 = 2 - 2
[[1,2,3,4],[2,3]]
=> [[1,2,2,3],[3,4]]
=> [[1,2,2,3,3,4]]
=> [[1,2,2,3,3,4]]
=> 0 = 2 - 2
[[1,2,4,4],[2,3]]
=> [[1,2,2,3],[4,4]]
=> [[1,2,2,3,4,4]]
=> [[1,2,2,3,4,4]]
=> 0 = 2 - 2
[[1,2,4,4],[3,3]]
=> [[1,2,3,3],[4,4]]
=> [[1,2,3,3,4,4]]
=> [[1,2,3,3,4,4]]
=> 0 = 2 - 2
[[2,2,4,4],[3,3]]
=> [[2,2,3,3],[4,4]]
=> [[2,2,3,3,4,4]]
=> [[2,2,3,3,4,4]]
=> 0 = 2 - 2
[[1,1,1,4],[2],[3]]
=> [[1,1,1,2],[3],[4]]
=> [[1,1,1,2,3],[4]]
=> [[1,1,1,2,3,4]]
=> 0 = 2 - 2
[[1,1,2,4],[2],[3]]
=> [[1,1,2,2],[3],[4]]
=> [[1,1,2,2,3],[4]]
=> [[1,1,2,2,3,4]]
=> 0 = 2 - 2
[[1,1,3,4],[2],[3]]
=> [[1,1,2,3],[3],[4]]
=> [[1,1,2,3,3],[4]]
=> [[1,1,2,3,3,4]]
=> 0 = 2 - 2
[[1,1,4,4],[2],[3]]
=> [[1,1,2,4],[3],[4]]
=> [[1,1,2,3,4],[4]]
=> [[1,1,2,3,4,4]]
=> 0 = 2 - 2
[[1,2,2,4],[2],[3]]
=> [[1,2,2,2],[3],[4]]
=> [[1,2,2,2,3],[4]]
=> [[1,2,2,2,3,4]]
=> 0 = 2 - 2
[[1,2,3,4],[2],[3]]
=> [[1,2,2,3],[3],[4]]
=> [[1,2,2,3,3],[4]]
=> [[1,2,2,3,3,4]]
=> 0 = 2 - 2
[[1,2,4,4],[2],[3]]
=> [[1,2,2,4],[3],[4]]
=> [[1,2,2,3,4],[4]]
=> [[1,2,2,3,4,4]]
=> 0 = 2 - 2
[[1,3,3,4],[2],[3]]
=> [[1,2,3,3],[3],[4]]
=> [[1,2,3,3,3],[4]]
=> [[1,2,3,3,3,4]]
=> 0 = 2 - 2
[[1,3,4,4],[2],[3]]
=> [[1,2,3,4],[3],[4]]
=> [[1,2,3,3,4],[4]]
=> [[1,2,3,3,4,4]]
=> 0 = 2 - 2
[[1,4,4,4],[2],[3]]
=> [[1,2,4,4],[3],[4]]
=> [[1,2,3,4,4],[4]]
=> [[1,2,3,4,4,4]]
=> 0 = 2 - 2
[[1,1,3],[2,2],[4]]
=> [[1,1,2,2],[3,4]]
=> [[1,1,2,2,3,4]]
=> [[1,1,2,2,3,4]]
=> 0 = 2 - 2
[[1,1,4],[2,2],[3]]
=> [[1,1,2,2],[3],[4]]
=> [[1,1,2,2,3],[4]]
=> [[1,1,2,2,3,4]]
=> 0 = 2 - 2
[[1,1,4],[2,2],[4]]
=> [[1,1,2,2],[4,4]]
=> [[1,1,2,2,4,4]]
=> [[1,1,2,2,4,4]]
=> 0 = 2 - 2
[[1,1,3],[2,3],[4]]
=> [[1,1,2,3],[3,4]]
=> [[1,1,2,3,3,4]]
=> [[1,1,2,3,3,4]]
=> 0 = 2 - 2
[[1,1,4],[2,3],[3]]
=> [[1,1,2,3],[3],[4]]
=> [[1,1,2,3,3],[4]]
=> [[1,1,2,3,3,4]]
=> 0 = 2 - 2
[[1,1,4],[2,3],[4]]
=> [[1,1,2,3],[4,4]]
=> [[1,1,2,3,4,4]]
=> [[1,1,2,3,4,4]]
=> 0 = 2 - 2
[[1,1,4],[2,4],[3]]
=> [[1,1,2,4],[3],[4]]
=> [[1,1,2,3,4],[4]]
=> [[1,1,2,3,4,4]]
=> 0 = 2 - 2
[[1,1,4],[3,3],[4]]
=> [[1,1,3,3],[4,4]]
=> [[1,1,3,3,4,4]]
=> [[1,1,3,3,4,4]]
=> 0 = 2 - 2
[[1,2,3],[2,3],[4]]
=> [[1,2,2,3],[3,4]]
=> [[1,2,2,3,3,4]]
=> [[1,2,2,3,3,4]]
=> 0 = 2 - 2
[[1,2,4],[2,3],[3]]
=> [[1,2,2,3],[3],[4]]
=> [[1,2,2,3,3],[4]]
=> [[1,2,2,3,3,4]]
=> 0 = 2 - 2
[[1,2,4],[2,3],[4]]
=> [[1,2,2,3],[4,4]]
=> [[1,2,2,3,4,4]]
=> [[1,2,2,3,4,4]]
=> 0 = 2 - 2
[[1,2,4],[2,4],[3]]
=> [[1,2,2,4],[3],[4]]
=> [[1,2,2,3,4],[4]]
=> [[1,2,2,3,4,4]]
=> 0 = 2 - 2
[[1,2,4],[3,3],[4]]
=> [[1,2,3,3],[4,4]]
=> [[1,2,3,3,4,4]]
=> [[1,2,3,3,4,4]]
=> 0 = 2 - 2
[[1,3,4],[2,4],[3]]
=> [[1,2,3,4],[3],[4]]
=> [[1,2,3,3,4],[4]]
=> [[1,2,3,3,4,4]]
=> 0 = 2 - 2
[[2,2,4],[3,3],[4]]
=> [[2,2,3,3],[4,4]]
=> [[2,2,3,3,4,4]]
=> [[2,2,3,3,4,4]]
=> 0 = 2 - 2
[[1,1,1],[2],[3],[4]]
=> [[1,1,1,2],[3],[4]]
=> [[1,1,1,2,3],[4]]
=> [[1,1,1,2,3,4]]
=> 0 = 2 - 2
[[1,1,2],[2],[3],[4]]
=> [[1,1,2,2],[3],[4]]
=> [[1,1,2,2,3],[4]]
=> [[1,1,2,2,3,4]]
=> 0 = 2 - 2
[[1,1,3],[2],[3],[4]]
=> [[1,1,2,3],[3],[4]]
=> [[1,1,2,3,3],[4]]
=> [[1,1,2,3,3,4]]
=> 0 = 2 - 2
Description
The major index of a semistandard tableau obtained by standardizing.
The following 181 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001684The reduced word complexity of a permutation. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001846The number of elements which do not have a complement in the lattice. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001330The hat guessing number of a graph. St001864The number of excedances of a signed permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001861The number of Bruhat lower covers of a permutation. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000693The modular (standard) major index of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001371The length of the longest Yamanouchi prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001424The number of distinct squares in a binary word. St001485The modular major index of a binary word. St001524The degree of symmetry of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001862The number of crossings of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001892The flag excedance statistic of a signed permutation. St001894The depth of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001721The degree of a binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000016The number of attacking pairs of a standard tableau. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000294The number of distinct factors of a binary word. St001885The number of binary words with the same proper border set. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral. St001618The cardinality of the Frattini sublattice of a lattice. St001889The size of the connectivity set of a signed permutation. St001893The flag descent of a signed permutation. St001822The number of alignments of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000135The number of lucky cars of the parking function. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001867The number of alignments of type EN of a signed permutation. St001927Sparre Andersen's number of positives of a signed permutation. St000540The sum of the entries of a parking function minus its length. St000165The sum of the entries of a parking function. St001852The size of the conjugacy class of the signed permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000632The jump number of the poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000907The number of maximal antichains of minimal length in a poset. St000942The number of critical left to right maxima of the parking functions. St001399The distinguishing number of a poset. St001423The number of distinct cubes in a binary word. St001433The flag major index of a signed permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001768The number of reduced words of a signed permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001819The flag Denert index of a signed permutation. St001821The sorting index of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St000068The number of minimal elements in a poset. St000080The rank of the poset. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St000850The number of 1/2-balanced pairs in a poset. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001209The pmaj statistic of a parking function. St001396Number of triples of incomparable elements in a finite poset. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001557The number of inversions of the second entry of a permutation. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001770The number of facets of a certain subword complex associated with the signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001926Sparre Andersen's position of the maximum of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000528The height of a poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000906The length of the shortest maximal chain in a poset. St001301The first Betti number of the order complex associated with the poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001817The number of flag weak exceedances of a signed permutation. St001851The number of Hecke atoms of a signed permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000643The size of the largest orbit of antichains under Panyushev complementation. St000680The Grundy value for Hackendot on posets. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St000912The number of maximal antichains in a poset. St001268The size of the largest ordinal summand in the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001397Number of pairs of incomparable elements in a finite poset. St001472The permanent of the Coxeter matrix of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001718The number of non-empty open intervals in a poset. St001854The size of the left Kazhdan-Lusztig cell, St000072The number of circled entries. St000073The number of boxed entries. St000077The number of boxed and circled entries. St000189The number of elements in the poset. St001343The dimension of the reduced incidence algebra of a poset. St001434The number of negative sum pairs of a signed permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001717The largest size of an interval in a poset. St001902The number of potential covers of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000656The number of cuts of a poset. St001848The atomic length of a signed permutation. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001782The order of rowmotion on the set of order ideals of a poset. St001858The number of covering elements of a signed permutation in absolute order. St001865The number of alignments of a signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St000070The number of antichains in a poset. St001779The order of promotion on the set of linear extensions of a poset. St000641The number of non-empty boolean intervals in a poset. St000639The number of relations in a poset. St000958The number of Bruhat factorizations of a permutation. St001853The size of the two-sided Kazhdan-Lusztig cell, St001664The number of non-isomorphic subposets of a poset. St001709The number of homomorphisms to the three element chain of a poset. St001815The number of order preserving surjections from a poset to a total order. St001813The product of the sizes of the principal order filters in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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