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Matching statistic: St000678
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[5,4],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,4,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[5,4,1],[4,1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,4,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[3,2,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[4,3,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
Description
The number of up steps after the last double rise of a Dyck path.
Matching statistic: St001185
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001185: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[5,4],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,4,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[5,4,1],[4,1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,4,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[[3,2,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[5,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,5,2,1],[5,2,1]]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,5,3,1],[5,3,1]]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[6,5,3,2],[5,3,2]]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[5,5,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,2,1],[4,2,2,1]]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,2,2,1],[4,2,1,1]]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[6,5,4,1],[5,4,1]]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[6,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[[5,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 1
[[5,4,4,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,2,1],[4,3,1,1]]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[6,5,4,3],[5,4,3]]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[5,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,3,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,2,2],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[4,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[4,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
[[4,4,3,2,2,1],[3,3,2,1,1]]
=> [3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
[[4,3,3,3,2,1],[3,2,2,2,1]]
=> [3,2,2,2,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[4,3,3,2,2,1],[3,2,2,1,1]]
=> [3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[4,3,2,2,2,1],[3,2,1,1,1]]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[5,4,3,2,1,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[7,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[7,5,2,1],[5,2,1]]
=> [5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[7,5,3,1],[5,3,1]]
=> [5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[7,5,3,2],[5,3,2]]
=> [5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[6,5,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,2,1],[4,2,2,1]]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,2,2,1],[4,2,1,1]]
=> [4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[7,5,4,1],[5,4,1]]
=> [5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[7,5,4,2],[5,4,2]]
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> ? = 2 - 1
[[6,5,4,2,1],[4,4,2,1]]
=> [4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> ? = 3 - 1
[[6,4,4,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,2,1],[4,3,1,1]]
=> [4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> ? = 5 - 1
[[7,5,4,3],[5,4,3]]
=> [5,4,3]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[6,5,4,3,1],[4,4,3,1]]
=> [4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,4,3,1],[4,3,3,1]]
=> [4,3,3,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,3,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,5,4,3,2],[4,4,3,2]]
=> [4,4,3,2]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,4,3,2],[4,3,3,2]]
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,3,2],[4,3,2,2]]
=> [4,3,2,2]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> ? = 3 - 1
[[6,4,3,2,2],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,4,4,3,2,1],[3,3,3,2,1]]
=> [3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[5,4,3,3,2,1],[3,3,2,2,1]]
=> [3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,0,0,1,0,0,0]
=> ? = 4 - 1
Description
The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra.
Matching statistic: St001232
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 88%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 88%
Values
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[5,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[5,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[4,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[5,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[5,4],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,4,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[4,3,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[[5,4,1],[4,1]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,4,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[3,2,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[5,4,2],[4,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[3,3,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[3,3,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,4,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[3,2,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[[4,3,2,1],[3,1,1]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
[[4,3,1,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[[5,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[4,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[4,3,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[5,4,3],[4,3]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
[[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[3,3,2,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[4,4,3,1],[3,3,1]]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[3,2,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[[4,3,3,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,3,2,1],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[5,4,3,1],[4,3,1]]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[[3,3,3,2],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[3,3,2,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,4,3,2],[3,3,2]]
=> [3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[3,2,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[[4,3,3,2],[3,2,2]]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 2 - 1
[[4,3,2,2],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> ? = 2 - 1
[[3,3,3,2,1],[2,2,2,1]]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[3,3,2,2,1],[2,2,1,1]]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[3,3,2,1,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,4,3,2,1],[3,3,2,1]]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[4,3,3,2,1],[3,2,2,1]]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,3,2,2,1],[3,2,1,1]]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> ? = 4 - 1
[[4,3,2,1,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,3,2,1],[4,3,2,1]]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 3 - 1
[[5,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[6,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[5,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,4,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[5,4,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[6,4,2],[4,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 2 - 1
[[4,4,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[5,4,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[6,4,2,1],[4,2,1]]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> ? = 3 - 1
[[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 2 - 1
[[5,4,3],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 3 - 1
[[5,3,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 2 - 1
[[6,4,3],[4,3]]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ? = 3 - 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001491
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 25%
Values
[[3,2],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[2,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3,2,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,2],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[3,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,2,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,3],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,3],[3]]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[[3,3,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,2,1],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,3,1],[3,1]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[2,2,2],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3,3,2],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[3,2,2],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3,2],[3,2]]
=> [3,2]
=> 10100 => 00100 => ? = 2 - 1
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => 01010 => ? = 3 - 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => 00110 => ? = 3 - 1
[[3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 001010 => ? = 3 - 1
[[5,2],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[5,2,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[5,3],[3]]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[[3,3,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,3,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,2,1],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[5,3,1],[3,1]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[3,2,2],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,3,2],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[4,2,2],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[5,3,2],[3,2]]
=> [3,2]
=> 10100 => 00100 => ? = 2 - 1
[[3,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[3,2,1,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,3,2,1],[2,2,1]]
=> [2,2,1]
=> 11010 => 01010 => ? = 3 - 1
[[4,2,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => 00110 => ? = 3 - 1
[[4,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[5,3,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 001010 => ? = 3 - 1
[[4,4],[3]]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[[5,4],[4]]
=> [4]
=> 10000 => 00000 => ? = 1 - 1
[[3,3,1],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,4,1],[3,1]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[4,3,1],[3]]
=> [3]
=> 1000 => 0000 => ? = 1 - 1
[[5,4,1],[4,1]]
=> [4,1]
=> 100010 => 000010 => ? = 2 - 1
[[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,4,2],[3,2]]
=> [3,2]
=> 10100 => 00100 => ? = 2 - 1
[[3,2,2],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,3,2],[3,1]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[5,4,2],[4,2]]
=> [4,2]
=> 100100 => 000100 => ? = 2 - 1
[[3,3,2,1],[2,1,1]]
=> [2,1,1]
=> 10110 => 00110 => ? = 3 - 1
[[3,3,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,4,2,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 001010 => ? = 3 - 1
[[3,2,1,1],[2]]
=> [2]
=> 100 => 000 => ? = 1 - 1
[[4,3,2,1],[3,1,1]]
=> [3,1,1]
=> 100110 => 000110 => ? = 3 - 1
[[4,3,1,1],[3,1]]
=> [3,1]
=> 10010 => 00010 => ? = 2 - 1
[[5,4,2,1],[4,2,1]]
=> [4,2,1]
=> 1001010 => 0001010 => ? = 3 - 1
[[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[4,4,3],[3,3]]
=> [3,3]
=> 11000 => 01000 => ? = 3 - 1
[[4,3,3],[3,2]]
=> [3,2]
=> 10100 => 00100 => ? = 2 - 1
[[5,4,3],[4,3]]
=> [4,3]
=> 101000 => 001000 => ? = 3 - 1
[[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3,3,3,1],[2,2,1]]
=> [2,2,1]
=> 11010 => 01010 => ? = 3 - 1
[[3,3,2,1],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[4,4,3,1],[3,3,1]]
=> [3,3,1]
=> 110010 => 010010 => ? = 4 - 1
[[3,2,2,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3,3,1],[3,2,1]]
=> [3,2,1]
=> 101010 => 001010 => ? = 3 - 1
[[4,3,2,1],[3,2]]
=> [3,2]
=> 10100 => 00100 => ? = 2 - 1
[[5,4,3,1],[4,3,1]]
=> [4,3,1]
=> 1010010 => 0010010 => ? = 4 - 1
[[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[3,3,3,2],[2,2,2]]
=> [2,2,2]
=> 11100 => 01100 => ? = 2 - 1
[[3,3,2,2],[2,2,1]]
=> [2,2,1]
=> 11010 => 01010 => ? = 3 - 1
[[4,4,3,2],[3,3,2]]
=> [3,3,2]
=> 110100 => 010100 => ? = 2 - 1
[[3,2,2,2],[2,1,1]]
=> [2,1,1]
=> 10110 => 00110 => ? = 3 - 1
[[4,3,3,2],[3,2,2]]
=> [3,2,2]
=> 101100 => 001100 => ? = 2 - 1
[[4,3,2,2],[3,2,1]]
=> [3,2,1]
=> 101010 => 001010 => ? = 3 - 1
[[5,4,3,2],[4,3,2]]
=> [4,3,2]
=> 1010100 => 0010100 => ? = 2 - 1
[[2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[2,2,1,1,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[3,2,1,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[5,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[6,2,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[5,3,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,2,2],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[5,3,2],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[5,2,2],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[4,2,1,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[5,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,4,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,3,2],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,4,2],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[4,3,2],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[3,3,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 0110 => 2 = 3 - 1
[[3,3,1,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
[[4,3,1,1],[2,1]]
=> [2,1]
=> 1010 => 0010 => 1 = 2 - 1
[[4,3,3],[2,2]]
=> [2,2]
=> 1100 => 0100 => 1 = 2 - 1
[[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 010 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
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