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Your data matches 6 different statistics following compositions of up to 3 maps.
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Matching statistic: St000335
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
Description
The difference of lower and upper interactions.
An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Matching statistic: St000613
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000613: Set partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00220: Set partitions —Yip⟶ Set partitions
St000613: Set partitions ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2,3,5},{4}}
=> 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1,2,3,4,5},{6}}
=> 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> {{1,2,3,4,6},{5}}
=> 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> {{1,2,3,5,6},{4}}
=> 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> {{1,2,3,5},{4},{6}}
=> 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> {{1,2,4,6},{3},{5}}
=> 1 = 2 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> {{1,2,4},{3},{5},{6}}
=> 1 = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> {{1},{2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> {{1,2,3,4,5,6},{7}}
=> {{1,2,3,4,5,6},{7}}
=> 0 = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> {{1,2,3,4,5},{6,7}}
=> {{1,2,3,4,5,7},{6}}
=> 1 = 2 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> {{1,2,3,4,5},{6},{7}}
=> {{1,2,3,4,5},{6},{7}}
=> 0 = 1 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> {{1,2,3,4,6,7},{5}}
=> 0 = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> {{1,2,3,4,6},{5},{7}}
=> 1 = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> {{1,2,3,5,6},{4},{7}}
=> 0 = 1 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> {{1,2,3,5,7},{4},{6}}
=> 1 = 2 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> {{1,2,3,5},{4},{6},{7}}
=> 1 = 2 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> {{1,2,4,6},{3},{5},{7}}
=> 1 = 2 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> {{1,2,4},{3},{5},{6},{7}}
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> {{1,2},{3},{4},{5},{6},{7}}
=> {{1,2},{3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> {{1,2,3,4,5,6,7},{8}}
=> {{1,2,3,4,5,6,7},{8}}
=> 0 = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> {{1,2,3,4,5,6},{7,8}}
=> {{1,2,3,4,5,6,8},{7}}
=> 1 = 2 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> {{1,2,3,4,5,6},{7},{8}}
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 1 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> {{1,2,3,4,5},{6,7,8}}
=> {{1,2,3,4,5,7,8},{6}}
=> 0 = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> {{1,2,3,4,5},{6,7},{8}}
=> {{1,2,3,4,5,7},{6},{8}}
=> ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> {{1,2,3,4,5},{6},{7},{8}}
=> {{1,2,3,4,5},{6},{7},{8}}
=> ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> {{1,2,3,4,6,7,8},{5}}
=> 0 = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> {{1,2,3,4,6,7},{5},{8}}
=> ? = 1 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> {{1,2,3,4,6,8},{5},{7}}
=> ? = 2 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> {{1,2,3,4,6},{5},{7},{8}}
=> ? = 2 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> {{1,2,3,5,6,8},{4},{7}}
=> ? = 2 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> {{1,2,3,5,6},{4},{7},{8}}
=> ? = 1 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> {{1,2,3,5,7},{4},{6},{8}}
=> ? = 2 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> {{1,2,3,5},{4},{6},{7},{8}}
=> ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> {{1,2,3},{4},{5},{6},{7},{8}}
=> ? = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> {{1,2,4,6,8},{3},{5},{7}}
=> ? = 3 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> {{1,2,4,6},{3},{5},{7},{8}}
=> ? = 2 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> {{1,2},{3,4},{5},{6},{7},{8}}
=> {{1,2,4},{3},{5},{6},{7},{8}}
=> ? = 2 - 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> {{1,2},{3},{4},{5},{6},{7},{8}}
=> ? = 1 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> {{1,2,3,4,5,6,7,8},{9}}
=> {{1,2,3,4,5,6,7,8},{9}}
=> ? = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> {{1,2,3,4,5,6,7},{8,9}}
=> {{1,2,3,4,5,6,7,9},{8}}
=> ? = 2 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> {{1,2,3,4,5,6,7},{8},{9}}
=> {{1,2,3,4,5,6,7},{8},{9}}
=> ? = 1 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> {{1,2,3,4,5,6},{7,8,9}}
=> {{1,2,3,4,5,6,8,9},{7}}
=> ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> {{1,2,3,4,5,6},{7,8},{9}}
=> {{1,2,3,4,5,6,8},{7},{9}}
=> ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> {{1,2,3,4,5,6},{7},{8},{9}}
=> {{1,2,3,4,5,6},{7},{8},{9}}
=> ? = 1 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> {{1,2,3,4,5},{6,7,8,9}}
=> {{1,2,3,4,5,7,8,9},{6}}
=> ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> {{1,2,3,4,5},{6,7,8},{9}}
=> {{1,2,3,4,5,7,8},{6},{9}}
=> ? = 1 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> {{1,2,3,4,5},{6,7},{8,9}}
=> {{1,2,3,4,5,7,9},{6},{8}}
=> ? = 2 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> {{1,2,3,4,5},{6,7},{8},{9}}
=> {{1,2,3,4,5,7},{6},{8},{9}}
=> ? = 2 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> {{1,2,3,4},{5,6,7,8},{9}}
=> {{1,2,3,4,6,7,8},{5},{9}}
=> ? = 1 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> {{1,2,3,4,6,7,9},{5},{8}}
=> ? = 2 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> {{1,2,3,4,6,7},{5},{8},{9}}
=> ? = 1 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> {{1,2,3,4,6,8},{5},{7},{9}}
=> ? = 2 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> {{1,2,3,4,6},{5},{7},{8},{9}}
=> ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> {{1,2,3,4},{5},{6},{7},{8},{9}}
=> {{1,2,3,4},{5},{6},{7},{8},{9}}
=> ? = 1 - 1
[3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> {{1,2,3,5,6,8,9},{4},{7}}
=> ? = 1 - 1
[3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> {{1,2,3,5,6,8},{4},{7},{9}}
=> ? = 2 - 1
[3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> {{1,2,3,5,6},{4},{7},{8},{9}}
=> ? = 1 - 1
[3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> {{1,2,3},{4,5},{6,7},{8,9}}
=> {{1,2,3,5,7,9},{4},{6},{8}}
=> ? = 3 - 1
[3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> {{1,2,3,5,7},{4},{6},{8},{9}}
=> ? = 2 - 1
[3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> {{1,2,3},{4,5},{6},{7},{8},{9}}
=> {{1,2,3,5},{4},{6},{7},{8},{9}}
=> ? = 2 - 1
[3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> {{1,2,3},{4},{5},{6},{7},{8},{9}}
=> ? = 1 - 1
[2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> {{1,2,4,6,8},{3},{5},{7},{9}}
=> ? = 3 - 1
[2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> {{1,2},{3,4},{5,6},{7},{8},{9}}
=> {{1,2,4,6},{3},{5},{7},{8},{9}}
=> ? = 2 - 1
[9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> {{1,2,3,4,5,6,7,8,9},{10}}
=> ? = 1 - 1
[8,2]
=> [[1,2,3,4,5,6,7,8],[9,10]]
=> {{1,2,3,4,5,6,7,8},{9,10}}
=> {{1,2,3,4,5,6,7,8,10},{9}}
=> ? = 2 - 1
[8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> {{1,2,3,4,5,6,7,8},{9},{10}}
=> ? = 1 - 1
[7,3]
=> [[1,2,3,4,5,6,7],[8,9,10]]
=> {{1,2,3,4,5,6,7},{8,9,10}}
=> {{1,2,3,4,5,6,7,9,10},{8}}
=> ? = 1 - 1
[7,2,1]
=> [[1,2,3,4,5,6,7],[8,9],[10]]
=> {{1,2,3,4,5,6,7},{8,9},{10}}
=> {{1,2,3,4,5,6,7,9},{8},{10}}
=> ? = 2 - 1
[7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> {{1,2,3,4,5,6,7},{8},{9},{10}}
=> {{1,2,3,4,5,6,7},{8},{9},{10}}
=> ? = 1 - 1
[6,4]
=> [[1,2,3,4,5,6],[7,8,9,10]]
=> {{1,2,3,4,5,6},{7,8,9,10}}
=> {{1,2,3,4,5,6,8,9,10},{7}}
=> ? = 1 - 1
[6,3,1]
=> [[1,2,3,4,5,6],[7,8,9],[10]]
=> {{1,2,3,4,5,6},{7,8,9},{10}}
=> {{1,2,3,4,5,6,8,9},{7},{10}}
=> ? = 1 - 1
Description
The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block.
Matching statistic: St001644
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 1
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 2
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 2
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 2
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 2
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 2
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 1
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
=> ? = 2
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
=> ? = 2
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 1
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 2
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
=> ? = 2
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
=> ? = 2
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> ? = 1
[8,1]
=> [[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
[7,2]
=> [[7,2],[]]
=> ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
=> ([(0,7),(1,3),(1,4),(2,5),(2,6),(3,7),(4,8),(5,8),(6,8)],9)
=> ? = 2
[7,1,1]
=> [[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
[6,3]
=> [[6,3],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 1
[6,2,1]
=> [[6,2,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9)
=> ([(0,8),(1,6),(2,3),(2,6),(3,7),(4,7),(4,8),(5,7),(5,8)],9)
=> ? = 2
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
[5,4]
=> [[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[5,3,1]
=> [[5,3,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 1
[5,2,2]
=> [[5,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 2
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9)
=> ([(0,6),(1,5),(2,5),(2,8),(3,7),(3,8),(4,7),(4,8),(6,7)],9)
=> ? = 2
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
[4,4,1]
=> [[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1
[4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 1
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 2
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9)
=> ([(0,6),(1,5),(2,5),(2,8),(3,7),(3,8),(4,7),(4,8),(6,7)],9)
=> ? = 2
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
[3,3,3]
=> [[3,3,3],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1
[3,3,2,1]
=> [[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 1
[3,2,2,2]
=> [[3,2,2,2],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 3
[3,2,2,1,1]
=> [[3,2,2,1,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 2
[3,2,1,1,1,1]
=> [[3,2,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9)
=> ([(0,8),(1,6),(2,3),(2,6),(3,7),(4,7),(4,8),(5,7),(5,8)],9)
=> ? = 2
[3,1,1,1,1,1,1]
=> [[3,1,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St001330
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00185: Skew partitions —cell poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 3 = 2 + 1
[2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1]
=> [[4,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,2]
=> [[3,2],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,1,1]
=> [[3,1,1],[]]
=> ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[2,2,1]
=> [[2,2,1],[]]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2 = 1 + 1
[5,1]
=> [[5,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[4,1,1]
=> [[4,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[3,3]
=> [[3,3],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
[3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
[3,1,1,1]
=> [[3,1,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(5,1)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[2,2,2]
=> [[2,2,2],[]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 1
[2,2,1,1]
=> [[2,2,1,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> ? = 2 + 1
[2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 2 = 1 + 1
[6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[5,2]
=> [[5,2],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[5,1,1]
=> [[5,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[4,3]
=> [[4,3],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[4,2,1]
=> [[4,2,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 2 + 1
[4,1,1,1]
=> [[4,1,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[3,3,1]
=> [[3,3,1],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 1 + 1
[3,2,2]
=> [[3,2,2],[]]
=> ([(0,3),(0,4),(2,6),(3,1),(3,5),(4,2),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
[3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> ? = 2 + 1
[3,1,1,1,1]
=> [[3,1,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(6,1)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[2,2,2,1]
=> [[2,2,2,1],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
[2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(4,1),(5,3),(5,6)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 2 = 1 + 1
[7,1]
=> [[7,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
=> ? = 2 + 1
[6,1,1]
=> [[6,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[5,2,1]
=> [[5,2,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
=> ? = 2 + 1
[5,1,1,1]
=> [[5,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[4,4]
=> [[4,4],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[4,3,1]
=> [[4,3,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 1 + 1
[4,2,2]
=> [[4,2,2],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[4,2,1,1]
=> [[4,2,1,1],[]]
=> ([(0,5),(0,6),(3,2),(4,1),(5,3),(5,7),(6,4),(6,7)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 2 + 1
[4,1,1,1,1]
=> [[4,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(7,3)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[3,3,2]
=> [[3,3,2],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7)],8)
=> ([(0,3),(0,7),(1,2),(1,7),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[3,3,1,1]
=> [[3,3,1,1],[]]
=> ([(0,4),(0,5),(1,7),(3,2),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 1 + 1
[3,2,2,1]
=> [[3,2,2,1],[]]
=> ([(0,4),(0,5),(3,2),(3,7),(4,3),(4,6),(5,1),(5,6),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,4),(3,7),(4,5),(4,6),(5,7)],8)
=> ? = 2 + 1
[3,2,1,1,1]
=> [[3,2,1,1,1],[]]
=> ([(0,5),(0,6),(3,4),(4,2),(5,3),(5,7),(6,1),(6,7)],8)
=> ([(0,7),(1,5),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6)],8)
=> ? = 2 + 1
[3,1,1,1,1,1]
=> [[3,1,1,1,1,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(7,1)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ([(0,4),(1,2),(1,5),(2,7),(3,5),(3,6),(4,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,7),(4,7),(5,7)],8)
=> ? = 2 + 1
[2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ([(0,2),(0,7),(3,4),(4,6),(5,3),(6,1),(7,5)],8)
=> ([(0,7),(1,6),(2,3),(2,4),(3,5),(4,6),(5,7)],8)
=> 2 = 1 + 1
[8,1]
=> [[8,1],[]]
=> ([(0,2),(0,8),(3,5),(4,3),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[7,2]
=> [[7,2],[]]
=> ([(0,2),(0,7),(2,8),(3,4),(4,6),(5,3),(6,1),(7,5),(7,8)],9)
=> ([(0,7),(1,3),(1,4),(2,5),(2,6),(3,7),(4,8),(5,8),(6,8)],9)
=> ? = 2 + 1
[7,1,1]
=> [[7,1,1],[]]
=> ([(0,7),(0,8),(3,4),(4,6),(5,3),(6,2),(7,5),(8,1)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[6,3]
=> [[6,3],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(4,8),(5,1),(6,4),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[6,2,1]
=> [[6,2,1],[]]
=> ([(0,6),(0,7),(3,5),(4,3),(5,2),(6,4),(6,8),(7,1),(7,8)],9)
=> ([(0,8),(1,6),(2,3),(2,6),(3,7),(4,7),(4,8),(5,7),(5,8)],9)
=> ? = 2 + 1
[6,1,1,1]
=> [[6,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[5,4]
=> [[5,4],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,8),(4,1),(4,7),(5,3),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 1
[5,3,1]
=> [[5,3,1],[]]
=> ([(0,5),(0,6),(3,4),(3,8),(4,2),(5,3),(5,7),(6,1),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[5,2,2]
=> [[5,2,2],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 2 + 1
[5,2,1,1]
=> [[5,2,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9)
=> ([(0,6),(1,5),(2,5),(2,8),(3,7),(3,8),(4,7),(4,8),(6,7)],9)
=> ? = 2 + 1
[5,1,1,1,1]
=> [[5,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,6),(5,2),(6,1),(7,3),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[4,4,1]
=> [[4,4,1],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 1 + 1
[4,3,2]
=> [[4,3,2],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[4,3,1,1]
=> [[4,3,1,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[4,2,2,1]
=> [[4,2,2,1],[]]
=> ([(0,5),(0,6),(3,1),(4,2),(4,8),(5,3),(5,7),(6,4),(6,7),(7,8)],9)
=> ([(0,8),(1,4),(2,6),(2,7),(3,5),(3,8),(4,7),(5,6),(5,7),(6,8)],9)
=> ? = 2 + 1
[4,2,1,1,1]
=> [[4,2,1,1,1],[]]
=> ([(0,6),(0,7),(3,4),(4,1),(5,2),(6,5),(6,8),(7,3),(7,8)],9)
=> ([(0,6),(1,5),(2,5),(2,8),(3,7),(3,8),(4,7),(4,8),(6,7)],9)
=> ? = 2 + 1
[4,1,1,1,1,1]
=> [[4,1,1,1,1,1],[]]
=> ([(0,7),(0,8),(3,5),(4,3),(5,2),(6,1),(7,6),(8,4)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 1 + 1
[3,3,3]
=> [[3,3,3],[]]
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,3,2,1]
=> [[3,3,2,1],[]]
=> ([(0,4),(0,5),(2,7),(3,1),(3,8),(4,2),(4,6),(5,3),(5,6),(6,7),(6,8)],9)
=> ([(0,7),(1,3),(1,8),(2,7),(2,8),(3,5),(4,5),(4,6),(5,8),(6,7),(6,8)],9)
=> ? = 2 + 1
[3,3,1,1,1]
=> [[3,3,1,1,1],[]]
=> ([(0,5),(0,6),(2,8),(3,4),(4,1),(5,3),(5,7),(6,2),(6,7),(7,8)],9)
=> ([(0,5),(1,2),(1,6),(2,8),(3,5),(3,7),(4,6),(4,7),(6,8),(7,8)],9)
=> ? = 1 + 1
[3,2,2,2]
=> [[3,2,2,2],[]]
=> ([(0,4),(0,5),(2,7),(3,2),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(8,7)],9)
=> ([(0,8),(1,4),(1,8),(2,3),(2,6),(3,7),(4,5),(4,6),(5,7),(5,8),(6,7)],9)
=> ? = 3 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000732
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000732: Permutations ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [[1],[2]]
=> [[1,2]]
=> [1,2] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> [2,1,3] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> [1,2,3] => 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [2,4,1,3] => 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0 = 1 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> [2,4,1,3,5] => 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> [4,3,2,6,1,5] => 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => 1 = 2 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0 = 1 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [[1,6],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6] => ? = 2 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => ? = 1 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [[1,5],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5] => ? = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7] => ? = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => ? = 1 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> [3,6,2,5,1,4,7] => ? = 1 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> [3,2,5,7,1,4,6] => ? = 2 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> [3,2,5,1,4,6,7] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => ? = 1 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => ? = 2 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => ? = 2 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 0 = 1 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8] => ? = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [[1,7],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7] => ? = 2 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => ? = 1 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [[1,6],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6] => ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8] => ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5] => ? = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8] => ? = 1 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7] => ? = 2 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => ? = 1 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> [3,6,2,5,8,1,4,7] => ? = 2 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> [3,6,2,5,1,4,7,8] => ? = 1 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> [3,2,5,7,1,4,6,8] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> [3,2,5,1,4,6,7,8] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => ? = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> [2,4,6,8,1,3,5,7] => ? = 3 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> [2,4,6,1,3,5,7,8] => ? = 2 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> [2,4,1,3,5,6,7,8] => ? = 2 - 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => ? = 1 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1,9] => ? = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [[1,8],[2,9],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,9,1,8] => ? = 2 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => ? = 1 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [[1,7],[2,8],[3,9],[4],[5],[6]]
=> [6,5,4,3,9,2,8,1,7] => ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> [6,5,4,3,2,8,1,7,9] => ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [[1,6],[2,7],[3,8],[4,9],[5]]
=> [5,4,9,3,8,2,7,1,6] => ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> [5,4,3,8,2,7,1,6,9] => ? = 1 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> [5,4,3,2,7,9,1,6,8] => ? = 2 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> [5,4,3,2,7,1,6,8,9] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => ? = 1 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> [4,8,3,7,2,6,1,5,9] => ? = 1 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> [4,3,7,2,6,9,1,5,8] => ? = 2 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> [4,3,7,2,6,1,5,8,9] => ? = 1 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> [4,3,2,6,8,1,5,7,9] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> [4,3,2,6,1,5,7,8,9] => ? = 2 - 1
[4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8,9] => ? = 1 - 1
Description
The number of double deficiencies of a permutation.
A double deficiency is an index $\sigma(i)$ such that $i > \sigma(i) > \sigma(\sigma(i))$.
Matching statistic: St000367
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000367: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000367: Permutations ⟶ ℤResult quality: 4% ●values known / values provided: 4%●distinct values known / distinct values provided: 33%
Values
[1,1]
=> [[1],[2]]
=> [2,1] => [2,1] => 0 = 1 - 1
[2,1]
=> [[1,2],[3]]
=> [3,1,2] => [2,3,1] => 0 = 1 - 1
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => [3,1,2] => 0 = 1 - 1
[3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => [2,4,3,1] => 1 = 2 - 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => [2,4,1,3] => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => [4,1,2,3] => 0 = 1 - 1
[4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
[3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,3,5,4,1] => 1 = 2 - 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [2,3,5,1,4] => 0 = 1 - 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [2,5,4,1,3] => 1 = 2 - 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [2,5,1,3,4] => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,1,2,3,4] => 0 = 1 - 1
[5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => 0 = 1 - 1
[4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [2,3,4,6,5,1] => 1 = 2 - 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [2,3,4,6,1,5] => 0 = 1 - 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,3,6,4,5,1] => 0 = 1 - 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [2,3,6,5,1,4] => 1 = 2 - 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [2,3,6,1,4,5] => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [2,6,4,1,5,3] => 1 = 2 - 1
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => [2,6,4,1,3,5] => 1 = 2 - 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => [2,6,1,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,1,2,3,4,5] => 0 = 1 - 1
[6,1]
=> [[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1 - 1
[5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 2 - 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [2,3,4,5,7,1,6] => ? = 1 - 1
[4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [2,3,4,7,5,6,1] => ? = 1 - 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => [2,3,4,7,6,1,5] => ? = 2 - 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => [2,3,4,7,1,5,6] => ? = 1 - 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => [2,3,7,5,6,1,4] => ? = 1 - 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => [2,3,7,5,1,6,4] => ? = 2 - 1
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => [2,3,7,5,1,4,6] => ? = 2 - 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 1 - 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => [2,7,4,1,6,3,5] => ? = 2 - 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => [2,7,4,1,3,5,6] => ? = 2 - 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => [2,7,1,3,4,5,6] => ? = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,1,2,3,4,5,6] => ? = 1 - 1
[7,1]
=> [[1,2,3,4,5,6,7],[8]]
=> [8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,8,1] => ? = 1 - 1
[6,2]
=> [[1,2,3,4,5,6],[7,8]]
=> [7,8,1,2,3,4,5,6] => [2,3,4,5,6,8,7,1] => ? = 2 - 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => [2,3,4,5,6,8,1,7] => ? = 1 - 1
[5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> [6,7,8,1,2,3,4,5] => [2,3,4,5,8,6,7,1] => ? = 1 - 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => [2,3,4,5,8,7,1,6] => ? = 2 - 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => [2,3,4,5,8,1,6,7] => ? = 1 - 1
[4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => [2,3,4,8,5,6,7,1] => ? = 1 - 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => [2,3,4,8,6,7,1,5] => ? = 1 - 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => [2,3,4,8,6,1,7,5] => ? = 2 - 1
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => [2,3,4,8,6,1,5,7] => ? = 2 - 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => [2,3,4,8,1,5,6,7] => ? = 1 - 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => [2,3,8,5,6,1,7,4] => ? = 2 - 1
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => [2,3,8,5,6,1,4,7] => ? = 1 - 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => [2,3,8,5,1,7,4,6] => ? = 2 - 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => [2,3,8,5,1,4,6,7] => ? = 2 - 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => [2,3,8,1,4,5,6,7] => ? = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => [2,8,4,1,6,3,7,5] => ? = 3 - 1
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => [2,8,4,1,6,3,5,7] => ? = 2 - 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => [2,8,4,1,3,5,6,7] => ? = 2 - 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => [2,8,1,3,4,5,6,7] => ? = 1 - 1
[8,1]
=> [[1,2,3,4,5,6,7,8],[9]]
=> [9,1,2,3,4,5,6,7,8] => [2,3,4,5,6,7,8,9,1] => ? = 1 - 1
[7,2]
=> [[1,2,3,4,5,6,7],[8,9]]
=> [8,9,1,2,3,4,5,6,7] => [2,3,4,5,6,7,9,8,1] => ? = 2 - 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => [2,3,4,5,6,7,9,1,8] => ? = 1 - 1
[6,3]
=> [[1,2,3,4,5,6],[7,8,9]]
=> [7,8,9,1,2,3,4,5,6] => [2,3,4,5,6,9,7,8,1] => ? = 1 - 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => [2,3,4,5,6,9,8,1,7] => ? = 2 - 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => [2,3,4,5,6,9,1,7,8] => ? = 1 - 1
[5,4]
=> [[1,2,3,4,5],[6,7,8,9]]
=> [6,7,8,9,1,2,3,4,5] => [2,3,4,5,9,6,7,8,1] => ? = 1 - 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => [2,3,4,5,9,7,8,1,6] => ? = 1 - 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => [2,3,4,5,9,7,1,8,6] => ? = 2 - 1
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => [2,3,4,5,9,7,1,6,8] => ? = 2 - 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => [2,3,4,5,9,1,6,7,8] => ? = 1 - 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => [2,3,4,9,6,7,8,1,5] => ? = 1 - 1
[4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [8,9,5,6,7,1,2,3,4] => [2,3,4,9,6,7,1,8,5] => ? = 2 - 1
[4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [9,8,5,6,7,1,2,3,4] => [2,3,4,9,6,7,1,5,8] => ? = 1 - 1
[4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,1,2,3,4] => [2,3,4,9,6,1,8,5,7] => ? = 2 - 1
[4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,1,2,3,4] => [2,3,4,9,6,1,5,7,8] => ? = 2 - 1
Description
The number of simsun double descents of a permutation.
The restriction of a permutation $\pi$ to $[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of $\pi$ of letters in $[k]$.
A simsun double descent of a permutation $\pi$ is a double descent of any restriction of $\pi$ to $[1,\ldots,k]$ for some $k$. (Note here that the same double descent can appear in multiple restrictions!)
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