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Your data matches 5 different statistics following compositions of up to 3 maps.
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Matching statistic: St000733
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Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [[1],[2]]
=> 2
[[1],[2]]
=> [[1,2]]
=> 1
[[1,2,3]]
=> [[1],[2],[3]]
=> 3
[[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[1,2],[3]]
=> [[1,3],[2]]
=> 1
[[1],[2],[3]]
=> [[1,2,3]]
=> 1
[[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 3
[[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 3
[[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 2
[[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 2
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1
[[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 4
[[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 4
[[1,2,3,5],[4]]
=> [[1,4],[2],[3],[5]]
=> 4
[[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[[1,3,5],[2,4]]
=> [[1,2],[3,4],[5]]
=> 3
[[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3
[[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 2
[[1,2,4],[3,5]]
=> [[1,3],[2,5],[4]]
=> 2
[[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
[[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 3
[[1,2,5],[3],[4]]
=> [[1,3,4],[2],[5]]
=> 3
[[1,3,4],[2],[5]]
=> [[1,2,5],[3],[4]]
=> 1
[[1,2,4],[3],[5]]
=> [[1,3,5],[2],[4]]
=> 1
[[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,3],[4,5]]
=> 2
[[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2
[[1,2],[3,5],[4]]
=> [[1,3,4],[2,5]]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,5],[3,4]]
=> 1
[[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 1
[[1,3],[2],[4],[5]]
=> [[1,2,4,5],[3]]
=> 1
[[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 5
[[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 5
[[1,2,3,5,6],[4]]
=> [[1,4],[2],[3],[5],[6]]
=> 5
[[1,2,3,4,6],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> 5
[[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
[[1,3,5,6],[2,4]]
=> [[1,2],[3,4],[5],[6]]
=> 4
[[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 4
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000504
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Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
St000504: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> 2
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> 3
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> 2
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 4
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> 3
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> 3
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> 2
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> 2
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 5
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> 4
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> 4
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> 4
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> 3
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> 3
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> 2
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> 2
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> 2
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> 3
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> 3
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> 3
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> 2
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> 2
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> 2
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> 2
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 6
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> 5
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5,6},{4}}
=> 5
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,4,5,6},{3}}
=> 5
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> 5
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,2,4,6},{3,5}}
=> 4
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,5,6},{3,4}}
=> 4
Description
The cardinality of the first block of a set partition.
The number of partitions of $\{1,\ldots,n\}$ into $k$ blocks in which the first block has cardinality $j+1$ is given by $\binom{n-1}{j}S(n-j-1,k-1)$, see [1, Theorem 1.1] and the references therein. Here, $S(n,k)$ are the ''Stirling numbers of the second kind'' counting all set partitions of $\{1,\ldots,n\}$ into $k$ blocks [2].
Matching statistic: St000382
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Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
Mp00128: Set partitions —to composition⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> [2] => 2
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> [1,1] => 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> [3] => 3
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> [2,1] => 2
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> [1,2] => 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,1,1] => 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [4] => 4
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> [3,1] => 3
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,1] => 3
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3] => 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2] => 2
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,2] => 2
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [2,1,1] => 2
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,2,1] => 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,1,2] => 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1] => 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> [5] => 5
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> [4,1] => 4
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> [4,1] => 4
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> [4,1] => 4
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> [1,4] => 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> [3,2] => 3
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> [3,2] => 3
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> [2,3] => 2
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> [2,3] => 2
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> [2,3] => 2
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> [3,1,1] => 3
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [3,1,1] => 3
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> [3,1,1] => 3
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> [1,3,1] => 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> [1,3,1] => 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> [1,1,3] => 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> [2,2,1] => 2
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> [2,1,2] => 2
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> [2,1,2] => 2
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> [1,2,2] => 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> [1,2,2] => 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1] => 2
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> [1,2,1,1] => 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> [1,1,2,1] => 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> [1,1,1,2] => 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> [6] => 6
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> [5,1] => 5
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5,6},{4}}
=> [5,1] => 5
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,4,5,6},{3}}
=> [5,1] => 5
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> [5,1] => 5
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> [1,5] => 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,2,4,6},{3,5}}
=> [4,2] => 4
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,5,6},{3,4}}
=> [4,2] => 4
Description
The first part of an integer composition.
Matching statistic: St000502
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(load all 2 compositions to match this statistic)
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00112: Set partitions —complement⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00112: Set partitions —complement⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> {{1,2}}
=> {{1,2}}
=> {{1,2}}
=> 1 = 2 - 1
[[1],[2]]
=> {{1},{2}}
=> {{1},{2}}
=> {{1},{2}}
=> 0 = 1 - 1
[[1,2,3]]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1,2,3}}
=> 2 = 3 - 1
[[1,3],[2]]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1 = 2 - 1
[[1,2],[3]]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 0 = 1 - 1
[[1],[2],[3]]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> 0 = 1 - 1
[[1,2,3,4]]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> 3 = 4 - 1
[[1,3,4],[2]]
=> {{1,3,4},{2}}
=> {{1,2,4},{3}}
=> {{1,2},{3,4}}
=> 2 = 3 - 1
[[1,2,4],[3]]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1},{2,3,4}}
=> 2 = 3 - 1
[[1,2,3],[4]]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,3},{2,4}}
=> 0 = 1 - 1
[[1,3],[2,4]]
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> {{1,4},{2,3}}
=> 1 = 2 - 1
[[1,2],[3,4]]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,2,4},{3}}
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> 0 = 1 - 1
[[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> 0 = 1 - 1
[[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> {{1,2,3,4,5}}
=> 4 = 5 - 1
[[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> {{1,2,3,5},{4}}
=> {{1,2,3},{4,5}}
=> 3 = 4 - 1
[[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> {{1,2,4,5},{3}}
=> {{1,2},{3,4,5}}
=> 3 = 4 - 1
[[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> {{1,3,4,5},{2}}
=> {{1},{2,3,4,5}}
=> 3 = 4 - 1
[[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> {{1},{2,3,4,5}}
=> {{1,3,5},{2,4}}
=> 0 = 1 - 1
[[1,3,5],[2,4]]
=> {{1,3,5},{2,4}}
=> {{1,3,5},{2,4}}
=> {{1,4,5},{2,3}}
=> 2 = 3 - 1
[[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> {{1,4,5},{2,3}}
=> {{1,3,4,5},{2}}
=> 2 = 3 - 1
[[1,3,4],[2,5]]
=> {{1,3,4},{2,5}}
=> {{1,4},{2,3,5}}
=> {{1,3,4},{2,5}}
=> 1 = 2 - 1
[[1,2,4],[3,5]]
=> {{1,2,4},{3,5}}
=> {{1,3},{2,4,5}}
=> {{1,4},{2,3,5}}
=> 1 = 2 - 1
[[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> {{1,2},{3,4,5}}
=> {{1,2,4},{3,5}}
=> 1 = 2 - 1
[[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> {{1,2,5},{3},{4}}
=> {{1,2},{3},{4,5}}
=> 2 = 3 - 1
[[1,3,5],[2],[4]]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> 2 = 3 - 1
[[1,2,5],[3],[4]]
=> {{1,2,5},{3},{4}}
=> {{1,4,5},{2},{3}}
=> {{1},{2},{3,4,5}}
=> 2 = 3 - 1
[[1,3,4],[2],[5]]
=> {{1,3,4},{2},{5}}
=> {{1},{2,3,5},{4}}
=> {{1,3,5},{2},{4}}
=> 0 = 1 - 1
[[1,2,4],[3],[5]]
=> {{1,2,4},{3},{5}}
=> {{1},{2,4,5},{3}}
=> {{1},{2,4},{3,5}}
=> 0 = 1 - 1
[[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> {{1},{2},{3,4,5}}
=> {{1,4},{2,5},{3}}
=> 0 = 1 - 1
[[1,4],[2,5],[3]]
=> {{1,4},{2,5},{3}}
=> {{1,4},{2,5},{3}}
=> {{1},{2,5},{3,4}}
=> 1 = 2 - 1
[[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> {{1,4},{2},{3,5}}
=> {{1,5},{2},{3,4}}
=> 1 = 2 - 1
[[1,2],[3,5],[4]]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1,5},{2,3},{4}}
=> 1 = 2 - 1
[[1,3],[2,4],[5]]
=> {{1,3},{2,4},{5}}
=> {{1},{2,4},{3,5}}
=> {{1,5},{2,4},{3}}
=> 0 = 1 - 1
[[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1,3},{2,5},{4}}
=> 0 = 1 - 1
[[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1},{2},{3},{4,5}}
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> 0 = 1 - 1
[[1,3],[2],[4],[5]]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> 0 = 1 - 1
[[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> 0 = 1 - 1
[[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> {{1,2,3,4,5,6}}
=> 5 = 6 - 1
[[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> {{1,2,3,4,6},{5}}
=> {{1,2,3,4},{5,6}}
=> 4 = 5 - 1
[[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> {{1,2,3,5,6},{4}}
=> {{1,2,3},{4,5,6}}
=> 4 = 5 - 1
[[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> {{1,2,4,5,6},{3}}
=> {{1,2},{3,4,5,6}}
=> 4 = 5 - 1
[[1,2,3,4,6],[5]]
=> {{1,2,3,4,6},{5}}
=> {{1,3,4,5,6},{2}}
=> {{1},{2,3,4,5,6}}
=> 4 = 5 - 1
[[1,2,3,4,5],[6]]
=> {{1,2,3,4,5},{6}}
=> {{1},{2,3,4,5,6}}
=> {{1,3,5},{2,4,6}}
=> 0 = 1 - 1
[[1,3,5,6],[2,4]]
=> {{1,3,5,6},{2,4}}
=> {{1,2,4,6},{3,5}}
=> {{1,2,5,6},{3,4}}
=> 3 = 4 - 1
[[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> {{1,2,5,6},{3,4}}
=> {{1,2,4,5,6},{3}}
=> 3 = 4 - 1
[[1,2,4,5,6,7,8],[3]]
=> {{1,2,4,5,6,7,8},{3}}
=> {{1,2,3,4,5,7,8},{6}}
=> {{1,2,3,4,5},{6,7,8}}
=> ? = 7 - 1
[[1,2,3,5,6,7,8],[4]]
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 7 - 1
[[1,2,3,4,6,7,8],[5]]
=> {{1,2,3,4,6,7,8},{5}}
=> {{1,2,3,5,6,7,8},{4}}
=> {{1,2,3},{4,5,6,7,8}}
=> ? = 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}}
=> {{1,3,5,7},{2,4,6,8}}
=> ? = 1 - 1
[[1,3,4,6,7,8],[2,5]]
=> {{1,3,4,6,7,8},{2,5}}
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,2,3,7,8},{4,5,6}}
=> ? = 6 - 1
[[1,3,4,5,7,8],[2,6]]
=> {{1,3,4,5,7,8},{2,6}}
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,2,7,8},{3,4,5,6}}
=> ? = 6 - 1
[[1,2,4,5,7,8],[3,6]]
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,2,4,5,7,8},{3,6}}
=> {{1,2,6,7,8},{3,4,5}}
=> ? = 6 - 1
[[1,3,4,5,6,8],[2,7]]
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,3,4,5,6,8},{2,7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ? = 6 - 1
[[1,2,4,5,6,8],[3,7]]
=> {{1,2,4,5,6,8},{3,7}}
=> {{1,3,4,5,7,8},{2,6}}
=> {{1,6,7,8},{2,3,4,5}}
=> ? = 6 - 1
[[1,2,3,5,6,8],[4,7]]
=> {{1,2,3,5,6,8},{4,7}}
=> {{1,3,4,6,7,8},{2,5}}
=> {{1,5,6,7,8},{2,3,4}}
=> ? = 6 - 1
[[1,3,4,5,6,7],[2,8]]
=> {{1,3,4,5,6,7},{2,8}}
=> {{1,7},{2,3,4,5,6,8}}
=> {{1,3,5,8},{2,4,6,7}}
=> ? = 2 - 1
[[1,2,4,5,6,7],[3,8]]
=> {{1,2,4,5,6,7},{3,8}}
=> {{1,6},{2,3,4,5,7,8}}
=> {{1,3,5,6,8},{2,4,7}}
=> ? = 2 - 1
[[1,2,3,5,6,7],[4,8]]
=> {{1,2,3,5,6,7},{4,8}}
=> {{1,5},{2,3,4,6,7,8}}
=> {{1,3,6,8},{2,4,5,7}}
=> ? = 2 - 1
[[1,2,3,4,6,7],[5,8]]
=> {{1,2,3,4,6,7},{5,8}}
=> {{1,4},{2,3,5,6,7,8}}
=> {{1,3,4,6,8},{2,5,7}}
=> ? = 2 - 1
[[1,2,3,4,5,7],[6,8]]
=> {{1,2,3,4,5,7},{6,8}}
=> {{1,3},{2,4,5,6,7,8}}
=> {{1,4,6,8},{2,3,5,7}}
=> ? = 2 - 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}}
=> {{1,2,4,6,8},{3,5,7}}
=> ? = 2 - 1
[[1,2,3,4,6,8],[5],[7]]
=> {{1,2,3,4,6,8},{5},{7}}
=> {{1,3,5,6,7,8},{2},{4}}
=> {{1},{2,3},{4,5,6,7,8}}
=> ? = 6 - 1
[[1,3,4,5,6,7],[2],[8]]
=> {{1,3,4,5,6,7},{2},{8}}
=> {{1},{2,3,4,5,6,8},{7}}
=> {{1,3,5},{2,4,6,8},{7}}
=> ? = 1 - 1
[[1,2,4,5,6,7],[3],[8]]
=> {{1,2,4,5,6,7},{3},{8}}
=> {{1},{2,3,4,5,7,8},{6}}
=> {{1,3,5,7},{2,4},{6,8}}
=> ? = 1 - 1
[[1,2,3,5,6,7],[4],[8]]
=> {{1,2,3,5,6,7},{4},{8}}
=> {{1},{2,3,4,6,7,8},{5}}
=> {{1,3},{2,4,6,8},{5,7}}
=> ? = 1 - 1
[[1,2,3,4,6,7],[5],[8]]
=> {{1,2,3,4,6,7},{5},{8}}
=> {{1},{2,3,5,6,7,8},{4}}
=> {{1,3,5,7},{2},{4,6,8}}
=> ? = 1 - 1
[[1,2,3,4,5,7],[6],[8]]
=> {{1,2,3,4,5,7},{6},{8}}
=> {{1},{2,4,5,6,7,8},{3}}
=> {{1},{2,4,6,8},{3,5,7}}
=> ? = 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}}
=> {{1,4,7},{2,5,8},{3,6}}
=> ? = 1 - 1
Description
The number of successions of a set partitions.
This is the number of indices $i$ such that $i$ and $i+1$ belonging to the same block.
Matching statistic: St001390
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 75%
Mp00069: Permutations —complement⟶ Permutations
St001390: Permutations ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 75%
Values
[[1,2]]
=> [1,2] => [2,1] => 2
[[1],[2]]
=> [2,1] => [1,2] => 1
[[1,2,3]]
=> [1,2,3] => [3,2,1] => 3
[[1,3],[2]]
=> [2,1,3] => [2,3,1] => 2
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => 1
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => 1
[[1,2,3,4]]
=> [1,2,3,4] => [4,3,2,1] => 4
[[1,3,4],[2]]
=> [2,1,3,4] => [3,4,2,1] => 3
[[1,2,4],[3]]
=> [3,1,2,4] => [2,4,3,1] => 3
[[1,2,3],[4]]
=> [4,1,2,3] => [1,4,3,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => 2
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5,4,3,2,1] => 5
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [4,5,3,2,1] => 4
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,5,4,2,1] => 4
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,5,4,3,1] => 4
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [1,5,4,3,2] => 1
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,2,5,3,1] => 3
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,2,5,4,1] => 3
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [4,1,5,3,2] => 2
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,5,4,2] => 2
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [2,1,5,4,3] => 2
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,4,5,2,1] => 3
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,4,5,3,1] => 3
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [2,3,5,4,1] => 3
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [1,4,5,3,2] => 1
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,3,5,4,2] => 1
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [3,4,1,5,2] => 2
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => 2
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [2,3,1,5,4] => 2
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [1,4,2,5,3] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,3,2,5,4] => 1
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,3,4,5,1] => 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,3,4,5,2] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,4,5,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,3,5,4] => 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,3,4,5] => 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6,5,4,3,2,1] => 6
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [5,6,4,3,2,1] => 5
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [4,6,5,3,2,1] => 5
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [3,6,5,4,2,1] => 5
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,6,5,4,3,1] => 5
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 1
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [5,3,6,4,2,1] => 4
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => 4
[[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 7
[[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => [6,7,5,4,3,2,1] => ? = 6
[[1,2,4,5,6,7],[3]]
=> [3,1,2,4,5,6,7] => [5,7,6,4,3,2,1] => ? = 6
[[1,2,3,5,6,7],[4]]
=> [4,1,2,3,5,6,7] => [4,7,6,5,3,2,1] => ? = 6
[[1,2,3,4,6,7],[5]]
=> [5,1,2,3,4,6,7] => [3,7,6,5,4,2,1] => ? = 6
[[1,2,3,4,5,7],[6]]
=> [6,1,2,3,4,5,7] => [2,7,6,5,4,3,1] => ? = 6
[[1,2,3,4,5,6],[7]]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => ? = 1
[[1,3,5,6,7],[2,4]]
=> [2,4,1,3,5,6,7] => [6,4,7,5,3,2,1] => ? = 5
[[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => [5,4,7,6,3,2,1] => ? = 5
[[1,3,4,6,7],[2,5]]
=> [2,5,1,3,4,6,7] => [6,3,7,5,4,2,1] => ? = 5
[[1,2,4,6,7],[3,5]]
=> [3,5,1,2,4,6,7] => [5,3,7,6,4,2,1] => ? = 5
[[1,2,3,6,7],[4,5]]
=> [4,5,1,2,3,6,7] => [4,3,7,6,5,2,1] => ? = 5
[[1,3,4,5,7],[2,6]]
=> [2,6,1,3,4,5,7] => [6,2,7,5,4,3,1] => ? = 5
[[1,2,4,5,7],[3,6]]
=> [3,6,1,2,4,5,7] => [5,2,7,6,4,3,1] => ? = 5
[[1,2,3,5,7],[4,6]]
=> [4,6,1,2,3,5,7] => [4,2,7,6,5,3,1] => ? = 5
[[1,2,3,4,7],[5,6]]
=> [5,6,1,2,3,4,7] => [3,2,7,6,5,4,1] => ? = 5
[[1,3,4,5,6],[2,7]]
=> [2,7,1,3,4,5,6] => [6,1,7,5,4,3,2] => ? = 2
[[1,2,4,5,6],[3,7]]
=> [3,7,1,2,4,5,6] => [5,1,7,6,4,3,2] => ? = 2
[[1,2,3,5,6],[4,7]]
=> [4,7,1,2,3,5,6] => [4,1,7,6,5,3,2] => ? = 2
[[1,2,3,4,6],[5,7]]
=> [5,7,1,2,3,4,6] => [3,1,7,6,5,4,2] => ? = 2
[[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => [2,1,7,6,5,4,3] => ? = 2
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [5,6,7,4,3,2,1] => ? = 5
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [4,6,7,5,3,2,1] => ? = 5
[[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => [4,5,7,6,3,2,1] => ? = 5
[[1,3,4,6,7],[2],[5]]
=> [5,2,1,3,4,6,7] => [3,6,7,5,4,2,1] => ? = 5
[[1,2,4,6,7],[3],[5]]
=> [5,3,1,2,4,6,7] => [3,5,7,6,4,2,1] => ? = 5
[[1,2,3,6,7],[4],[5]]
=> [5,4,1,2,3,6,7] => [3,4,7,6,5,2,1] => ? = 5
[[1,3,4,5,7],[2],[6]]
=> [6,2,1,3,4,5,7] => [2,6,7,5,4,3,1] => ? = 5
[[1,2,4,5,7],[3],[6]]
=> [6,3,1,2,4,5,7] => [2,5,7,6,4,3,1] => ? = 5
[[1,2,3,5,7],[4],[6]]
=> [6,4,1,2,3,5,7] => [2,4,7,6,5,3,1] => ? = 5
[[1,2,3,4,7],[5],[6]]
=> [6,5,1,2,3,4,7] => [2,3,7,6,5,4,1] => ? = 5
[[1,3,4,5,6],[2],[7]]
=> [7,2,1,3,4,5,6] => [1,6,7,5,4,3,2] => ? = 1
[[1,2,4,5,6],[3],[7]]
=> [7,3,1,2,4,5,6] => [1,5,7,6,4,3,2] => ? = 1
[[1,2,3,5,6],[4],[7]]
=> [7,4,1,2,3,5,6] => [1,4,7,6,5,3,2] => ? = 1
[[1,2,3,4,6],[5],[7]]
=> [7,5,1,2,3,4,6] => [1,3,7,6,5,4,2] => ? = 1
[[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => [1,2,7,6,5,4,3] => ? = 1
[[1,3,5,7],[2,4,6]]
=> [2,4,6,1,3,5,7] => [6,4,2,7,5,3,1] => ? = 4
[[1,2,5,7],[3,4,6]]
=> [3,4,6,1,2,5,7] => [5,4,2,7,6,3,1] => ? = 4
[[1,3,4,7],[2,5,6]]
=> [2,5,6,1,3,4,7] => [6,3,2,7,5,4,1] => ? = 4
[[1,2,4,7],[3,5,6]]
=> [3,5,6,1,2,4,7] => [5,3,2,7,6,4,1] => ? = 4
[[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => [4,3,2,7,6,5,1] => ? = 4
[[1,3,5,6],[2,4,7]]
=> [2,4,7,1,3,5,6] => [6,4,1,7,5,3,2] => ? = 3
[[1,2,5,6],[3,4,7]]
=> [3,4,7,1,2,5,6] => [5,4,1,7,6,3,2] => ? = 3
[[1,3,4,6],[2,5,7]]
=> [2,5,7,1,3,4,6] => [6,3,1,7,5,4,2] => ? = 3
[[1,2,4,6],[3,5,7]]
=> [3,5,7,1,2,4,6] => [5,3,1,7,6,4,2] => ? = 3
[[1,2,3,6],[4,5,7]]
=> [4,5,7,1,2,3,6] => [4,3,1,7,6,5,2] => ? = 3
[[1,3,4,5],[2,6,7]]
=> [2,6,7,1,3,4,5] => [6,2,1,7,5,4,3] => ? = 3
[[1,2,4,5],[3,6,7]]
=> [3,6,7,1,2,4,5] => [5,2,1,7,6,4,3] => ? = 3
[[1,2,3,5],[4,6,7]]
=> [4,6,7,1,2,3,5] => [4,2,1,7,6,5,3] => ? = 3
[[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => [3,2,1,7,6,5,4] => ? = 3
Description
The number of bumps occurring when Schensted-inserting the letter 1 of a permutation.
For a given permutation $\pi$, this is the index of the row containing $\pi^{-1}(1)$ of the recording tableau of $\pi$ (obtained by [[Mp00070]]).
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