Your data matches 30 different statistics following compositions of up to 3 maps.
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Matching statistic: St000733
Mapping
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => 1 = 0 + 1
[[1],[2]]
 => 2 = 1 + 1
[[1,2,3]]
 => 1 = 0 + 1
[[1,3],[2]]
 => 1 = 0 + 1
[[1,2],[3]]
 => 2 = 1 + 1
[[1],[2],[3]]
 => 3 = 2 + 1
[[1,2,3,4]]
 => 1 = 0 + 1
[[1,3,4],[2]]
 => 1 = 0 + 1
[[1,2,4],[3]]
 => 1 = 0 + 1
[[1,2,3],[4]]
 => 2 = 1 + 1
[[1,3],[2,4]]
 => 2 = 1 + 1
[[1,2],[3,4]]
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => 1 = 0 + 1
[[1,3],[2],[4]]
 => 3 = 2 + 1
[[1,2],[3],[4]]
 => 3 = 2 + 1
[[1],[2],[3],[4]]
 => 4 = 3 + 1
[[1,2,3,4,5]]
 => 1 = 0 + 1
[[1,3,4,5],[2]]
 => 1 = 0 + 1
[[1,2,4,5],[3]]
 => 1 = 0 + 1
[[1,2,3,5],[4]]
 => 1 = 0 + 1
[[1,2,3,4],[5]]
 => 2 = 1 + 1
[[1,3,5],[2,4]]
 => 1 = 0 + 1
[[1,2,5],[3,4]]
 => 1 = 0 + 1
[[1,3,4],[2,5]]
 => 2 = 1 + 1
[[1,2,4],[3,5]]
 => 2 = 1 + 1
[[1,2,3],[4,5]]
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => 1 = 0 + 1
[[1,3,5],[2],[4]]
 => 1 = 0 + 1
[[1,2,5],[3],[4]]
 => 1 = 0 + 1
[[1,3,4],[2],[5]]
 => 3 = 2 + 1
[[1,2,4],[3],[5]]
 => 3 = 2 + 1
[[1,2,3],[4],[5]]
 => 3 = 2 + 1
[[1,4],[2,5],[3]]
 => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => 2 = 1 + 1
[[1,2],[3,5],[4]]
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => 3 = 2 + 1
[[1,2],[3,4],[5]]
 => 3 = 2 + 1
[[1,5],[2],[3],[4]]
 => 1 = 0 + 1
[[1,4],[2],[3],[5]]
 => 4 = 3 + 1
[[1,3],[2],[4],[5]]
 => 4 = 3 + 1
[[1,2],[3],[4],[5]]
 => 4 = 3 + 1
[[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[[1,2,3,4,5,6]]
 => 1 = 0 + 1
[[1,3,4,5,6],[2]]
 => 1 = 0 + 1
[[1,2,4,5,6],[3]]
 => 1 = 0 + 1
[[1,2,3,5,6],[4]]
 => 1 = 0 + 1
[[1,2,3,4,6],[5]]
 => 1 = 0 + 1
[[1,2,3,4,5],[6]]
 => 2 = 1 + 1
[[1,3,5,6],[2,4]]
 => 1 = 0 + 1
[[1,2,5,6],[3,4]]
 => 1 = 0 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000504
(load all 4 compositions to match this statistic)
Mapping
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%
Values
[[1,2]]
 => {{1,2}}
 => {{1,2}}
 => 2 = 1 + 1
[[1],[2]]
 => {{1},{2}}
 => {{1},{2}}
 => 1 = 0 + 1
[[1,2,3]]
 => {{1,2,3}}
 => {{1,2,3}}
 => 3 = 2 + 1
[[1,3],[2]]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => 2 = 1 + 1
[[1,2],[3]]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => 1 = 0 + 1
[[1],[2],[3]]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 1 = 0 + 1
[[1,2,3,4]]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 4 = 3 + 1
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 3 = 2 + 1
[[1,2,4],[3]]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => 3 = 2 + 1
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 1 = 0 + 1
[[1,3],[2,4]]
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 2 = 1 + 1
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => 2 = 1 + 1
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => 2 = 1 + 1
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 1 = 0 + 1
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => 1 = 0 + 1
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 1 = 0 + 1
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 5 = 4 + 1
[[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => 4 = 3 + 1
[[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => 4 = 3 + 1
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => 1 = 0 + 1
[[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => {{1,3,5},{2,4}}
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => 3 = 2 + 1
[[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => 2 = 1 + 1
[[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => 2 = 1 + 1
[[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => {{1,2,5},{3},{4}}
 => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => {{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => {{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => 1 = 0 + 1
[[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => 1 = 0 + 1
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => {{1,4},{2,5},{3}}
 => {{1,4},{2,5},{3}}
 => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => 2 = 1 + 1
[[1,2],[3,5],[4]]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => 1 = 0 + 1
[[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => 1 = 0 + 1
[[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => 1 = 0 + 1
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => 1 = 0 + 1
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => 6 = 5 + 1
[[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => {{1,2,3,4,6},{5}}
 => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => {{1,2,4,5,6},{3}}
 => {{1,2,3,5,6},{4}}
 => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => {{1,2,4,5,6},{3}}
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => {{1,3,4,5,6},{2}}
 => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => {{1},{2,3,4,5,6}}
 => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => {{1,3,5,6},{2,4}}
 => {{1,2,4,6},{3,5}}
 => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => {{1,2,5,6},{3,4}}
 => 4 = 3 + 1
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: St000502
(load all 2 compositions to match this statistic)
Mapping
Mp00284: Standard tableaux —rows⟶ Set partitions
Mp00091: Set partitions —rotate increasing⟶ Set partitions
Mp00174: Set partitions —dual major index to intertwining number⟶ Set partitions
St000502: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
 => {{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 1
[[1],[2]]
 => {{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
[[1,2,3]]
 => {{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 2
[[1,3],[2]]
 => {{1,3},{2}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 1
[[1,2],[3]]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
[[1],[2],[3]]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
[[1,2,3,4]]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 3
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 2
[[1,2,4],[3]]
 => {{1,2,4},{3}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 2
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => {{1,3},{2,4}}
 => 0
[[1,3],[2,4]]
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 1
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => {{1,4},{2,3}}
 => {{1,3,4},{2}}
 => 1
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 1
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 0
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => 4
[[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => {{1,2,4,5},{3}}
 => {{1,2},{3,4,5}}
 => 3
[[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => {{1,2,3,5},{4}}
 => {{1,2,3},{4,5}}
 => 3
[[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => {{1,2,3,4},{5}}
 => {{1,2,3,4},{5}}
 => 3
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => {{1,3,5},{2,4}}
 => 0
[[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => {{1,2,4},{3,5}}
 => {{1,2,5},{3,4}}
 => 2
[[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => {{1,2,3},{4,5}}
 => {{1,2,3,5},{4}}
 => 2
[[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => {{1,3},{2,4,5}}
 => {{1,4},{2,3,5}}
 => 1
[[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => {{1,4},{2,3,5}}
 => {{1,3,4},{2,5}}
 => 1
[[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => {{1,5},{2,3,4}}
 => {{1,3},{2,4,5}}
 => 1
[[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => {{1,2,5},{3},{4}}
 => {{1,2},{3},{4,5}}
 => 2
[[1,3,5],[2],[4]]
 => {{1,3,5},{2},{4}}
 => {{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 2
[[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => {{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 2
[[1,3,4],[2],[5]]
 => {{1,3,4},{2},{5}}
 => {{1},{2,4,5},{3}}
 => {{1},{2,4},{3,5}}
 => 0
[[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => {{1},{2,3,5},{4}}
 => {{1,3,5},{2},{4}}
 => 0
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => {{1},{2,3,4},{5}}
 => {{1,3},{2,4},{5}}
 => 0
[[1,4],[2,5],[3]]
 => {{1,4},{2,5},{3}}
 => {{1,3},{2,5},{4}}
 => {{1},{2,3,5},{4}}
 => 1
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => {{1,3},{2,4},{5}}
 => {{1,4},{2,3},{5}}
 => 1
[[1,2],[3,5],[4]]
 => {{1,2},{3,5},{4}}
 => {{1,4},{2,3},{5}}
 => {{1,3,4},{2},{5}}
 => 1
[[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => {{1,5},{2,4},{3}}
 => 0
[[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => {{1,3},{2,5},{4}}
 => 0
[[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => {{1,2},{3},{4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => {{1},{2},{3,5},{4}}
 => 0
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => {{1},{2,4},{3},{5}}
 => {{1},{2,4},{3},{5}}
 => 0
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => {{1},{2,3},{4},{5}}
 => {{1,3},{2},{4},{5}}
 => 0
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0
[[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
[[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => {{1,2,4,5,6},{3}}
 => {{1,2},{3,4,5,6}}
 => 4
[[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
[[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => {{1,2,3,4,6},{5}}
 => {{1,2,3,4},{5,6}}
 => 4
[[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => {{1,2,3,4,5},{6}}
 => {{1,2,3,4,5},{6}}
 => 4
[[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
[[1,3,5,6],[2,4]]
 => {{1,3,5,6},{2,4}}
 => {{1,2,4,6},{3,5}}
 => {{1,2,5,6},{3,4}}
 => 3
[[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => {{1,2,3,6},{4,5}}
 => {{1,2,3,5,6},{4}}
 => 3
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: St000382
(load all 2 compositions to match this statistic)
Mapping
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%
Values
[[1,2]]
 => {{1,2}}
 => {{1,2}}
 => [2] => 2 = 1 + 1
[[1],[2]]
 => {{1},{2}}
 => {{1},{2}}
 => [1,1] => 1 = 0 + 1
[[1,2,3]]
 => {{1,2,3}}
 => {{1,2,3}}
 => [3] => 3 = 2 + 1
[[1,3],[2]]
 => {{1,3},{2}}
 => {{1,3},{2}}
 => [2,1] => 2 = 1 + 1
[[1,2],[3]]
 => {{1,2},{3}}
 => {{1},{2,3}}
 => [1,2] => 1 = 0 + 1
[[1],[2],[3]]
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,1,1] => 1 = 0 + 1
[[1,2,3,4]]
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => [4] => 4 = 3 + 1
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => {{1,2,4},{3}}
 => [3,1] => 3 = 2 + 1
[[1,2,4],[3]]
 => {{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [3,1] => 3 = 2 + 1
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => {{1},{2,3,4}}
 => [1,3] => 1 = 0 + 1
[[1,3],[2,4]]
 => {{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [2,2] => 2 = 1 + 1
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => {{1,2},{3,4}}
 => [2,2] => 2 = 1 + 1
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => {{1,4},{2},{3}}
 => [2,1,1] => 2 = 1 + 1
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,2,1] => 1 = 0 + 1
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => {{1},{2},{3,4}}
 => [1,1,2] => 1 = 0 + 1
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => {{1,2,3,4,5}}
 => [5] => 5 = 4 + 1
[[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => {{1,2,3,5},{4}}
 => [4,1] => 4 = 3 + 1
[[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => {{1,2,4,5},{3}}
 => [4,1] => 4 = 3 + 1
[[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => {{1,3,4,5},{2}}
 => [4,1] => 4 = 3 + 1
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => {{1},{2,3,4,5}}
 => [1,4] => 1 = 0 + 1
[[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => {{1,3,5},{2,4}}
 => [3,2] => 3 = 2 + 1
[[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => {{1,4,5},{2,3}}
 => [3,2] => 3 = 2 + 1
[[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => {{1,4},{2,3,5}}
 => [2,3] => 2 = 1 + 1
[[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => {{1,3},{2,4,5}}
 => [2,3] => 2 = 1 + 1
[[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => {{1,2},{3,4,5}}
 => [2,3] => 2 = 1 + 1
[[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => {{1,2,5},{3},{4}}
 => [3,1,1] => 3 = 2 + 1
[[1,3,5],[2],[4]]
 => {{1,3,5},{2},{4}}
 => {{1,3,5},{2},{4}}
 => [3,1,1] => 3 = 2 + 1
[[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => {{1,4,5},{2},{3}}
 => [3,1,1] => 3 = 2 + 1
[[1,3,4],[2],[5]]
 => {{1,3,4},{2},{5}}
 => {{1},{2,3,5},{4}}
 => [1,3,1] => 1 = 0 + 1
[[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => {{1},{2,4,5},{3}}
 => [1,3,1] => 1 = 0 + 1
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => {{1},{2},{3,4,5}}
 => [1,1,3] => 1 = 0 + 1
[[1,4],[2,5],[3]]
 => {{1,4},{2,5},{3}}
 => {{1,4},{2,5},{3}}
 => [2,2,1] => 2 = 1 + 1
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => {{1,4},{2},{3,5}}
 => [2,1,2] => 2 = 1 + 1
[[1,2],[3,5],[4]]
 => {{1,2},{3,5},{4}}
 => {{1,3},{2},{4,5}}
 => [2,1,2] => 2 = 1 + 1
[[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => {{1},{2,4},{3,5}}
 => [1,2,2] => 1 = 0 + 1
[[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => {{1},{2,3},{4,5}}
 => [1,2,2] => 1 = 0 + 1
[[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => {{1,5},{2},{3},{4}}
 => [2,1,1,1] => 2 = 1 + 1
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => {{1},{2,5},{3},{4}}
 => [1,2,1,1] => 1 = 0 + 1
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => {{1},{2},{3,5},{4}}
 => [1,1,2,1] => 1 = 0 + 1
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => {{1},{2},{3},{4,5}}
 => [1,1,1,2] => 1 = 0 + 1
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => 1 = 0 + 1
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => {{1,2,3,4,5,6}}
 => [6] => 6 = 5 + 1
[[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => {{1,2,3,4,6},{5}}
 => [5,1] => 5 = 4 + 1
[[1,2,4,5,6],[3]]
 => {{1,2,4,5,6},{3}}
 => {{1,2,3,5,6},{4}}
 => [5,1] => 5 = 4 + 1
[[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => {{1,2,4,5,6},{3}}
 => [5,1] => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => {{1,3,4,5,6},{2}}
 => [5,1] => 5 = 4 + 1
[[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => {{1},{2,3,4,5,6}}
 => [1,5] => 1 = 0 + 1
[[1,3,5,6],[2,4]]
 => {{1,3,5,6},{2,4}}
 => {{1,2,4,6},{3,5}}
 => [4,2] => 4 = 3 + 1
[[1,2,5,6],[3,4]]
 => {{1,2,5,6},{3,4}}
 => {{1,2,5,6},{3,4}}
 => [4,2] => 4 = 3 + 1
Description
The first part of an integer composition.
Matching statistic: St001604
(load all 3 compositions to match this statistic)
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 29% ●values known / values provided: 51%●distinct values known / distinct values provided: 29%
Values
[[1,2]]
 => [2]
 => [1,1]
 => [1]
 => ? ∊ {0,1}
[[1],[2]]
 => [1,1]
 => [2]
 => []
 => ? ∊ {0,1}
[[1,2,3]]
 => [3]
 => [1,1,1]
 => [1,1]
 => ? ∊ {0,0,1,2}
[[1,3],[2]]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,2}
[[1,2],[3]]
 => [2,1]
 => [3]
 => []
 => ? ∊ {0,0,1,2}
[[1],[2],[3]]
 => [1,1,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,1,2}
[[1,2,3,4]]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,3,4],[2]]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2,3],[4]]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,3],[2,4]]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2],[3,4]]
 => [2,2]
 => [4]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,4],[2],[3]]
 => [2,1,1]
 => [2,2]
 => [2]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,3],[2],[4]]
 => [2,1,1]
 => [2,2]
 => [2]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2],[3],[4]]
 => [2,1,1]
 => [2,2]
 => [2]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [3,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2,3,4,5]]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,3,4,5],[2]]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,4,5],[3]]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,5],[4]]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,4],[5]]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 0
[[1,3,5],[2,4]]
 => [3,2]
 => [5]
 => []
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3,4]]
 => [3,2]
 => [5]
 => []
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2,5]]
 => [3,2]
 => [5]
 => []
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3,5]]
 => [3,2]
 => [5]
 => []
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4,5]]
 => [3,2]
 => [5]
 => []
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [4,1]
 => [1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,2,1]
 => [2,1]
 => 0
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [3,1,1]
 => [1,1]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [3,2]
 => [2]
 => ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,4,5,6]]
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[[1,3,4,5,6],[2]]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,4,5,6],[3]]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,3,5,6],[4]]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,3,4,6],[5]]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,2,3,4,5],[6]]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,3,5,6],[2,4]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,5,6],[3,4]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,3,4,6],[2,5]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,4,6],[3,5]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,6],[4,5]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,3,4,5],[2,6]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,4,5],[3,6]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,5],[4,6]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,2,3,4],[5,6]]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 0
[[1,4,5,6],[2],[3]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,3,5,6],[2],[4]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,5,6],[3],[4]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,3,4,6],[2],[5]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,4,6],[3],[5]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,6],[4],[5]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,3,4,5],[2],[6]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,4,5],[3],[6]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,5],[4],[6]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,4],[5],[6]]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 0
[[1,3,5],[2,4,6]]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5],[3,4,6]]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4],[2,5,6]]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4],[3,5,6]]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3],[4,5,6]]
 => [3,3]
 => [6]
 => []
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,6],[2,5],[3]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,6],[2,5],[4]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,6],[3,5],[4]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,6],[2,4],[5]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,6],[3,4],[5]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,5],[2,6],[3]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5],[2,6],[4]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5],[3,6],[4]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4],[2,6],[5]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4],[3,6],[5]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3],[4,6],[5]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5],[2,4],[6]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5],[3,4],[6]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4],[2,5],[6]]
 => [3,2,1]
 => [5,1]
 => [1]
 => ? ∊ {0,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [3,3]
 => [3]
 => 1
[[1,4],[2,5],[3,6]]
 => [2,2,2]
 => [2,2,2]
 => [2,2]
 => 1
[[1,3],[2,5],[4,6]]
 => [2,2,2]
 => [2,2,2]
 => [2,2]
 => 1
[[1,2],[3,5],[4,6]]
 => [2,2,2]
 => [2,2,2]
 => [2,2]
 => 1
[[1,3],[2,4],[5,6]]
 => [2,2,2]
 => [2,2,2]
 => [2,2]
 => 1
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group. This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001876
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001876: Lattices ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 71%
Values
[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1}
[[1],[2]]
 => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,1}
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,1,2}
[[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,2}
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,1,2}
[[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => 3
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,4,5],[3],[6]]
 => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,3,5],[4],[6]]
 => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => ([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => 3
[[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => 4
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,6],[3,4],[5]]
 => [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,5],[3,6],[4]]
 => [4,3,6,1,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,4],[2,6],[5]]
 => [5,2,6,1,3,4] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,4],[3,6],[5]]
 => [5,3,6,1,2,4] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => ([(0,5),(1,5),(2,3),(3,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3,5],[2,4],[6]]
 => [6,2,4,1,3,5] => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => ([(1,5),(2,3),(2,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4],[3,5],[6]]
 => [6,3,5,1,2,4] => ([(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,4,6],[2],[3],[5]]
 => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,6],[2],[4],[5]]
 => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,4,5],[2],[3],[6]]
 => [6,3,2,1,4,5] => ([(1,5),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,3,5],[2],[4],[6]]
 => [6,4,2,1,3,5] => ([(1,5),(2,4),(3,4),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,5],[3],[4],[6]]
 => [6,4,3,1,2,5] => ([(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,5}
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,4],[2,5],[3,6]]
 => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2],[3,5],[4,6]]
 => [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,3],[2,4],[5,6]]
 => [5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
Description
The number of 2-regular simple modules in the incidence algebra of the lattice.
Matching statistic: St001570
(load all 2 compositions to match this statistic)
Mapping
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001570: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
 => [2] => [1] => ([],1)
 => ? ∊ {0,1}
[[1],[2]]
 => [2] => [1] => ([],1)
 => ? ∊ {0,1}
[[1,2,3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,0,1,2}
[[1,3],[2]]
 => [2,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,1,2}
[[1,2],[3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,0,1,2}
[[1],[2],[3]]
 => [3] => [1] => ([],1)
 => ? ∊ {0,0,1,2}
[[1,2,3,4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,3,4],[2]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,2,3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,3],[2,4]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,2],[3,4]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,4],[2],[3]]
 => [3,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,3],[2],[4]]
 => [2,2] => [2] => ([],2)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [4] => [1] => ([],1)
 => ? ∊ {0,0,0,0,1,1,1,2,2,3}
[[1,2,3,4,5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4,5],[2]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2,4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,2,5],[3,4]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2,5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3,5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4,5]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4,5],[2],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,2,5],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2,5],[3]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,5],[4]]
 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,2],[3,5],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3,4],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,5],[2],[3],[4]]
 => [4,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2],[3],[5]]
 => [3,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2],[4],[5]]
 => [2,3] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1],[2],[3],[4],[5]]
 => [5] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,4,5,6]]
 => [6] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,3,4,5,6],[2]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,4,5,6],[3]]
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,3,5,6],[4]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,3,4,6],[5]]
 => [5,1] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,3,4,5],[6]]
 => [6] => [1] => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,3,5,6],[2,4]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,2,5,6],[3,4]]
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,3,4,6],[2,5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,6],[3,5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,3,6],[4,5]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,3,4,5],[2,6]]
 => [2,4] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,4,5],[3,6]]
 => [3,3] => [2] => ([],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,2,3,5],[4,6]]
 => [4,2] => [1,1] => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,4,4,4,4,5}
[[1,3,5,6],[2],[4]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3,4,6],[2],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5],[2,4,6]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3,4],[2,5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4],[3,5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,4,6],[2,5],[3]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,6],[2,5],[4]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3,6],[2,4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,6],[3,4],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5],[2,6],[4]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3,4],[2,6],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4],[3,6],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5],[2,4],[6]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,4,6],[2],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,6],[2],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5],[2],[4],[6]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3],[2,5],[4,6]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3],[2,4],[5,6]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2],[3,4],[5,6]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,4],[2,6],[3],[5]]
 => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3],[2,6],[4],[5]]
 => [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3],[2,5],[4],[6]]
 => [2,2,2] => [3] => ([],3)
 => 3
[[1,3,5,6,7],[2,4]]
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,3,4,6,7],[2,5]]
 => [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,6,7],[3,5]]
 => [3,2,2] => [1,2] => ([(1,2)],3)
 => 2
[[1,3,4,5,7],[2,6]]
 => [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,5,7],[3,6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,2,3,5,7],[4,6]]
 => [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5,6,7],[2],[4]]
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,3,4,6,7],[2],[5]]
 => [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,6,7],[3],[5]]
 => [3,2,2] => [1,2] => ([(1,2)],3)
 => 2
[[1,3,4,5,7],[2],[6]]
 => [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,5,7],[3],[6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,2,3,5,7],[4],[6]]
 => [4,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,3,5,7],[2,4,6]]
 => [2,2,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2
[[1,2,5,7],[3,4,6]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,3,4,7],[2,5,6]]
 => [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,7],[3,5,6]]
 => [3,2,2] => [1,2] => ([(1,2)],3)
 => 2
[[1,3,5,6],[2,4,7]]
 => [2,2,3] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[[1,3,4,6],[2,5,7]]
 => [2,3,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,6],[3,5,7]]
 => [3,2,2] => [1,2] => ([(1,2)],3)
 => 2
[[1,3,4,5],[2,6,7]]
 => [2,4,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 0
[[1,2,4,5],[3,6,7]]
 => [3,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
Description
The minimal number of edges to add to make a graph Hamiltonian. A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Matching statistic: St000714
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000714: Integer partitions ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 57%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,1,2}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,1,2}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,1,2}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,1,2}
[[1,2,3,4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,3,4],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,2,3],[4]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3,4,5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4,5],[2]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4,5],[3]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,5],[4]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,4],[5]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2,4]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3,4]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,2,3,4,5,6]]
 => [6]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5,6],[2,4]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5,6],[3,4]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,6],[2,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,6],[3,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,6],[4,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5],[2,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5],[3,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5],[4,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,5}
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4],[2,5],[3,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 3
[[1,3],[2,5],[4,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 3
[[1,2],[3,5],[4,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 3
[[1,3],[2,4],[5,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 3
[[1,2],[3,4],[5,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 3
[[1,5],[2,6],[3],[4]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,4],[2,6],[3],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,3],[2,6],[4],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,2],[3,6],[4],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,4],[2,5],[3],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,3],[2,5],[4],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,2],[3,5],[4],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,3],[2,4],[5],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,2],[3,4],[5],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 0
[[1,5,6,7],[2],[3],[4]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,6,7],[2],[3],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,6,7],[2],[4],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,6,7],[3],[4],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,5,7],[2],[3],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,5,7],[2],[4],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,5,7],[3],[4],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4,7],[2],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,4,7],[3],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,3,7],[4],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,4,5,6],[2],[3],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,5,6],[2],[4],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,2,5,6],[3],[4],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
[[1,3,4,6],[2],[5],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
Description
The number of semistandard Young tableau of given shape, with entries at most 2. This is also the dimension of the corresponding irreducible representation of $GL_2$.
Matching statistic: St000941
Mapping
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 46%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [2]
 => []
 => ?
 => ? ∊ {0,1}
[[1],[2]]
 => [1,1]
 => [1]
 => []
 => ? ∊ {0,1}
[[1,2,3]]
 => [3]
 => []
 => ?
 => ? ∊ {0,0,1,2}
[[1,3],[2]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,1,2}
[[1,2],[3]]
 => [2,1]
 => [1]
 => []
 => ? ∊ {0,0,1,2}
[[1],[2],[3]]
 => [1,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,1,2}
[[1,2,3,4]]
 => [4]
 => []
 => ?
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,3,4],[2]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2,3],[4]]
 => [3,1]
 => [1]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,3],[2,4]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2],[3,4]]
 => [2,2]
 => [2]
 => []
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,4],[2],[3]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,3],[2],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1,2],[3],[4]]
 => [2,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,1,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3,4,5]]
 => [5]
 => []
 => ?
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4,5],[2]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4,5],[3]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,5],[4]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3,4],[5]]
 => [4,1]
 => [1]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2,4]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3,4]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4,5]]
 => [3,2]
 => [2]
 => []
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4,5],[2],[3]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,5],[3],[4]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [3,1,1]
 => [1,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,4],[2,5],[3]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3,5],[4]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,3],[2,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,2],[3,4],[5]]
 => [2,2,1]
 => [2,1]
 => [1]
 => ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[[1,5],[2],[3],[4]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2],[3],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3],[2],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2],[3],[4],[5]]
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1],[2],[3],[4],[5]]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,2,3,4,5,6]]
 => [6]
 => []
 => ?
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5,6],[2]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5,6],[3]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5,6],[4]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,6],[5]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,4,5],[6]]
 => [5,1]
 => [1]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5,6],[2,4]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5,6],[3,4]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,6],[2,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,6],[3,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,6],[4,5]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,5],[2,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5],[3,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5],[4,6]]
 => [4,2]
 => [2]
 => []
 => ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,5,6],[2],[3],[4]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4,6],[2],[3],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,6],[2],[4],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,6],[3],[4],[5]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4,5],[2],[3],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,5],[2],[4],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,5],[3],[4],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4],[2],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,4],[3],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3],[4],[5],[6]]
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4],[2,5],[3,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1,3],[2,5],[4,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1,2],[3,5],[4,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1,3],[2,4],[5,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1,2],[3,4],[5,6]]
 => [2,2,2]
 => [2,2]
 => [2]
 => 0
[[1,5],[2,6],[3],[4]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,4],[2,6],[3],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,3],[2,6],[4],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,2],[3,6],[4],[5]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,4],[2,5],[3],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,3],[2,5],[4],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,2],[3,5],[4],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,3],[2,4],[5],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,2],[3,4],[5],[6]]
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0
[[1,6],[2],[3],[4],[5]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,5],[2],[3],[4],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,4],[2],[3],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,3],[2],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1,2],[3],[4],[5],[6]]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
[[1],[2],[3],[4],[5],[6]]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 1
[[1,5,6,7],[2],[3],[4]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4,6,7],[2],[3],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,6,7],[2],[4],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,6,7],[3],[4],[5]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4,5,7],[2],[3],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,5,7],[2],[4],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,5,7],[3],[4],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4,7],[2],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,4,7],[3],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,3,7],[4],[5],[6]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,4,5,6],[2],[3],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,5,6],[2],[4],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,2,5,6],[3],[4],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
[[1,3,4,6],[2],[5],[7]]
 => [4,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0
Description
The number of characters of the symmetric group whose value on the partition is even.
Matching statistic: St001877
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001877: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 43%
Values
[[1,2]]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => ? ∊ {0,1}
[[1],[2]]
 => [2,1] => ([],2)
 => ([],1)
 => ? ∊ {0,1}
[[1,2,3]]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2]]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => ? ∊ {0,1,2}
[[1,2],[3]]
 => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => ? ∊ {0,1,2}
[[1],[2],[3]]
 => [3,2,1] => ([],3)
 => ([],1)
 => ? ∊ {0,1,2}
[[1,2,3,4]]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,4],[2]]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,4],[3]]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,3],[4]]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3],[2,4]]
 => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2],[3,4]]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,4],[2],[3]]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,3],[2],[4]]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2],[3],[4]]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => ? ∊ {0,1,1,2,2,3}
[[1],[2],[3],[4]]
 => [4,3,2,1] => ([],4)
 => ([],1)
 => ? ∊ {0,1,1,2,2,3}
[[1,2,3,4,5]]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => ([],5)
 => ([],1)
 => ? ∊ {0,0,0,1,1,1,1,2,2,2,2,3,3,3,4}
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => 0
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => ([(1,5),(3,4),(4,2),(5,3)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => ([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5,6],[3,4]]
 => [3,4,1,2,5,6] => ([(0,4),(1,3),(3,5),(4,5),(5,2)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4,6],[2,5]]
 => [2,5,1,3,4,6] => ([(0,4),(1,2),(1,4),(2,5),(3,5),(4,3)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4,6],[3,5]]
 => [3,5,1,2,4,6] => ([(0,3),(1,2),(1,4),(2,5),(3,4),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3,6],[4,5]]
 => [4,5,1,2,3,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,3,4,5],[2,6]]
 => [2,6,1,3,4,5] => ([(0,5),(1,3),(1,5),(4,2),(5,4)],6)
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0
[[1,2,4,5],[3,6]]
 => [3,6,1,2,4,5] => ([(0,4),(1,3),(1,5),(4,5),(5,2)],6)
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => 1
[[1,2,3,5],[4,6]]
 => [4,6,1,2,3,5] => ([(0,4),(1,3),(1,5),(2,5),(4,2)],6)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[1,2,3,4],[5,6]]
 => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,5,6],[2],[3]]
 => [3,2,1,4,5,6] => ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5,6],[2],[4]]
 => [4,2,1,3,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5,6],[3],[4]]
 => [4,3,1,2,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4,6],[2],[5]]
 => [5,2,1,3,4,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,6],[3],[5]]
 => [5,3,1,2,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,6],[4],[5]]
 => [5,4,1,2,3,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3,4,5],[2],[6]]
 => [6,2,1,3,4,5] => ([(1,5),(2,5),(3,4),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4,5],[3],[6]]
 => [6,3,1,2,4,5] => ([(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3,5],[4],[6]]
 => [6,4,1,2,3,5] => ([(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,3,4],[5],[6]]
 => [6,5,1,2,3,4] => ([(2,3),(3,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => ([(0,4),(1,2),(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => 1
[[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => ([(0,3),(1,4),(3,5),(4,2),(4,5)],6)
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => 2
[[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => ([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => 2
[[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(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)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,6],[2,5],[3]]
 => [3,2,5,1,4,6] => ([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,6],[2,5],[4]]
 => [4,2,5,1,3,6] => ([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,6],[3,5],[4]]
 => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,6],[2,4],[5]]
 => [5,2,4,1,3,6] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,6],[3,4],[5]]
 => [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,4,5],[2,6],[3]]
 => [3,2,6,1,4,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,5],[2,6],[4]]
 => [4,2,6,1,3,5] => ([(0,4),(1,3),(1,5),(2,3),(2,4),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,5],[3,6],[4]]
 => [4,3,6,1,2,5] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,4],[2,6],[5]]
 => [5,2,6,1,3,4] => ([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,4],[3,6],[5]]
 => [5,3,6,1,2,4] => ([(0,4),(1,4),(1,5),(2,3),(3,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3],[4,6],[5]]
 => [5,4,6,1,2,3] => ([(0,5),(1,5),(2,3),(3,4)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,3,5],[2,4],[6]]
 => [6,2,4,1,3,5] => ([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2,5],[3,4],[6]]
 => [6,3,4,1,2,5] => ([(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,3,4],[2,5],[6]]
 => [6,2,5,1,3,4] => ([(1,5),(2,3),(2,5),(5,4)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2,4],[3,5],[6]]
 => [6,3,5,1,2,4] => ([(1,4),(2,3),(2,5),(4,5)],6)
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1
[[1,2,3],[4,5],[6]]
 => [6,4,5,1,2,3] => ([(1,3),(2,4),(4,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,5,6],[2],[3],[4]]
 => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,4,6],[2],[3],[5]]
 => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,3,6],[2],[4],[5]]
 => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([],1)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,6],[3],[4],[5]]
 => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,1)],2)
 => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,5}
[[1,2,3],[4],[5],[6]]
 => [6,5,4,1,2,3] => ([(3,4),(4,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,4],[2,5],[3,6]]
 => [3,6,2,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,3],[2,5],[4,6]]
 => [4,6,2,5,1,3] => ([(0,4),(1,4),(1,5),(2,3),(2,5)],6)
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[1,2],[3,5],[4,6]]
 => [4,6,3,5,1,2] => ([(0,5),(1,3),(2,4),(2,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,3],[2,4],[5,6]]
 => [5,6,2,4,1,3] => ([(0,5),(1,3),(2,4),(2,5)],6)
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => 2
[[1,3],[2,4],[5],[6]]
 => [6,5,2,4,1,3] => ([(2,5),(3,4),(3,5)],6)
 => ([(0,2),(2,1)],3)
 => 0
[[1,2],[3,4],[5],[6]]
 => [6,5,3,4,1,2] => ([(2,5),(3,4)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[[1,2,3,4,5,6,7]]
 => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => 0
Description
Number of indecomposable injective modules with projective dimension 2.
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St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000054The first entry of the permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000314The number of left-to-right-maxima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000699The toughness times the least common multiple of 1,. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001429The number of negative entries in a signed permutation. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.