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Your data matches 28 different statistics following compositions of up to 3 maps.
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Matching statistic: St000160
Mp00225: Semistandard tableaux —weight⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> [1,1]
=> 2
[[2,2]]
=> [2]
=> 1
[[1],[2]]
=> [1,1]
=> 2
[[1,3]]
=> [1,1]
=> 2
[[2,3]]
=> [1,1]
=> 2
[[3,3]]
=> [2]
=> 1
[[1],[3]]
=> [1,1]
=> 2
[[2],[3]]
=> [1,1]
=> 2
[[1,1,2]]
=> [2,1]
=> 1
[[1,2,2]]
=> [2,1]
=> 1
[[2,2,2]]
=> [3]
=> 1
[[1,1],[2]]
=> [2,1]
=> 1
[[1,2],[2]]
=> [2,1]
=> 1
[[1,4]]
=> [1,1]
=> 2
[[2,4]]
=> [1,1]
=> 2
[[3,4]]
=> [1,1]
=> 2
[[4,4]]
=> [2]
=> 1
[[1],[4]]
=> [1,1]
=> 2
[[2],[4]]
=> [1,1]
=> 2
[[3],[4]]
=> [1,1]
=> 2
[[1,1,3]]
=> [2,1]
=> 1
[[1,2,3]]
=> [1,1,1]
=> 3
[[1,3,3]]
=> [2,1]
=> 1
[[2,2,3]]
=> [2,1]
=> 1
[[2,3,3]]
=> [2,1]
=> 1
[[3,3,3]]
=> [3]
=> 1
[[1,1],[3]]
=> [2,1]
=> 1
[[1,2],[3]]
=> [1,1,1]
=> 3
[[1,3],[2]]
=> [1,1,1]
=> 3
[[1,3],[3]]
=> [2,1]
=> 1
[[2,2],[3]]
=> [2,1]
=> 1
[[2,3],[3]]
=> [2,1]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> 3
[[1,1,1,2]]
=> [3,1]
=> 1
[[1,1,2,2]]
=> [2,2]
=> 2
[[1,2,2,2]]
=> [3,1]
=> 1
[[2,2,2,2]]
=> [4]
=> 1
[[1,1,1],[2]]
=> [3,1]
=> 1
[[1,1,2],[2]]
=> [2,2]
=> 2
[[1,2,2],[2]]
=> [3,1]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> 2
[[1,5]]
=> [1,1]
=> 2
[[2,5]]
=> [1,1]
=> 2
[[3,5]]
=> [1,1]
=> 2
[[4,5]]
=> [1,1]
=> 2
[[5,5]]
=> [2]
=> 1
[[1],[5]]
=> [1,1]
=> 2
[[2],[5]]
=> [1,1]
=> 2
[[3],[5]]
=> [1,1]
=> 2
[[4],[5]]
=> [1,1]
=> 2
Description
The multiplicity of the smallest part of a partition.
This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$.
The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences
\begin{align*}
spt(5n+4) &\equiv 0\quad \pmod{5}\\\
spt(7n+5) &\equiv 0\quad \pmod{7}\\\
spt(13n+6) &\equiv 0\quad \pmod{13},
\end{align*}
analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St001603
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 60%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001603: Integer partitions ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 60%
Values
[[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2}
[[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [1,1]
=> [1]
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1,1],[2,2,2]]
=> [3,3]
=> [3]
=> 1
[[1],[2],[3],[5]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[2],[4],[5]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1],[3],[4],[5]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[2],[3],[4],[5]]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[[1,1],[2,2],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2,3],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[2,3],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,3],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[2,2],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[2,2],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[2,3],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> 1
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[[1,1,1],[2,2,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,1],[2,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,1],[3,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,2],[2,2,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,2],[2,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,2],[3,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,2,2],[2,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,2,2],[3,3,3]]
=> [3,3]
=> [3]
=> 1
[[2,2,2],[3,3,3]]
=> [3,3]
=> [3]
=> 1
[[1,1,1],[2,2],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1,1],[2,3],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1,2],[2,2],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1,2],[2,3],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1,3],[2,2],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1,3],[2,3],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,2,2],[2,3],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,2,3],[2,3],[3]]
=> [3,2,1]
=> [2,1]
=> 1
[[1,1],[2,2],[3,3]]
=> [2,2,2]
=> [2,2]
=> 2
[[1,1,1,1],[2,2,2]]
=> [4,3]
=> [3]
=> 1
[[1,1,1,2],[2,2,2]]
=> [4,3]
=> [3]
=> 1
Description
The number of colourings of a polygon such that the multiplicities of a colour are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000284
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000284: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The Plancherel distribution on integer partitions.
This is defined as the distribution induced by the RSK shape of the uniform distribution on permutations. In other words, this is the size of the preimage of the map 'Robinson-Schensted tableau shape' from permutations to integer partitions.
Equivalently, this is given by the square of the number of standard Young tableaux of the given shape.
Matching statistic: St000510
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000510: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of invariant oriented cycles when acting with a permutation of given cycle type.
Matching statistic: St000515
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000515: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 2
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 2
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
Description
The number of invariant set partitions when acting with a permutation of given cycle type.
Matching statistic: St000668
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 2
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000681
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St000698
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer $k$, one associates a $k$-core to a partition by repeatedly removing all rim hooks of size $k$.
This statistic counts the $2$-rim hooks that are removed in this process to obtain a $2$-core.
Matching statistic: St000704
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 20%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 1
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
Matching statistic: St000707
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000707: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {1,2,2}
[[1,3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2,3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[3,3]]
=> [[3,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1],[3]]
=> [[1,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[2],[3]]
=> [[2,3]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1}
[[1,4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2,4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3,4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[4,4]]
=> [[4,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1],[4]]
=> [[1,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[2],[4]]
=> [[2,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[3],[4]]
=> [[3,4]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2}
[[1,5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2,5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3,5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4,5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[5,5]]
=> [[5,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1],[5]]
=> [[1,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[2],[5]]
=> [[2,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[3],[5]]
=> [[3,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[4],[5]]
=> [[4,5]]
=> [2]
=> []
=> ? ∊ {1,2,2,2,2,2,2,2,2}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[3]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[2],[4]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,5],[3],[4]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,5],[3],[4]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[5]]
=> [[1,2],[3],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[5]]
=> [[1,2],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[5]]
=> [[1,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[5]]
=> [[2,3],[4],[5]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,3,3],[2,2]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1,3],[2,2],[3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,1],[2,2],[3,3]]
=> [[1,1,2,2],[3,3]]
=> [4,2]
=> [2]
=> 2
[[1,6],[2],[3]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[4]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[2],[5]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[4]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[3],[5]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,6],[4],[5]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[4]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[3],[5]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2,6],[4],[5]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3,6],[4],[5]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[3],[6]]
=> [[1,2],[3],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[4],[6]]
=> [[1,2],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[2],[5],[6]]
=> [[1,2],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[4],[6]]
=> [[1,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[3],[5],[6]]
=> [[1,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1],[4],[5],[6]]
=> [[1,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[4],[6]]
=> [[2,3],[4],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[3],[5],[6]]
=> [[2,3],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[2],[4],[5],[6]]
=> [[2,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[3],[4],[5],[6]]
=> [[3,4],[5],[6]]
=> [2,1,1]
=> [1,1]
=> 1
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 2
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 1
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 1
Description
The product of the factorials of the parts.
The following 18 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000260The radius of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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