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Your data matches 17 different statistics following compositions of up to 3 maps.
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Matching statistic: St001803
St001803: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> 0
[[1,2]]
=> 0
[[1],[2]]
=> 1
[[1,2,3]]
=> 0
[[1,3],[2]]
=> 0
[[1,2],[3]]
=> 1
[[1],[2],[3]]
=> 2
[[1,2,3,4]]
=> 0
[[1,3,4],[2]]
=> 0
[[1,2,4],[3]]
=> 0
[[1,2,3],[4]]
=> 1
[[1,3],[2,4]]
=> 0
[[1,2],[3,4]]
=> 1
[[1,4],[2],[3]]
=> 0
[[1,3],[2],[4]]
=> 1
[[1,2],[3],[4]]
=> 1
[[1],[2],[3],[4]]
=> 3
[[1,2,3,4,5]]
=> 0
[[1,3,4,5],[2]]
=> 0
[[1,2,4,5],[3]]
=> 0
[[1,2,3,5],[4]]
=> 0
[[1,2,3,4],[5]]
=> 1
[[1,3,5],[2,4]]
=> 0
[[1,2,5],[3,4]]
=> 0
[[1,3,4],[2,5]]
=> 0
[[1,2,4],[3,5]]
=> 0
[[1,2,3],[4,5]]
=> 1
[[1,4,5],[2],[3]]
=> 0
[[1,3,5],[2],[4]]
=> 0
[[1,2,5],[3],[4]]
=> 0
[[1,3,4],[2],[5]]
=> 1
[[1,2,4],[3],[5]]
=> 1
[[1,2,3],[4],[5]]
=> 1
[[1,4],[2,5],[3]]
=> 0
[[1,3],[2,5],[4]]
=> 1
[[1,2],[3,5],[4]]
=> 2
[[1,3],[2,4],[5]]
=> 1
[[1,2],[3,4],[5]]
=> 2
[[1,5],[2],[3],[4]]
=> 0
[[1,4],[2],[3],[5]]
=> 1
[[1,3],[2],[4],[5]]
=> 1
[[1,2],[3],[4],[5]]
=> 1
[[1],[2],[3],[4],[5]]
=> 4
[[1,2,3,4,5,6]]
=> 0
[[1,3,4,5,6],[2]]
=> 0
[[1,2,4,5,6],[3]]
=> 0
[[1,2,3,5,6],[4]]
=> 0
[[1,2,3,4,6],[5]]
=> 0
[[1,2,3,4,5],[6]]
=> 1
[[1,3,5,6],[2,4]]
=> 0
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001604
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 70%●distinct values known / distinct values provided: 38%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 70%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [1]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [2]
=> [2]
=> []
=> ? ∊ {0,1}
[[1],[2]]
=> [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,1}
[[1,2,3]]
=> [3]
=> [2,1]
=> [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]
=> [1,1,1]
=> [1,1]
=> ? ∊ {0,0,1,2}
[[1,2,3,4]]
=> [4]
=> [2,2]
=> [2]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5]]
=> [5]
=> [2,2,1]
=> [2,1]
=> 0
[[1,3,4,5],[2]]
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [3,2]
=> [2]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [5]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> [2,2,2]
=> [2,2]
=> 1
[[1,3,4,5,6],[2]]
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [2,2,1,1]
=> [2,1,1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4,6]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5,6]]
=> [4,2]
=> [4,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,5,6],[2],[3]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2],[4]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2],[5]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2],[6]]
=> [4,1,1]
=> [4,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,4,6]]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
[[1,2,5],[3,4,6]]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
[[1,3,4],[2,5,6]]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
[[1,2,4],[3,5,6]]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
[[1,2,3],[4,5,6]]
=> [3,3]
=> [3,2,1]
=> [2,1]
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,6],[2,5],[4]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,6],[3,5],[4]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,6],[2,4],[5]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,4,5],[2,6],[3]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,5],[2,6],[4]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,5],[3,6],[4]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,4],[2,6],[5]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,3],[4,6],[5]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,5],[2,4],[6]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,5],[3,4],[6]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,3,4],[2,5],[6]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,4],[3,5],[6]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,2,3],[4,5],[6]]
=> [3,2,1]
=> [3,3]
=> [3]
=> 1
[[1,5,6],[2],[3],[4]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,4,6],[2],[3],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,6],[2],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,6],[3],[4],[5]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,4,5],[2],[3],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,5],[3],[4],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,4],[3],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,3],[4],[5],[6]]
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,6],[2],[3],[4],[5]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
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: St000566
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 63%●distinct values known / distinct values provided: 38%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 63%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[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]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[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]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[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]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[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]
=> 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]
=> 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 ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
Matching statistic: St000621
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 63%●distinct values known / distinct values provided: 25%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 63%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[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]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[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]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[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]
=> 1
[[1,3],[2,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,5],[4,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,3],[2,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[1,2],[3,4],[5,6]]
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
[[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]
=> 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]
=> 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 standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000941
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: 38% ●values known / values provided: 63%●distinct values known / distinct values provided: 38%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000941: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 63%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[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]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,4}
[[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]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[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: St000940
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 53%●distinct values known / distinct values provided: 38%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 38% ●values known / values provided: 53%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [1] => [1]
=> []
=> ? = 0
[[1,2]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[[1],[2]]
=> [2,1] => [2]
=> []
=> ? ∊ {0,1}
[[1,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[[1,3],[2]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,2}
[[1,2],[3]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,1,2}
[[1],[2],[3]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,1,2}
[[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,2]
=> [2]
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 0
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,2]
=> [2]
=> 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2]
=> [2]
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,2,1]
=> [2,1]
=> 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [4,2]
=> [2]
=> 0
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 0
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 0
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [3,3]
=> [3]
=> 0
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [4,2]
=> [2]
=> 0
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,2,2]
=> [2,2]
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [4,1,1]
=> [1,1]
=> 0
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [3,2,1]
=> [2,1]
=> 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,2]
=> [2]
=> 0
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [4,2]
=> [2]
=> 0
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [3,2,1]
=> [2,1]
=> 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 0
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [4,1,1]
=> [1,1]
=> 0
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [4,2]
=> [2]
=> 0
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St001124
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 53%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 53%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1] => [1]
=> []
=> ? = 0
[[1,2]]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[[1],[2]]
=> [2,1] => [2]
=> []
=> ? ∊ {0,1}
[[1,2,3]]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 0
[[1,3],[2]]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,1,2}
[[1,2],[3]]
=> [3,1,2] => [3]
=> []
=> ? ∊ {0,1,2}
[[1],[2],[3]]
=> [3,2,1] => [2,1]
=> [1]
=> ? ∊ {0,1,2}
[[1,2,3,4]]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [2,2]
=> [2]
=> 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,3}
[[1,2],[3],[4]]
=> [4,3,1,2] => [4]
=> []
=> ? ∊ {1,1,1,1,3}
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,2]
=> [2]
=> 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 0
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,2]
=> [2]
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 1
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [3,2]
=> [2]
=> 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [2,2,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,2]
=> [2]
=> 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,2,1]
=> [2,1]
=> 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [3,2,1]
=> [2,1]
=> 1
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [4,2]
=> [2]
=> 0
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [3,3]
=> [3]
=> 0
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [4,2]
=> [2]
=> 0
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [3,3]
=> [3]
=> 0
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [4,2]
=> [2]
=> 0
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [2,2,2]
=> [2,2]
=> 0
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [4,1,1]
=> [1,1]
=> 0
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [4,1,1]
=> [1,1]
=> 0
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [3,2,1]
=> [2,1]
=> 1
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [4,2]
=> [2]
=> 0
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [4,2]
=> [2]
=> 0
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [3,2,1]
=> [2,1]
=> 1
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [3,3]
=> [3]
=> 0
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [2,2,1,1]
=> [2,1,1]
=> 1
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [3,2,1]
=> [2,1]
=> 1
[[1,3,6],[2],[4],[5]]
=> [5,4,2,1,3,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,6],[3],[4],[5]]
=> [5,4,3,1,2,6] => [4,1,1]
=> [1,1]
=> 0
[[1,4,5],[2],[3],[6]]
=> [6,3,2,1,4,5] => [4,2]
=> [2]
=> 0
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3],[4],[6]]
=> [6,4,3,1,2,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3],[2,5],[4],[6]]
=> [6,4,2,5,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2],[3,5],[4],[6]]
=> [6,4,3,5,1,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3],[2,4],[5],[6]]
=> [6,5,2,4,1,3] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
Description
The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St001629
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 38% ●values known / values provided: 43%●distinct values known / distinct values provided: 38%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001629: Integer compositions ⟶ ℤResult quality: 38% ●values known / values provided: 43%●distinct values known / distinct values provided: 38%
Values
[[1]]
=> [[1]]
=> [1] => [1] => ? = 0
[[1,2]]
=> [[1,2]]
=> [2] => [1] => ? ∊ {0,1}
[[1],[2]]
=> [[1,2]]
=> [2] => [1] => ? ∊ {0,1}
[[1,2,3]]
=> [[1,2,3]]
=> [3] => [1] => ? ∊ {0,0,1,2}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3] => [1] => ? ∊ {0,0,1,2}
[[1,2],[3]]
=> [[1,2,3]]
=> [3] => [1] => ? ∊ {0,0,1,2}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3] => [1] => ? ∊ {0,0,1,2}
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1] => [1,1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4] => [1] => ? ∊ {0,0,0,0,0,1,1,1,1,3}
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [5] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [3,3] => [2] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [4,2] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6] => [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [5,1] => [1,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,6],[4]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2,4],[6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5],[2],[4],[6]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3],[2,4],[5,6]]
=> [[1,2,4,6],[3,5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3],[2,6],[4],[5]]
=> [[1,2,4,6],[3],[5]]
=> [3,2,1] => [1,1,1] => 1
[[1,3,5,6,7],[2,4]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6,7],[2,5]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6,7],[3,5]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,6,7],[2],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6,7],[2],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6,7],[3],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,7],[2,4,6]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2,4,7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,5,7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,5,7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6,7],[2,5],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2,5],[4]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6,7],[3,5],[4]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2,4],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5,7],[2,6],[4]]
=> [[1,2,4,6],[3,7],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,7],[2,4],[6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,7],[5]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,7],[5]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5,6],[2,4],[7]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2,5],[7]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3,5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6,7],[2],[3],[5]]
=> [[1,2,5,7],[3],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6,7],[2],[4],[5]]
=> [[1,2,4,7],[3],[5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5,7],[2],[4],[6]]
=> [[1,2,4,6],[3],[5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,3,5,6],[2],[4],[7]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4,6],[2],[5],[7]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,4,6],[2,5,7],[3]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,5,7],[4]]
=> [[1,2,4,5,7],[3],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6],[3,5,7],[4]]
=> [[1,2,3,5,7],[4],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,4,7],[5]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
[[1,3,5],[2,6,7],[4]]
=> [[1,2,4,6,7],[3],[5]]
=> [3,2,2] => [1,2] => 0
[[1,3,5],[2,4,7],[6]]
=> [[1,2,4,6,7],[3,5]]
=> [3,2,2] => [1,2] => 0
[[1,3,4],[2,5,7],[6]]
=> [[1,2,4,5,7],[3,6]]
=> [3,3,1] => [2,1] => 0
[[1,2,4],[3,5,7],[6]]
=> [[1,2,3,5,7],[4,6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,5],[2,4,6],[7]]
=> [[1,2,4,6],[3,5,7]]
=> [3,2,2] => [1,2] => 0
[[1,3,7],[2,4],[5,6]]
=> [[1,2,4,6],[3,5],[7]]
=> [3,2,2] => [1,2] => 0
[[1,4,6],[2,5],[3,7]]
=> [[1,2,5,7],[3,6],[4]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,5],[4,7]]
=> [[1,2,4,5],[3,7],[6]]
=> [3,3,1] => [2,1] => 0
[[1,2,6],[3,5],[4,7]]
=> [[1,2,3,5],[4,7],[6]]
=> [4,2,1] => [1,1,1] => 1
[[1,3,6],[2,4],[5,7]]
=> [[1,2,4,7],[3,5],[6]]
=> [3,3,1] => [2,1] => 0
Description
The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles.
Matching statistic: St000455
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 13%●distinct values known / distinct values provided: 12%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 13%●distinct values known / distinct values provided: 12%
Values
[[1]]
=> [1] => [1] => ([],1)
=> ? = 0
[[1,2]]
=> [1,2] => [2] => ([],2)
=> ? ∊ {0,1}
[[1],[2]]
=> [2,1] => [2] => ([],2)
=> ? ∊ {0,1}
[[1,2,3]]
=> [1,2,3] => [3] => ([],3)
=> ? ∊ {0,1,2}
[[1,3],[2]]
=> [2,1,3] => [3] => ([],3)
=> ? ∊ {0,1,2}
[[1,2],[3]]
=> [3,1,2] => [3] => ([],3)
=> ? ∊ {0,1,2}
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 0
[[1,2,3,4]]
=> [1,2,3,4] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,3,4],[2]]
=> [2,1,3,4] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,2,4],[3]]
=> [3,1,2,4] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,2,3],[4]]
=> [4,1,2,3] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,3],[2,4]]
=> [2,4,1,3] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,2],[3,4]]
=> [3,4,1,2] => [4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,3] => ([(2,3)],4)
=> 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,3}
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [1,4] => ([(3,4)],5)
=> 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [1,4] => ([(3,4)],5)
=> 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [1,4] => ([(3,4)],5)
=> 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,3] => ([(2,4),(3,4)],5)
=> 0
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [1,4] => ([(3,4)],5)
=> 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,4}
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [2,5,1,3,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,6],[3,5]]
=> [3,5,1,2,4,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,6],[4,5]]
=> [4,5,1,2,3,6] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [2,6,1,3,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [3,6,1,2,4,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4,6]]
=> [4,6,1,2,3,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,3,5,6],[2],[4]]
=> [4,2,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,5,6],[3],[4]]
=> [4,3,1,2,5,6] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,4,6],[2],[5]]
=> [5,2,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,4,6],[3],[5]]
=> [5,3,1,2,4,6] => [1,5] => ([(4,5)],6)
=> 0
[[1,2,3,6],[4],[5]]
=> [5,4,1,2,3,6] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,4,5],[2],[6]]
=> [6,2,1,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,4,5],[3],[6]]
=> [6,3,1,2,4,5] => [1,5] => ([(4,5)],6)
=> 0
[[1,2,3,5],[4],[6]]
=> [6,4,1,2,3,5] => [1,5] => ([(4,5)],6)
=> 0
[[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,5],[2,4,6]]
=> [2,4,6,1,3,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3,4,6]]
=> [3,4,6,1,2,5] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5,6]]
=> [2,5,6,1,3,4] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4],[3,5,6]]
=> [3,5,6,1,2,4] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,6],[2,5],[3]]
=> [3,2,5,1,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,6],[3,5],[4]]
=> [4,3,5,1,2,6] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,6],[2,4],[5]]
=> [5,2,4,1,3,6] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,6],[3,4],[5]]
=> [5,3,4,1,2,6] => [1,5] => ([(4,5)],6)
=> 0
[[1,4,5],[2,6],[3]]
=> [3,2,6,1,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,3,5],[2,6],[4]]
=> [4,2,6,1,3,5] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,5],[3,6],[4]]
=> [4,3,6,1,2,5] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,4],[2,6],[5]]
=> [5,2,6,1,3,4] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,4],[3,6],[5]]
=> [5,3,6,1,2,4] => [1,5] => ([(4,5)],6)
=> 0
[[1,2,3],[4,6],[5]]
=> [5,4,6,1,2,3] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,5],[2,4],[6]]
=> [6,2,4,1,3,5] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,5],[3,4],[6]]
=> [6,3,4,1,2,5] => [1,5] => ([(4,5)],6)
=> 0
[[1,3,4],[2,5],[6]]
=> [6,2,5,1,3,4] => [2,4] => ([(3,5),(4,5)],6)
=> 0
[[1,2,4],[3,5],[6]]
=> [6,3,5,1,2,4] => [1,5] => ([(4,5)],6)
=> 0
[[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => [1,5] => ([(4,5)],6)
=> 0
[[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,4,6],[2],[3],[5]]
=> [5,3,2,1,4,6] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[[1,4],[2,5],[3,6]]
=> [3,6,2,5,1,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,5],[4,6]]
=> [4,6,2,5,1,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,5],[4,6]]
=> [4,6,3,5,1,2] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[[1,3],[2,4],[5,6]]
=> [5,6,2,4,1,3] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 0
[[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => [2,5] => ([(4,6),(5,6)],7)
=> 0
[[1,3,5,6,7],[2],[4]]
=> [4,2,1,3,5,6,7] => [2,5] => ([(4,6),(5,6)],7)
=> 0
[[1,2,5,6,7],[3],[4]]
=> [4,3,1,2,5,6,7] => [1,6] => ([(5,6)],7)
=> 0
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 25%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 25%
Values
[[1]]
=> [1] => ([],1)
=> 0
[[1,2]]
=> [2] => ([],2)
=> ? ∊ {0,1}
[[1],[2]]
=> [2] => ([],2)
=> ? ∊ {0,1}
[[1,2,3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1],[2],[3]]
=> [3] => ([],3)
=> ? ∊ {0,0,2}
[[1,2,3,4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,2,3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1],[2],[3],[4]]
=> [4] => ([],4)
=> ? ∊ {0,0,0,0,0,1,3}
[[1,2,3,4,5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3,4,5],[2]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2,3,4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3,5],[2,4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3,4]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3,4],[2,5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4,5],[2],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3,5],[2],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3,4],[2],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,4],[2,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3],[2,5],[4]]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[[1,2],[3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,3],[2,4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2],[3,4],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,5],[2],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[[1,4],[2],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,3],[2],[4],[5]]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1],[2],[3],[4],[5]]
=> [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,4}
[[1,2,3,4,5,6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5,6],[2]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5,6],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5,6],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2,4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3,4]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2,5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,6],[3,5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4,5]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,5],[2,6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4,6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5,6]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,5,6],[2],[3]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5,6],[2],[4]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5,6],[3],[4]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4,6],[2],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3,4,5],[2],[6]]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,4,5],[3],[6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,5],[4],[6]]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,3,4],[5],[6]]
=> [6] => ([],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,5],[2,4,6]]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,2,5],[3,4,6]]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,5}
[[1,3,4],[2,5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2,5],[3]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2,4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3,4],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,4],[2,6],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,4],[3,6],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,3],[4,6],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,4,6],[2],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3,6],[2],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2,6],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,3],[2,4],[5,6]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,4],[5,6]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,4],[2,6],[3],[5]]
=> [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,3],[2,6],[4],[5]]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1
[[1,2],[3,6],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,6],[2],[3],[4],[5]]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[[1,2,3,4,5,7],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2,6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4,6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,5],[6,7]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,4,5,7],[2],[6]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5,7],[3],[6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,5,7],[4],[6]]
=> [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,3,4,7],[5],[6]]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[[1,3,5,7],[2,4,6]]
=> [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,5,7],[3,4,6]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,3,4,5],[2,6,7]]
=> [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
[[1,2,4,5],[3,6,7]]
=> [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 7 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001948The number of augmented double ascents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001570The minimal number of edges to add to make a graph Hamiltonian. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
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