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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000739
St000739: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2]]
=> 1
[[2,2]]
=> 2
[[1],[2]]
=> 2
[[1,1,2]]
=> 1
[[1,2,2]]
=> 1
[[2,2,2]]
=> 2
[[1,1],[2]]
=> 2
[[1,2],[2]]
=> 2
[[1,1,3]]
=> 1
[[1,2,3]]
=> 1
[[1,3,3]]
=> 1
[[2,2,3]]
=> 2
[[2,3,3]]
=> 2
[[3,3,3]]
=> 3
[[1,1],[3]]
=> 3
[[1,2],[3]]
=> 3
[[1,3],[2]]
=> 2
[[1,3],[3]]
=> 3
[[2,2],[3]]
=> 3
[[2,3],[3]]
=> 3
[[1],[2],[3]]
=> 3
[[1,1,1,2]]
=> 1
[[1,1,2,2]]
=> 1
[[1,2,2,2]]
=> 1
[[2,2,2,2]]
=> 2
[[1,1,1],[2]]
=> 2
[[1,1,2],[2]]
=> 2
[[1,2,2],[2]]
=> 2
[[1,1],[2,2]]
=> 2
[[1,1,1,3]]
=> 1
[[1,1,2,3]]
=> 1
[[1,1,3,3]]
=> 1
[[1,2,2,3]]
=> 1
[[1,2,3,3]]
=> 1
[[1,3,3,3]]
=> 1
[[2,2,2,3]]
=> 2
[[2,2,3,3]]
=> 2
[[2,3,3,3]]
=> 2
[[3,3,3,3]]
=> 3
[[1,1,1],[3]]
=> 3
[[1,1,2],[3]]
=> 3
[[1,1,3],[2]]
=> 2
[[1,1,3],[3]]
=> 3
[[1,2,2],[3]]
=> 3
[[1,2,3],[2]]
=> 2
[[1,2,3],[3]]
=> 3
[[1,3,3],[2]]
=> 2
[[1,3,3],[3]]
=> 3
[[2,2,2],[3]]
=> 3
[[2,2,3],[3]]
=> 3
Description
The first entry in the last row of a semistandard tableau.
Matching statistic: St001605
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2}
[[2,2]]
=> [2]
=> []
=> ?
=> ? ∊ {1,2,2}
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,2,2}
[[1,1,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [3]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1]
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [4]
=> []
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,2],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,3],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,3],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[2,4],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[2,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[2,2],[3,3],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[2,2],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[2,3],[3,4],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,3],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,1],[2,2],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,3],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[2,5],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,3],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[3,5],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[4,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,1],[4,5],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,3],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[2,5],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,3],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
[[1,2],[3,5],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 2
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000260
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[[2,2]]
=> [1,2] => [1,2] => ([],2)
=> ? ∊ {2,2}
[[1],[2]]
=> [2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[[1,1,1,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,1,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[2,2,2,2]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,1,1],[2]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,1,2],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,2,2],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,2,2,2,2}
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3],[3]]
=> [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {1,1,1,1,1,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,1],[2,2],[4]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,3],[4]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,4],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[2,4],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[3,3],[4]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[3,4],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[2,3],[4]]
=> [5,2,4,1,3] => [5,2,1,4,3] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 1
[[1,2],[2,4],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,2],[2,4],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,2],[3,3],[4]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,3],[2,4],[3]]
=> [3,2,5,1,4] => [3,5,2,1,4] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,2],[3,4],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3],[2,4],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,3],[3,4],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[2,2],[3,3],[4]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[2,2],[3,4],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[2,3],[3,4],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[[1,1],[2,2],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,3],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,5],[3]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[2,4],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
[[1,1],[2,5],[4]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[2,5],[5]]
=> [4,3,5,1,2] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
[[1,1],[3,3],[5]]
=> [5,3,4,1,2] => [3,1,5,4,2] => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00107: Semistandard tableaux —catabolism⟶ Semistandard tableaux
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 40%
Values
[[1,2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[2,2]]
=> [[2,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[1],[2]]
=> [[1,2]]
=> [1,2] => ([],2)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [1,2,3] => ([],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [3,1,2] => ([(0,2),(1,2)],3)
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [1,2,3,4] => ([],4)
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[[1,1,4],[2],[3]]
=> [[1,1,2],[3],[4]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,4],[2],[3]]
=> [[1,2,2],[3],[4]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,4],[2],[3]]
=> [[1,2,3],[3],[4]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4,4],[2],[3]]
=> [[1,2,4],[3],[4]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1],[2],[3],[4]]
=> [[1,1,2],[3],[4]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[2],[3],[4]]
=> [[1,2,2],[3],[4]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2],[3],[4]]
=> [[1,2,3],[3],[4]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2],[3],[4]]
=> [[1,2,4],[3],[4]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[[1,1,5],[2],[3]]
=> [[1,1,2],[3],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,5],[2],[4]]
=> [[1,1,2],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,5],[3],[4]]
=> [[1,1,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[2],[3]]
=> [[1,2,2],[3],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[2],[4]]
=> [[1,2,2],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[3]]
=> [[1,2,3],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1],[2],[3],[5]]
=> [[1,1,2],[3],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1],[2],[4],[5]]
=> [[1,1,2],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1],[3],[4],[5]]
=> [[1,1,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[2],[3],[5]]
=> [[1,2,2],[3],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[2],[4],[5]]
=> [[1,2,2],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[3],[5]]
=> [[1,2,5],[3],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[2],[4],[5]]
=> [[1,2,5],[4],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,3],[3],[4],[5]]
=> [[1,3,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[1,5],[3],[4],[5]]
=> [[1,3,5],[4],[5]]
=> [4,3,1,2,5] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,2],[3],[4],[5]]
=> [[2,2,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[[2,3],[3],[4],[5]]
=> [[2,3,3],[4],[5]]
=> [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
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}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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
[[1,1,4],[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
[[1,3,5],[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
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 2
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[5]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[4],[5]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4],[5]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[5]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4],[5]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 5
[[1,5],[2],[3],[5]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4],[5]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4],[5]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4],[5]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4],[5]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4],[5]]
=> [[2,3,3],[4],[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: St000939
(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
St000939: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000939: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
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}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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
[[1,1,4],[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
[[1,3,5],[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]
=> 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
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[5]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[4],[5]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4],[5]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[5]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4],[5]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[1,5],[2],[3],[5]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4],[5]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4],[5]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4],[5]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4],[5]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4],[5]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
Description
The number of characters of the symmetric group whose value on the partition is positive.
Matching statistic: St000993
(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
St000993: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 60%
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}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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
[[1,1,4],[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
[[1,3,5],[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]
=> 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
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[5]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[4],[5]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4],[5]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[5]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4],[5]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[[1,5],[2],[3],[5]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4],[5]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4],[5]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4],[5]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4],[5]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4],[5]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001568
(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
St001568: 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
St001568: 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}
[[1,1,2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> [[2,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> [[1,1,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> [[1,2,2]]
=> [3]
=> []
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3,3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2,3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> [[3,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> [[1,1,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> [[1,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> [[2,2,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> [[2,3,3]]
=> [3]
=> []
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> ? ∊ {1,1,1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> [[2,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> [[1,1,1,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> [[1,2,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> [[1,1,2,2]]
=> [4]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3,3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2,3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> [[2,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> [[3,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> [[1,1,1,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> [[1,1,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> [[1,1,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> [[1,1,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> [[1,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> [[1,2,2],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[3]]
=> [[1,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> [[1,2,3],[3]]
=> [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[3]]
=> [[1,3,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> [[2,2,2,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> [[2,2,3,3]]
=> [4]
=> []
=> ? ∊ {1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[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
[[1,1,4],[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
[[1,3,5],[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]
=> 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
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[3]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[2],[4]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[2],[4]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3,5],[3],[4]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4,5],[3],[4]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5,5],[3],[4]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2,5],[3],[4]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3,5],[3],[4]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,4,5],[3],[4]]
=> [[2,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,5,5],[3],[4]]
=> [[2,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[3],[5]]
=> [[1,1,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[2],[4],[5]]
=> [[1,1,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,1],[3],[4],[5]]
=> [[1,1,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[3],[5]]
=> [[1,2,2],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[2],[4],[5]]
=> [[1,2,2],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[3],[5]]
=> [[1,2,3],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[[1,5],[2],[3],[5]]
=> [[1,2,5],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[2],[4],[5]]
=> [[1,2,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[2],[4],[5]]
=> [[1,2,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,3],[3],[4],[5]]
=> [[1,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,4],[3],[4],[5]]
=> [[1,3,4],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[1,5],[3],[4],[5]]
=> [[1,3,5],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,2],[3],[4],[5]]
=> [[2,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
[[2,3],[3],[4],[5]]
=> [[2,3,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 2
Description
The smallest positive integer that does not appear twice in the partition.
Matching statistic: St000456
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> ?
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2,3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,4],[2,3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2],[4]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2,4],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,2],[3,3],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,3],[2,4],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1,1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1,2],[2],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[2,3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 1
[[1,1],[2,4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,1],[2,5],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[3],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 1
[[1,2],[2],[4],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
[[1,3],[2],[3],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St000464
Values
[[1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,2,2}
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,2,2}
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,2,2}
[[1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,2,2,2}
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,3,3,3,3,3,3,3}
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,2,2,2,2,2}
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2,2,3]]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3,3]]
=> ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
=> ([(3,8),(4,7),(5,6),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3,3]]
=> ([(0,6),(1,7),(2,9),(4,8),(5,1),(5,9),(6,2),(6,5),(7,8),(8,3),(9,4),(9,7)],10)
=> ([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2,3]]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> ([(3,8),(4,7),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3,3]]
=> ([(0,6),(1,11),(2,8),(3,9),(4,5),(4,11),(5,3),(5,7),(6,1),(6,4),(7,8),(7,9),(8,10),(9,10),(11,2),(11,7)],12)
=> ([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3,3]]
=> ([(0,7),(1,13),(2,12),(3,9),(4,11),(5,6),(5,12),(6,4),(6,8),(7,2),(7,5),(8,11),(8,13),(10,9),(11,10),(12,1),(12,8),(13,3),(13,10)],14)
=> ([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[3,3,3,3]]
=> ([(0,8),(1,14),(3,13),(4,12),(5,11),(6,7),(6,12),(7,5),(7,9),(8,4),(8,6),(9,11),(9,13),(10,14),(11,10),(12,3),(12,9),(13,1),(13,10),(14,2)],15)
=> ([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,2,3],[3]]
=> ([(0,5),(0,6),(1,7),(2,7),(3,2),(4,1),(5,3),(6,4)],8)
=> ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,3,3],[3]]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([],5)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,2,3],[3]]
=> ([(0,6),(0,7),(1,9),(2,8),(3,5),(4,2),(5,1),(5,8),(6,3),(7,4),(8,9)],10)
=> ([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[2,3,3],[3]]
=> ([(0,9),(0,10),(1,11),(2,14),(3,12),(4,13),(5,4),(5,11),(6,5),(7,3),(8,1),(8,14),(9,6),(10,2),(10,8),(11,13),(13,12),(14,7)],15)
=> ([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> ?
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[2,3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([],3)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,2],[2],[3]]
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> ? ∊ {1,1,1,1,1,1,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3}
[[1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[[1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2,3],[2],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,1],[3,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,2],[2,4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,4],[2,3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,2],[2],[4]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[[1,1,3],[2],[4]]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2,2],[2],[4]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2],[2,4],[3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,2],[3,3],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,3],[2,4],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,1],[3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1,1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1,2],[2],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1],[2,3],[5]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> 2
[[1,1],[2,4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,1],[2,5],[4]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,2],[2],[3],[5]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
[[1,2],[2],[4],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
[[1,3],[2],[3],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
Description
The Schultz index of a connected graph.
This is
$$\sum_{\{u,v\}\subseteq V} (d(u)+d(v))d(u,v)$$
where $d(u)$ is the degree of vertex $u$ and $d(u,v)$ is the distance between vertices $u$ and $v$.
For trees on $n$ vertices, the Schultz index is related to the Wiener index via $S(T)=4W(T)-n(n-1)$ [2].
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St001545The second Elser number of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another 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. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001060The distinguishing index of a graph. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. 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. St000929The constant term of the character polynomial of an integer 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. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000259The diameter of a connected graph.
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