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Your data matches 94 different statistics following compositions of up to 3 maps.
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Matching statistic: St000681
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000681: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [2]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [1,1]
=> 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [2]
=> 1
Description
The Grundy value of Chomp on Ferrers diagrams.
Players take turns and choose a cell of the diagram, cutting off all cells below and to the right of this cell in English notation. The player who is left with the single cell partition looses. The traditional version is played on chocolate bars, see [1].
This statistic is the Grundy value of the partition, that is, the smallest non-negative integer which does not occur as value of a partition obtained by a single move.
Matching statistic: St001557
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00066: Permutations —inverse⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [3,2,4,1] => [1,3,4,2] => 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [1,3,2,4,5] => 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [1,3,2] => 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [1,3,4,2] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [1,3,2,4] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,3,4,2] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,3,4,2] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,3,4,2] => 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [1,3,2,4] => 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [3,2,4,1] => [1,3,4,2] => 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [1,3,2,4] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [1,4,2,3] => 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,4,5,1,2] => [1,3,5,2,4] => 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [3,4,5,1,6,2] => [1,3,5,6,2,4] => ? = 1
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [3,4,1,5,6,2] => [1,3,2,4,5,6] => ? = 1
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [3,4,1,5,2,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [3,1,4,5,6,2] => [1,3,4,5,6,2] => ? = 1
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [3,1,4,5,2,6] => [1,3,4,5,2,6] => ? = 1
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [3,1,4,2,5,6] => [1,3,4,2,5,6] => ? = 1
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [3,4,5,6,1,2] => [1,3,5,2,4,6] => ? = 1
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,4,5,1,2,6] => [1,3,5,2,4,6] => ? = 1
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [3,4,5,6,2,1] => [1,3,5,2,4,6] => ? = 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [3,4,5,2,6,1] => [1,3,5,6,2,4] => ? = 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [3,4,5,2,1,6] => [1,3,5,2,4,6] => ? = 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [3,4,2,5,6,1] => [1,3,2,4,5,6] => ? = 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [3,4,2,5,1,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [3,4,2,1,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [3,2,4,5,6,1] => [1,3,4,5,6,2] => ? = 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [3,2,4,5,1,6] => [1,3,4,5,2,6] => ? = 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [3,2,4,1,5,6] => [1,3,4,2,5,6] => ? = 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [1,3,2,4,5,6] => ? = 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [4,5,1,2,6,3] => [1,4,2,5,6,3] => ? = 0
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [4,5,1,6,2,3] => [1,4,6,3,2,5] => ? = 0
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [4,1,5,6,2,3] => [1,4,6,3,5,2] => ? = 0
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [1,4,2,5,3,6] => ? = 0
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [4,5,6,2,3,1] => [1,4,2,5,3,6] => ? = 2
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [4,5,6,2,1,3] => [1,4,2,5,3,6] => ? = 2
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [4,5,2,3,6,1] => [1,4,3,2,5,6] => ? = 2
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation π of length n,
#{2<k≤n∣π(2)>π(k)}.
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001960
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,4,1,5,2] => 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,4,1,3] => [3,1,4,2] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [3,4,2,1] => 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [3,2,4,1] => 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,4,1,5,2] => 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,5,1,2,4] => [3,4,1,5,2] => 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [3,6,1,2,4,5] => [3,4,1,5,6,2] => ? = 1
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [2,6,1,3,4,5] => [3,1,4,5,6,2] => ? = 1
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [2,5,1,3,4,6] => [3,1,4,5,2,6] => ? = 1
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [4,6,1,2,3,5] => [3,4,5,1,6,2] => ? = 1
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,5,1,2,4,6] => [3,4,1,5,2,6] => ? = 1
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [2,4,1,3,5,6] => [3,1,4,2,5,6] => ? = 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [6,5,1,2,3,4] => [3,4,5,6,2,1] => ? = 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [6,4,1,2,3,5] => [3,4,5,2,6,1] => ? = 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [3,4,5,2,1,6] => ? = 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [6,3,1,2,4,5] => [3,4,2,5,6,1] => ? = 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [3,4,2,5,1,6] => ? = 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [3,4,2,1,5,6] => ? = 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [6,2,1,3,4,5] => [3,2,4,5,6,1] => ? = 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [3,2,4,5,1,6] => ? = 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [3,2,4,1,5,6] => ? = 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [3,2,1,4,5,6] => ? = 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [2,4,6,1,3,5] => [4,1,5,2,6,3] => ? = 0
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [6,3,5,1,2,4] => [4,5,2,6,3,1] => ? = 2
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [4,3,6,1,2,5] => [4,5,2,1,6,3] => ? = 2
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [6,2,4,1,3,5] => [4,2,5,3,6,1] => ? = 2
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001434
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001434: Signed permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => 2 = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,1,-3] => 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => 2 = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => 2 = 1 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [4,6,1,2,3,5] => [4,5,1,6,2,-3] => ? = 1 + 1
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [3,6,1,2,4,5] => [4,1,5,6,2,-3] => ? = 1 + 1
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [3,5,1,2,4,6] => [4,1,5,2,6,-3] => ? = 1 + 1
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [2,6,1,3,4,5] => [1,4,5,6,2,-3] => ? = 1 + 1
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [2,5,1,3,4,6] => [1,4,5,2,6,-3] => ? = 1 + 1
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [2,4,1,3,5,6] => [1,4,2,5,6,-3] => ? = 1 + 1
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [5,6,1,2,3,4] => [4,5,6,1,2,-3] => ? = 1 + 1
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [4,5,1,2,3,6] => [4,5,1,2,6,-3] => ? = 1 + 1
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [3,4,1,2,5,6] => [4,1,2,5,6,-3] => ? = 1 + 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [6,5,1,2,3,4] => [4,5,6,2,1,-3] => ? = 1 + 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [6,4,1,2,3,5] => [4,5,2,6,1,-3] => ? = 1 + 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [5,4,1,2,3,6] => [4,5,2,1,6,-3] => ? = 1 + 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [6,3,1,2,4,5] => [4,2,5,6,1,-3] => ? = 1 + 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [5,3,1,2,4,6] => [4,2,5,1,6,-3] => ? = 1 + 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [4,3,1,2,5,6] => [4,2,1,5,6,-3] => ? = 1 + 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [6,2,1,3,4,5] => [2,4,5,6,1,-3] => ? = 1 + 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [5,2,1,3,4,6] => [2,4,5,1,6,-3] => ? = 1 + 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [4,2,1,3,5,6] => [2,4,1,5,6,-3] => ? = 1 + 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [3,2,1,4,5,6] => [2,1,4,5,6,-3] => ? = 1 + 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [3,4,6,1,2,5] => [5,1,2,6,3,-4] => ? = 0 + 1
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [3,5,6,1,2,4] => [5,1,6,2,3,-4] => ? = 0 + 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [2,5,6,1,3,4] => [1,5,6,2,3,-4] => ? = 0 + 1
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [4,5,6,1,2,3] => [5,6,1,2,3,-4] => ? = 0 + 1
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [6,4,5,1,2,3] => [5,6,2,3,1,-4] => ? = 2 + 1
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [5,4,6,1,2,3] => [5,6,2,1,3,-4] => ? = 2 + 1
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [6,3,4,1,2,5] => [5,2,3,6,1,-4] => ? = 2 + 1
Description
The number of negative sum pairs of a signed permutation.
The number of negative sum pairs of a signed permutation σ is:
nsp(σ)=|{1≤i<j≤n∣σ(i)+σ(j)<0}|,
see [1, Eq.(8.1)].
Matching statistic: St001668
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00069: Permutations —complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2 = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2 = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2 = 1 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [3,1,6,5,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [4,1,6,5,3,2] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [4,2,6,5,3,1] => ([(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [5,2,6,4,3,1] => ([(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [5,3,6,4,2,1] => ([(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5)],6)
=> ? = 1 + 1
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 1 + 1
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => ([(0,5),(5,1),(5,2),(5,3),(5,4)],6)
=> ? = 1 + 1
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => ([(0,4),(0,5),(5,1),(5,2),(5,3)],6)
=> ? = 1 + 1
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [2,3,6,5,4,1] => ([(1,5),(5,2),(5,3),(5,4)],6)
=> ? = 1 + 1
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => ([(0,3),(0,4),(0,5),(5,1),(5,2)],6)
=> ? = 1 + 1
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [2,4,6,5,3,1] => ([(1,4),(1,5),(5,2),(5,3)],6)
=> ? = 1 + 1
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [3,4,6,5,2,1] => ([(2,3),(3,4),(3,5)],6)
=> ? = 1 + 1
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [1,5,6,4,3,2] => ([(0,2),(0,3),(0,4),(0,5),(5,1)],6)
=> ? = 1 + 1
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [2,5,6,4,3,1] => ([(1,3),(1,4),(1,5),(5,2)],6)
=> ? = 1 + 1
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [3,5,6,4,2,1] => ([(2,3),(2,4),(4,5)],6)
=> ? = 1 + 1
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
=> ? = 1 + 1
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [4,3,1,6,5,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [4,2,1,6,5,3] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [5,2,1,6,4,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 0 + 1
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => ([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 2 + 1
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [2,3,1,6,5,4] => ([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ? = 2 + 1
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => ([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
Description
The number of points of the poset minus the width of the poset.
Matching statistic: St000193
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00005: Alternating sign matrices —transpose⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 1 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> 3 = 1 + 2
[[1],[2],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> 3 = 1 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[2,3],[3,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[3,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 1 + 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 1 + 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 1 + 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[1,0,0,0],[0,1,0,0]]
=> 3 = 1 + 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> 3 = 1 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> 4 = 2 + 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0]]
=> ? = 1 + 2
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> ? = 1 + 2
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 0 + 2
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2 + 2
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 2 + 2
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[0,0,0,0,0,1],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0]]
=> ? = 2 + 2
Description
The row of the unique '1' in the first column of the alternating sign matrix.
Matching statistic: St000199
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00069: Permutations —complement⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000199: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1],[2],[4]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,2,5,4,1] => [[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1],[2],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[3,4]]
=> [2,4,1,3] => [3,1,4,2] => [[0,1,0,0],[0,0,0,1],[1,0,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[3,3],[4,4]]
=> [3,4,1,2] => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 3 = 1 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 2 + 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [3,1,6,5,4,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [4,1,6,5,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [4,2,6,5,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [5,1,6,4,3,2] => [[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [5,2,6,4,3,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [5,3,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [2,1,6,5,4,3] => [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [3,2,6,5,4,1] => [[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [4,3,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [1,2,6,5,4,3] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [1,3,6,5,4,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [2,3,6,5,4,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [1,4,6,5,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [2,4,6,5,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [3,4,6,5,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [1,5,6,4,3,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [2,5,6,4,3,1] => [[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [3,5,6,4,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [4,5,6,3,2,1] => [[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [4,3,1,6,5,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [4,2,1,6,5,3] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [5,2,1,6,4,3] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [3,2,1,6,5,4] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [1,3,2,6,5,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 + 2
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [2,3,1,6,5,4] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 + 2
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [1,4,3,6,5,2] => [[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 + 2
Description
The column of the unique '1' in the last row of the alternating sign matrix.
Matching statistic: St000200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00004: Alternating sign matrices —rotate clockwise⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 60%
Values
[[1],[2],[3]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,2]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1],[2],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[3],[4]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2,3]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[3,3]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 1 + 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[1,0,0,0,0]]
=> 3 = 1 + 2
[[1],[2],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[3],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[3],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[2],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[3],[4],[5]]
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3 = 1 + 2
[[1,1],[2,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[3,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[3,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,2],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[3,4]]
=> [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[2,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[3,3],[4,4]]
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 1 + 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 1 + 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 1 + 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 3 = 1 + 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
=> [[1,0,0,0],[0,0,1,0],[0,0,0,1],[0,1,0,0]]
=> 3 = 1 + 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
=> 3 = 1 + 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4 = 2 + 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> 3 = 1 + 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1,1],[2,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,1],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2,3]]
=> [4,6,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,2]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2,3]]
=> [3,6,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2,3]]
=> [3,5,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2,2]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[2,3]]
=> [2,6,1,3,4,5] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0]]
=> [[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[2,3]]
=> [2,5,1,3,4,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[2,3]]
=> [2,4,1,3,5,6] => [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[2,2,2,2],[3,3]]
=> [5,6,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0]]
=> [[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[2,2,2,3],[3,3]]
=> [4,5,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[2,2,3,3],[3,3]]
=> [3,4,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1,1],[2],[3]]
=> [6,5,1,2,3,4] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 1 + 2
[[1,1,1,2],[2],[3]]
=> [6,4,1,2,3,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1,3],[2],[3]]
=> [5,4,1,2,3,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,2],[2],[3]]
=> [6,3,1,2,4,5] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,1,2,3],[2],[3]]
=> [5,3,1,2,4,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,3,3],[2],[3]]
=> [4,3,1,2,5,6] => [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,2,2],[2],[3]]
=> [6,2,1,3,4,5] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0]]
=> ? = 1 + 2
[[1,2,2,3],[2],[3]]
=> [5,2,1,3,4,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
=> [[0,1,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,2,3,3],[2],[3]]
=> [4,2,1,3,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,3,3,3],[2],[3]]
=> [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
=> ? = 1 + 2
[[1,1,1],[2,2,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,2],[2,2,3]]
=> [3,4,6,1,2,5] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 0 + 2
[[1,1,2],[2,3,3]]
=> [3,5,6,1,2,4] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,1,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,2,2],[2,3,3]]
=> [2,5,6,1,3,4] => [[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,1,0,0,0]]
=> ? = 0 + 2
[[1,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[2,2,2],[3,3,3]]
=> [4,5,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 0 + 2
[[1,1,1],[2,2],[3]]
=> [6,4,5,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 + 2
[[1,1,1],[2,3],[3]]
=> [5,4,6,1,2,3] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,1,0,0,0]]
=> [[0,1,0,0,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0]]
=> ? = 2 + 2
[[1,1,2],[2,2],[3]]
=> [6,3,4,1,2,5] => [[0,0,0,1,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0]]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,1,0,0,0,0]]
=> ? = 2 + 2
Description
The row of the unique '1' in the last column of the alternating sign matrix.
Matching statistic: St000454
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[[1],[2],[3]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,3]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,4]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[2,4]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[3,4]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,3],[3,4]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
[[1],[2],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[3],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[2],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[4],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[3],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[4],[5],[6]]
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[[1,1],[2,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[3,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[4,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[5,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[2,5]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[3,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,3],[2,5]]
=> [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,2],[4,5]]
=> [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 1
[[1,1],[2],[5]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[3],[5]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1],[4],[5]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[2],[5]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[3],[5]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2],[5]]
=> [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[2],[3]]
=> [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,2],[4],[5]]
=> [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
Description
The largest eigenvalue of a graph if it is integral.
If a graph is d-regular, then its largest eigenvalue equals d. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001232
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 80%
Values
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2,3],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,2],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,2],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,2],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,3,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[1,2],[2,3],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ? = 2 + 1
[[1,1,1,1],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1,2],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,2,2],[2,2]]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1 + 1
[[1,1,1],[2,2,2]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 0 + 1
[[1],[2],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[3],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[2],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[3],[4],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[3],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[4],[5],[6]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[3,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[4,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[5,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[3,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,3],[2,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,2],[4,5]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> ? = 1 + 1
[[1,1],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,1],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[3],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,3],[2],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,5],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[[1,2],[4],[5]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 1 + 1
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001644The dimension of a graph. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001937The size of the center of a parking function. St001896The number of right descents of a signed permutations. St001946The number of descents in a parking function. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001845The number of join irreducibles minus the rank of a lattice. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001490The number of connected components of a skew partition. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000910The number of maximal chains of minimal length in a poset. St000307The number of rowmotion orbits of a poset. St001631The number of simple modules S with dimExt1(S,A)=1 in the incidence algebra A of the poset. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St001754The number of tolerances of a finite lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000080The rank of the poset. St001857The number of edges in the reduced word graph of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001625The Möbius invariant of a lattice. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000417The size of the automorphism group of the ordered tree. St000679The pruning number of an ordered tree. St001058The breadth of the ordered tree. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001964The interval resolution global dimension of a poset. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000522The number of 1-protected nodes of a rooted tree. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000413The number of ordered trees with the same underlying unordered tree. St000521The number of distinct subtrees of an ordered tree. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000189The number of elements in the poset. St001171The vector space dimension of Ext1A(Io,A) when Io is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(xn). St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St000400The path length of an ordered tree. St000180The number of chains of a poset. St000529The number of permutations whose descent word is the given binary word. St000100The number of linear extensions of a poset. St000416The number of inequivalent increasing trees of an ordered tree. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St000422The energy of a graph, if it is integral.
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