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Your data matches 74 different statistics following compositions of up to 3 maps.
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Matching statistic: St000375
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Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => 0
[[1,2]]
=> [1,2] => 0
[[1],[2]]
=> [2,1] => 0
[[1,2,3]]
=> [1,2,3] => 0
[[1,3],[2]]
=> [2,1,3] => 0
[[1,2],[3]]
=> [3,1,2] => 0
[[1],[2],[3]]
=> [3,2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => 0
[[1,2],[3,4]]
=> [3,4,1,2] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[[1,2,3,4,5]]
=> [1,2,3,4,5] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => 0
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => 0
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => 0
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => 0
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => 0
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => 0
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => 1
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => 0
Description
The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001086
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
=> [1] => [1] => [] => 0
[[1,2]]
=> [1,2] => [1,2] => [1] => 0
[[1],[2]]
=> [2,1] => [2,1] => [1] => 0
[[1,2,3]]
=> [1,2,3] => [1,2,3] => [1,2] => 0
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1] => 0
[[1,2],[3]]
=> [3,1,2] => [2,3,1] => [2,1] => 0
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [2,1] => 0
[[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3] => 0
[[1,3,4],[2]]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3] => 0
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1] => 0
[[1,2,3],[4]]
=> [4,1,2,3] => [2,3,4,1] => [2,3,1] => 0
[[1,3],[2,4]]
=> [2,4,1,3] => [1,3,4,2] => [1,3,2] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => [3,1,2] => 0
[[1,4],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1] => 0
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,2,4,1] => [3,2,1] => 0
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,2,1] => 0
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [3,2,1] => 0
[[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4] => 0
[[1,3,4,5],[2]]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4] => 0
[[1,2,4,5],[3]]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,3,1,4] => 0
[[1,2,3,5],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [2,3,4,1] => 0
[[1,2,3,4],[5]]
=> [5,1,2,3,4] => [2,3,4,5,1] => [2,3,4,1] => 0
[[1,3,5],[2,4]]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,3,4,2] => 0
[[1,2,5],[3,4]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [3,1,4,2] => 1
[[1,3,4],[2,5]]
=> [2,5,1,3,4] => [1,3,4,5,2] => [1,3,4,2] => 0
[[1,2,4],[3,5]]
=> [3,5,1,2,4] => [3,1,4,5,2] => [3,1,4,2] => 1
[[1,2,3],[4,5]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [3,4,1,2] => 0
[[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4] => 0
[[1,3,5],[2],[4]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [3,2,4,1] => 0
[[1,2,5],[3],[4]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1] => 0
[[1,3,4],[2],[5]]
=> [5,2,1,3,4] => [3,2,4,5,1] => [3,2,4,1] => 0
[[1,2,4],[3],[5]]
=> [5,3,1,2,4] => [3,4,2,5,1] => [3,4,2,1] => 0
[[1,2,3],[4],[5]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [3,4,2,1] => 0
[[1,4],[2,5],[3]]
=> [3,2,5,1,4] => [2,1,4,5,3] => [2,1,4,3] => 1
[[1,3],[2,5],[4]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [2,4,1,3] => 0
[[1,2],[3,5],[4]]
=> [4,3,5,1,2] => [4,2,1,5,3] => [4,2,1,3] => 0
[[1,3],[2,4],[5]]
=> [5,2,4,1,3] => [2,4,5,3,1] => [2,4,3,1] => 1
[[1,2],[3,4],[5]]
=> [5,3,4,1,2] => [4,2,5,3,1] => [4,2,3,1] => 0
[[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => [4,3,2,1,5] => [4,3,2,1] => 0
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,2,5,1] => [4,3,2,1] => 0
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,3,5,2,1] => [4,3,2,1] => 0
[[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => [4,5,3,2,1] => [4,3,2,1] => 0
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [4,3,2,1] => 0
[[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,2,3,4,5] => 0
[[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => [2,1,3,4,5] => 0
[[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [2,3,1,4,5,6] => [2,3,1,4,5] => 0
[[1,2,3,5,6],[4]]
=> [4,1,2,3,5,6] => [2,3,4,1,5,6] => [2,3,4,1,5] => 0
[[1,2,3,4,6],[5]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [2,3,4,5,1] => 0
[[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => [2,3,4,5,6,1] => [2,3,4,5,1] => 0
[[1,3,5,6],[2,4]]
=> [2,4,1,3,5,6] => [1,3,4,2,5,6] => [1,3,4,2,5] => 0
Description
The number of occurrences of the consecutive pattern 132 in a permutation.
This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St000225
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(load all 4 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? = 0
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? = 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? = 0
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? = 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [2,2]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? = 2
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St001214
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001214: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? = 0
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? = 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? = 0
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? = 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? = 2
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
Description
The aft of an integer partition.
The aft is the size of the partition minus the length of the first row or column, whichever is larger.
See also [[St000784]].
Matching statistic: St001588
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001588: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 95%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> []
=> ?
=> ? = 0
[[1,2]]
=> [2]
=> []
=> ?
=> ? = 0
[[1],[2]]
=> [1,1]
=> [1]
=> [1]
=> 0
[[1,2,3]]
=> [3]
=> []
=> ?
=> ? = 0
[[1,3],[2]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1,2],[3]]
=> [2,1]
=> [1]
=> [1]
=> 0
[[1],[2],[3]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3,4]]
=> [4]
=> []
=> ?
=> ? = 0
[[1,3,4],[2]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [3,1]
=> [1]
=> [1]
=> 0
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 0
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 0
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2,3,4,5]]
=> [5]
=> []
=> ?
=> ? = 0
[[1,3,4,5],[2]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> [1]
=> 0
[[1,3,5],[2,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5],[3,4]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4],[2,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3],[4,5]]
=> [3,2]
=> [2]
=> [1,1]
=> 0
[[1,4,5],[2],[3]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,5],[2],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,3,4],[2],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> [2]
=> 0
[[1,4],[2,5],[3]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,5],[4]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,3],[2,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,2],[3,4],[5]]
=> [2,2,1]
=> [2,1]
=> [1,1,1]
=> 0
[[1,5],[2],[3],[4]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[[1,2,3,4,5,6]]
=> [6]
=> []
=> ?
=> ? = 2
[[1,3,4,5,6],[2]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> [1]
=> 0
[[1,3,5,6],[2,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,5,6],[3,4]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,6],[2,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,3,6],[4,5]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,3,4,5],[2,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
[[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> [1,1]
=> 0
Description
The number of distinct odd parts smaller than the largest even part in an integer partition.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2]
=> [2]
=> 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2]
=> [2]
=> 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2]
=> [2]
=> 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St001587
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2]
=> [2]
=> 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001657
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(load all 2 compositions to match this statistic)
Mp00106: Standard tableaux —catabolism⟶ Standard tableaux
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001657: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 87%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [[1]]
=> [1]
=> []
=> ? = 0
[[1,2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1],[2]]
=> [[1,2]]
=> [2]
=> []
=> ? ∊ {0,0}
[[1,2,3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1,3],[2]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2],[3]]
=> [[1,2,3]]
=> [3]
=> []
=> ? ∊ {0,0}
[[1],[2],[3]]
=> [[1,2],[3]]
=> [2,1]
=> [1]
=> 0
[[1,2,3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2,4],[3]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1,2,3],[4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,3],[2,4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3,4]]
=> [[1,2,3,4]]
=> [4]
=> []
=> ? ∊ {0,0,1}
[[1,4],[2],[3]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> [3,1]
=> [1]
=> 0
[[1,2],[3],[4]]
=> [[1,2,3],[4]]
=> [3,1]
=> [1]
=> 0
[[1],[2],[3],[4]]
=> [[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> 0
[[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4,5],[3]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,5],[4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,2,3,4],[5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,3,5],[2,4]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2,5],[3,4]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,3,4],[2,5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3,5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4,5]]
=> [[1,2,3,4,5]]
=> [5]
=> []
=> ? ∊ {0,1,1}
[[1,4,5],[2],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2,5],[3],[4]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3,4],[2],[5]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2,4],[3],[5]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,2,3],[4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,4],[2,5],[3]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2,5],[4]]
=> [[1,2,4,5],[3]]
=> [4,1]
=> [1]
=> 0
[[1,2],[3,5],[4]]
=> [[1,2,3,5],[4]]
=> [4,1]
=> [1]
=> 0
[[1,3],[2,4],[5]]
=> [[1,2,4],[3,5]]
=> [3,2]
=> [2]
=> 1
[[1,2],[3,4],[5]]
=> [[1,2,3,4],[5]]
=> [4,1]
=> [1]
=> 0
[[1,5],[2],[3],[4]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,4],[2],[3],[5]]
=> [[1,2,5],[3],[4]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,3],[2],[4],[5]]
=> [[1,2,4],[3],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1,2],[3],[4],[5]]
=> [[1,2,3],[4],[5]]
=> [3,1,1]
=> [1,1]
=> 0
[[1],[2],[3],[4],[5]]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5,6],[3]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5,6],[4]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,6],[5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4,5],[6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,3,5,6],[2,4]]
=> [[1,2,4,6],[3,5]]
=> [4,2]
=> [2]
=> 1
[[1,2,5,6],[3,4]]
=> [[1,2,3,4],[5,6]]
=> [4,2]
=> [2]
=> 1
[[1,3,4,6],[2,5]]
=> [[1,2,4,5],[3,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,4,6],[3,5]]
=> [[1,2,3,5],[4,6]]
=> [4,2]
=> [2]
=> 1
[[1,2,3,6],[4,5]]
=> [[1,2,3,4,5],[6]]
=> [5,1]
=> [1]
=> 0
[[1,3,4,5],[2,6]]
=> [[1,2,4,5,6],[3]]
=> [5,1]
=> [1]
=> 0
[[1,2,4,5],[3,6]]
=> [[1,2,3,5,6],[4]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,5],[4,6]]
=> [[1,2,3,4,6],[5]]
=> [5,1]
=> [1]
=> 0
[[1,2,3,4],[5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
[[1,4,5,6],[2],[3]]
=> [[1,2,5,6],[3],[4]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,5,6],[2],[4]]
=> [[1,2,4,6],[3],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,5,6],[3],[4]]
=> [[1,2,3,6],[4],[5]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,3,4,6],[2],[5]]
=> [[1,2,4,5],[3],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,4,6],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3,6],[4],[5]]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1]
=> [1,1]
=> 0
[[1,2,3],[4,5,6]]
=> [[1,2,3,4,5,6]]
=> [6]
=> []
=> ? ∊ {0,1,1,2}
Description
The number of twos in an integer partition.
The total number of twos in all partitions of $n$ is equal to the total number of singletons [[St001484]] in all partitions of $n-1$, see [1].
The following 64 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000379The number of Hamiltonian cycles in a graph. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000944The 3-degree of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000929The constant term of the character polynomial of an integer partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St000936The number of even values of the symmetric group character corresponding to the partition. St000068The number of minimal elements in a poset. St000699The toughness times the least common multiple of 1,. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001867The number of alignments of type EN of a signed permutation. St000914The sum of the values of the Möbius function of a poset. St001396Number of triples of incomparable elements in a finite poset. St001868The number of alignments of type NE of a signed permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. 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.
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