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Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St001867
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001867: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,3,5,4,2] => 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,4,5,3] => 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of alignments of type EN of a signed permutation.
An alignment of type EN of a signed permutation π∈Hn is a pair −n≤i≤j≤n, i,j≠0, such that one of the following conditions hold:
* $-i < 0 < -\pi(i) < \pi(j) < j$
* $i \leq\pi(i) < \pi(j) < j$.
Matching statistic: St000590
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
St000590: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> ? = 0
{{1,2}}
=> 0
{{1},{2}}
=> 0
{{1,2,3}}
=> 0
{{1,2},{3}}
=> 0
{{1,3},{2}}
=> 0
{{1},{2,3}}
=> 1
{{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> 0
{{1,2,3},{4}}
=> 0
{{1,2,4},{3}}
=> 0
{{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> 0
{{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> 1
{{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> 0
{{1},{2,3,4,5}}
=> 1
{{1},{2,3,4},{5}}
=> 1
{{1},{2,3,5},{4}}
=> 1
{{1},{2,3},{4,5}}
=> 3
{{1},{2,3},{4},{5}}
=> 1
{{1},{2,4,5},{3}}
=> 1
{{1},{2,4},{3,5}}
=> 2
{{1},{2,4},{3},{5}}
=> 1
{{1},{2,5},{3,4}}
=> 2
{{1},{2},{3,4,5}}
=> 2
{{1},{2},{3,4},{5}}
=> 2
{{1},{2,5},{3},{4}}
=> 1
{{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3},{4,5}}
=> 3
{{1},{2},{3},{4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000589
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000606
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000606: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000606: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block.
Matching statistic: St000611
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000611: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000611: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => {{1}}
=> ? = 0
{{1,2}}
=> [2,1] => [2,1] => {{1,2}}
=> 0
{{1},{2}}
=> [1,2] => [1,2] => {{1},{2}}
=> 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => {{1,3},{2}}
=> 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => {{1,3},{2}}
=> 0
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => {{1,4},{2},{3}}
=> 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,1,3,2] => {{1,4},{2},{3}}
=> 0
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => {{1,4},{2},{3}}
=> 0
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 0
{{1},{2,3,4}}
=> [1,3,4,2] => [1,4,2,3] => {{1},{2,4},{3}}
=> 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,5,2,3,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,4,2,3,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [1,5,2,4,3] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => {{1},{2,4},{3},{5}}
=> 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [1,2,5,3,4] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => {{1},{2},{3,4},{5}}
=> 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => {{1},{2},{3},{4},{5}}
=> 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal.
Matching statistic: St001868
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001868: Signed permutations ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> {{1}}
=> [1] => [1] => 0
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1,3},{2}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3},{4}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => 0
{{1,2,4},{3}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => 0
{{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => 1
{{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => 0
{{1,3,4},{2}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => 0
{{1,3},{2,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => 0
{{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => 0
{{1,4},{2,3}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => 1
{{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => 1
{{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => 0
{{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => 1
{{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => ? = 1
{{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [1,3,4,2,5] => 1
{{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
{{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 3
{{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [1,2,4,3,5] => 1
{{1},{2,4,5},{3}}
=> {{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => ? = 1
{{1},{2,4},{3,5}}
=> {{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => ? = 2
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => 1
{{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => ? = 2
{{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
{{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
{{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
{{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 2
{{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 3
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of alignments of type NE of a signed permutation.
An alignment of type NE of a signed permutation $\pi\in\mathfrak H_n$ is a pair $1 \leq i, j\leq n$ such that $\pi(i) < i < j \leq \pi(j)$.
Matching statistic: St001862
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 75%
Values
{{1}}
=> [1] => [1] => [1] => 0
{{1,2}}
=> [2,1] => [2,1] => [2,1] => 0
{{1},{2}}
=> [1,2] => [1,2] => [1,2] => 0
{{1,2,3}}
=> [2,3,1] => [3,1,2] => [3,1,2] => 0
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [3,2,1] => 0
{{1},{2,3}}
=> [1,3,2] => [2,3,1] => [2,3,1] => 1
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> [2,3,4,1] => [4,1,2,3] => [4,1,2,3] => 0
{{1,2,3},{4}}
=> [2,3,1,4] => [3,1,2,4] => [3,1,2,4] => 0
{{1,2,4},{3}}
=> [2,4,3,1] => [4,2,3,1] => [4,2,3,1] => 0
{{1,2},{3,4}}
=> [2,1,4,3] => [3,2,4,1] => [3,2,4,1] => 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
{{1,3,4},{2}}
=> [3,2,4,1] => [4,2,1,3] => [4,2,1,3] => 0
{{1,3},{2,4}}
=> [3,4,1,2] => [1,4,2,3] => [1,4,2,3] => 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0
{{1},{2,3,4}}
=> [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [2,3,1,4] => [2,3,1,4] => 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,1,3,2] => [4,1,3,2] => 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [3,4,2,1] => [3,4,2,1] => 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [2,3,4,1] => [2,3,4,1] => 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [3,5,2,4,1] => [3,5,2,4,1] => ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [3,4,2,5,1] => [3,4,2,5,1] => ? = 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [2,3,5,1,4] => [2,3,5,1,4] => ? = 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [2,3,4,1,5] => [2,3,4,1,5] => ? = 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [3,4,5,2,1] => [3,4,5,2,1] => ? = 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [2,3,4,5,1] => [2,3,4,5,1] => ? = 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
Description
The number of crossings of a signed permutation.
A crossing of a signed permutation $\pi$ is a pair $(i, j)$ of indices such that
* $i < j \leq \pi(i) < \pi(j)$, or
* $-i < j \leq -\pi(i) < \pi(j)$, or
* $i > j > \pi(i) > \pi(j)$.
Matching statistic: St001857
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00220: Set partitions —Yip⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001857: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
{{1}}
=> {{1}}
=> [1] => [1] => 0
{{1,2}}
=> {{1,2}}
=> [2,1] => [2,1] => 0
{{1},{2}}
=> {{1},{2}}
=> [1,2] => [1,2] => 0
{{1,2,3}}
=> {{1,2,3}}
=> [2,3,1] => [2,3,1] => 0
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1,3] => [2,1,3] => 0
{{1,3},{2}}
=> {{1},{2,3}}
=> [1,3,2] => [1,3,2] => 0
{{1},{2,3}}
=> {{1,3},{2}}
=> [3,2,1] => [3,2,1] => 1
{{1},{2},{3}}
=> {{1},{2},{3}}
=> [1,2,3] => [1,2,3] => 0
{{1,2,3,4}}
=> {{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => ? = 0
{{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => ? = 0
{{1,2,4},{3}}
=> {{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => ? = 0
{{1,2},{3,4}}
=> {{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => ? = 1
{{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => ? = 0
{{1,3,4},{2}}
=> {{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => ? = 0
{{1,3},{2,4}}
=> {{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => ? = 0
{{1,3},{2},{4}}
=> {{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => ? = 0
{{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => ? = 0
{{1},{2,3,4}}
=> {{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => ? = 1
{{1},{2,3},{4}}
=> {{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => ? = 1
{{1,4},{2},{3}}
=> {{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => ? = 0
{{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => ? = 1
{{1},{2},{3,4}}
=> {{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => ? = 2
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => ? = 0
{{1},{2,3,4,5}}
=> {{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => ? = 1
{{1},{2,3,4},{5}}
=> {{1,3,4},{2},{5}}
=> [3,2,4,1,5] => [3,2,4,1,5] => ? = 1
{{1},{2,3,5},{4}}
=> {{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => ? = 1
{{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> [3,2,5,4,1] => [3,2,5,4,1] => ? = 3
{{1},{2,3},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [3,2,1,4,5] => [3,2,1,4,5] => ? = 1
{{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [1,4,3,5,2] => ? = 1
{{1},{2,4},{3,5}}
=> {{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [1,4,3,2,5] => ? = 1
{{1},{2,5},{3,4}}
=> {{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => ? = 2
{{1},{2},{3,4,5}}
=> {{1,4,5},{2},{3}}
=> [4,2,3,5,1] => [4,2,3,5,1] => ? = 2
{{1},{2},{3,4},{5}}
=> {{1,4},{2},{3},{5}}
=> [4,2,3,1,5] => [4,2,3,1,5] => ? = 2
{{1},{2,5},{3},{4}}
=> {{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [1,2,5,4,3] => ? = 1
{{1},{2},{3,5},{4}}
=> {{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [1,5,3,4,2] => ? = 2
{{1},{2},{3},{4,5}}
=> {{1,5},{2},{3},{4}}
=> [5,2,3,4,1] => [5,2,3,4,1] => ? = 3
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 0
Description
The number of edges in the reduced word graph of a signed permutation.
The reduced word graph of a signed permutation $\pi$ has the reduced words of $\pi$ as vertices and an edge between two reduced words if they differ by exactly one braid move.
Matching statistic: St000102
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000102: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1] => [[1]]
=> [[1]]
=> 0
{{1,2}}
=> [2,1] => [[0,1],[1,0]]
=> [[1,2],[2]]
=> 0
{{1},{2}}
=> [1,2] => [[1,0],[0,1]]
=> [[1,1],[2]]
=> 0
{{1,2,3}}
=> [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
=> [[1,1,3],[2,3],[3]]
=> ? = 0
{{1,2},{3}}
=> [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
=> [[1,1,2],[2,2],[3]]
=> ? = 0
{{1,3},{2}}
=> [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
=> [[1,2,3],[2,3],[3]]
=> ? = 0
{{1},{2,3}}
=> [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
=> [[1,1,1],[2,3],[3]]
=> ? = 1
{{1},{2},{3}}
=> [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
=> [[1,1,1],[2,2],[3]]
=> ? = 0
{{1,2,3,4}}
=> [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,4],[2,2,4],[3,4],[4]]
=> ? = 0
{{1,2,3},{4}}
=> [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,3],[2,2,3],[3,3],[4]]
=> ? = 0
{{1,2,4},{3}}
=> [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,4],[2,3,4],[3,4],[4]]
=> ? = 0
{{1,2},{3,4}}
=> [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,2],[2,2,2],[3,4],[4]]
=> ? = 1
{{1,2},{3},{4}}
=> [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,2],[2,2,2],[3,3],[4]]
=> ? = 0
{{1,3,4},{2}}
=> [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
=> [[1,1,2,4],[2,2,4],[3,4],[4]]
=> ? = 0
{{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 0
{{1,3},{2},{4}}
=> [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [[1,1,2,3],[2,2,3],[3,3],[4]]
=> ? = 0
{{1,4},{2,3}}
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ? = 0
{{1},{2,3,4}}
=> [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> [[1,1,1,1],[2,2,4],[3,4],[4]]
=> ? = 1
{{1},{2,3},{4}}
=> [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,3],[3,3],[4]]
=> ? = 1
{{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [[1,1,1,1],[2,3,4],[3,4],[4]]
=> ? = 1
{{1},{2},{3,4}}
=> [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [[1,1,1,1],[2,2,2],[3,4],[4]]
=> ? = 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [[1,1,1,1],[2,2,2],[3,3],[4]]
=> ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ? = 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ? = 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
=> ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ? = 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ? = 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => [[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]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ? = 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
=> ? = 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,1],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ? = 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
=> ? = 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
=> ? = 0
Description
The charge of a semistandard tableau.
Matching statistic: St001964
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 25%
Values
{{1}}
=> [1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
{{1,2}}
=> [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1},{2}}
=> [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
{{1,2,3}}
=> [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0
{{1,2},{3}}
=> [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 0
{{1,3},{2}}
=> [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 0
{{1,2,3,4}}
=> [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1,2,4},{3}}
=> [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 0
{{1,3,4},{2}}
=> [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 0
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ? = 0
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 0
{{1,4},{2,3}}
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
=> ? = 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
=> ? = 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
=> ? = 2
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
=> ? = 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
=> ? = 1
{{1},{2,3,4},{5}}
=> [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
=> ? = 1
{{1},{2,3,5},{4}}
=> [1,3,5,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
=> ? = 3
{{1},{2,3},{4},{5}}
=> [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 1
{{1},{2,4,5},{3}}
=> [1,4,3,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 1
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
=> ? = 2
{{1},{2,4},{3},{5}}
=> [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,10),(4,9),(5,9),(5,10),(7,6),(8,6),(9,11),(10,11),(11,7),(11,8)],12)
=> ? = 1
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
=> ? = 2
{{1},{2},{3,4,5}}
=> [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
=> ? = 2
{{1},{2},{3,4},{5}}
=> [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 2
{{1},{2,5},{3},{4}}
=> [1,5,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 1
{{1},{2},{3,5},{4}}
=> [1,2,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
=> ? = 2
{{1},{2},{3},{4,5}}
=> [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
=> ? = 3
{{1},{2},{3},{4},{5}}
=> [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
=> ? = 0
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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