searching the database
Your data matches 78 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000708
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000708: 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
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000708: 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]
=> [2]
=> 2
[[1,1],[2,2]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[4]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[2,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[2,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1,1],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1,1,2],[2,2]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[3],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[2],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[3],[4],[5]]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[2,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,3],[3,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,3],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 2
[[1,1,1],[2,3]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[[1,1,1],[3,3]]
=> [3,2]
=> [2]
=> [1,1]
=> 1
Description
The product of the parts of an integer partition.
Matching statistic: St001864
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001864: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 3
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1
Description
The number of excedances of a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, this is $\lvert\{i\in[n] \mid \pi(i) > i\}\rvert$.
Matching statistic: St001904
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00239: Permutations —Corteel⟶ Permutations
Mp00305: Permutations —parking function⟶ Parking functions
St001904: Parking functions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1],[2,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,3,2,1] => [4,3,2,1] => 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,2,1] => [3,4,2,1] => 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,1,2] => [3,4,1,2] => 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,5,2,3,1] => [4,5,2,3,1] => ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,5,2,1,4] => [3,5,2,1,4] => ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [3,4,2,1,5] => ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,4,2,1] => [3,5,4,2,1] => ? = 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,5,3,2,1] => [4,5,3,2,1] => ? = 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,5,4,3,1] => [2,5,4,3,1] => ? = 3
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,5,2,3,4,1] => [6,5,2,3,4,1] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,4,2,3,1,6] => [5,4,2,3,1,6] => ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,2,1,5,6] => [4,3,2,1,5,6] => ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 2
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 1
Description
The length of the initial strictly increasing segment of a parking function.
Matching statistic: St001862
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001862: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => [3,2,4,1] => 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,5,2,3] => [4,1,5,2,3] => ? = 2 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,1,5,2,4] => [3,1,5,2,4] => ? = 2 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 2 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,3,1,5,2] => [4,3,1,5,2] => ? = 3 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,3,1,4,2] => [5,3,1,4,2] => ? = 3 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,5,1,4,3] => [2,5,1,4,3] => ? = 3 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,5,2,3,4] => [6,1,5,2,3,4] => ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,1,4,2,3,6] => [5,1,4,2,3,6] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,1,3,2,5,6] => [4,1,3,2,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,5,2,4,3] => [6,1,5,2,4,3] => ? = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 1 - 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,1,4,2,3] => [5,1,4,2,3] => ? = 1 - 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 1 - 1
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: St001882
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,2,1,3] => [4,2,1,3] => 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [4,3,1,2] => [4,3,1,2] => 0 = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [3,4,1,2] => [3,4,1,2] => 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,4,2,1] => [3,4,2,1] => 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => ? = 2 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [3,5,1,2,4] => [3,5,1,2,4] => ? = 2 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,1,2,5] => [3,4,1,2,5] => ? = 2 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,1,3,4] => [2,5,1,3,4] => ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [3,5,4,1,2] => [3,5,4,1,2] => ? = 3 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [3,4,5,1,2] => [3,4,5,1,2] => ? = 3 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,5,1,3] => [2,4,5,1,3] => ? = 3 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,5,1,2,3,4] => [6,5,1,2,3,4] => ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,4,1,2,3,6] => [5,4,1,2,3,6] => ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [4,3,1,2,5,6] => [4,3,1,2,5,6] => ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [5,4,6,1,2,3] => [5,4,6,1,2,3] => ? = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 1 = 2 - 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,2],[2,4]]
=> [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,3],[2,4]]
=> [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,2],[4,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,4],[2,4]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[3,4]]
=> [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
[[1,1,4],[3,3]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,3],[4,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,1,4],[3,4]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,1,4],[4,4]]
=> [3,4,1,2,5] => [4,3,1,2,5] => [4,3,1,2,5] => ? = 1 - 1
[[1,2,2],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,2],[3,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,3],[2,4]]
=> [2,5,1,3,4] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 1 - 1
[[1,2,4],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,2],[4,4]]
=> [4,5,1,2,3] => [5,4,1,2,3] => [5,4,1,2,3] => ? = 1 - 1
[[1,2,4],[2,4]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 - 1
[[1,2,3],[3,4]]
=> [3,5,1,2,4] => [5,3,1,2,4] => [5,3,1,2,4] => ? = 1 - 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation.
For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St000633
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000633: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,4],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
Description
The size of the automorphism group of a poset.
A poset automorphism is a permutation of the elements of the poset preserving the order relation.
Matching statistic: St001399
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001399: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,1],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,4],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1
[[1,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[1,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
[[2,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 1
Description
The distinguishing number of a poset.
This is the minimal number of colours needed to colour the vertices of a poset, such that only the trivial automorphism of the poset preserves the colouring.
See also [[St000469]], which is the same concept for graphs.
Matching statistic: St000850
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000850: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000850: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 1 = 2 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 1
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1 - 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 1 - 1
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 1 - 1
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1 - 1
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,3],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,4],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
Description
The number of 1/2-balanced pairs in a poset.
A pair of elements $x,y$ of a poset is $\alpha$-balanced if the proportion of linear extensions where $x$ comes before $y$ is between $\alpha$ and $1-\alpha$.
Kislitsyn [1] conjectured that every poset which is not a chain has a $1/3$-balanced pair. Brightwell, Felsner and Trotter [2] show that at least a $(1-\sqrt 5)/10$-balanced pair exists in posets which are not chains.
Olson and Sagan [3] exhibit various posets that have a $1/2$-balanced pair.
Matching statistic: St001534
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001534: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001534: Posets ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,2],[2,3]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,2],[2],[3]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,2],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,3],[2,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[2,3],[3,4]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,2],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[2,3],[3],[4]]
=> [4,2,1,3] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 0 = 2 - 2
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,2],[2,3]]
=> [3,5,1,2,4] => [3,1,5,4,2] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 1 - 2
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,2,2],[2,3]]
=> [2,5,1,3,4] => [5,4,3,1,2] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 - 2
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 2
[[1,1,2],[2],[3]]
=> [5,3,1,2,4] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 2
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 2
[[1,2,2],[2],[3]]
=> [5,2,1,3,4] => [2,5,4,3,1] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 2
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 2
[[1,3,3],[2],[3]]
=> [3,2,1,4,5] => [2,3,1,4,5] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 2 - 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 3 - 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 3 - 2
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,6),(2,9),(2,11),(3,6),(3,9),(3,10),(4,7),(4,9),(4,10),(4,11),(5,7),(5,9),(5,10),(5,11),(6,13),(7,12),(7,13),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,8),(13,8)],14)
=> ? = 3 - 2
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,3,1,6,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1 - 2
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,3,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
=> ? = 1 - 2
[[1,1,2,2],[2,2]]
=> [3,4,1,2,5,6] => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,14),(1,18),(2,13),(2,14),(2,18),(3,12),(3,14),(3,18),(4,11),(4,14),(4,18),(5,8),(5,9),(5,10),(5,16),(6,5),(6,11),(6,12),(6,13),(6,18),(8,15),(8,19),(9,15),(9,19),(10,15),(10,19),(11,8),(11,16),(11,17),(12,9),(12,16),(12,17),(13,10),(13,16),(13,17),(14,17),(15,7),(16,15),(16,19),(17,19),(18,16),(18,17),(19,7)],20)
=> ? = 1 - 2
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
=> ? = 1 - 2
[[1],[2],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[3],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[2],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[3],[4],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[3],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[4],[5],[6]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 2 - 2
[[1,1],[2,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,1],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,2],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[3,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,3],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[4,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,4],[2,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,2],[5,5]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 2
[[1,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[1,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[2,3],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[2,4],[3,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
[[2,4],[4,5]]
=> [2,4,1,3] => [4,3,1,2] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> -1 = 1 - 2
Description
The alternating sum of the coefficients of the Poincare polynomial of the poset cone.
For a poset $P$ on $\{1,\dots,n\}$, let $\mathcal K_P = \{\vec x\in\mathbb R^n| x_i < x_j \text{ for } i < _P j\}$. Furthermore let $\mathcal L(\mathcal A)$ be the intersection lattice of the braid arrangement $A_{n-1}$ and let $\mathcal L^{int} = \{ X \in \mathcal L(\mathcal A) | X \cap \mathcal K_P \neq \emptyset \}$.
Then the Poincare polynomial of the poset cone is $Poin(t) = \sum_{X\in\mathcal L^{int}} |\mu(0, X)| t^{codim X}$.
This statistic records its $Poin(-1)$.
Matching statistic: St001491
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 33%
Values
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => 2
[[1,1],[2,2]]
=> [2,2]
=> 1100 => 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => 2
[[1],[3],[4]]
=> [1,1,1]
=> 1110 => 2
[[2],[3],[4]]
=> [1,1,1]
=> 1110 => 2
[[1,1],[2,3]]
=> [2,2]
=> 1100 => 1
[[1,1],[3,3]]
=> [2,2]
=> 1100 => 1
[[1,2],[2,3]]
=> [2,2]
=> 1100 => 1
[[1,2],[3,3]]
=> [2,2]
=> 1100 => 1
[[2,2],[3,3]]
=> [2,2]
=> 1100 => 1
[[1,1],[2],[3]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[2],[3]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2],[3]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,1,1],[2,2]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,2],[2,2]]
=> [3,2]
=> 10100 => ? = 1
[[1],[2],[5]]
=> [1,1,1]
=> 1110 => 2
[[1],[3],[5]]
=> [1,1,1]
=> 1110 => 2
[[1],[4],[5]]
=> [1,1,1]
=> 1110 => 2
[[2],[3],[5]]
=> [1,1,1]
=> 1110 => 2
[[2],[4],[5]]
=> [1,1,1]
=> 1110 => 2
[[3],[4],[5]]
=> [1,1,1]
=> 1110 => 2
[[1,1],[2,4]]
=> [2,2]
=> 1100 => 1
[[1,1],[3,4]]
=> [2,2]
=> 1100 => 1
[[1,1],[4,4]]
=> [2,2]
=> 1100 => 1
[[1,2],[2,4]]
=> [2,2]
=> 1100 => 1
[[1,2],[3,4]]
=> [2,2]
=> 1100 => 1
[[1,3],[2,4]]
=> [2,2]
=> 1100 => 1
[[1,2],[4,4]]
=> [2,2]
=> 1100 => 1
[[1,3],[3,4]]
=> [2,2]
=> 1100 => 1
[[1,3],[4,4]]
=> [2,2]
=> 1100 => 1
[[2,2],[3,4]]
=> [2,2]
=> 1100 => 1
[[2,2],[4,4]]
=> [2,2]
=> 1100 => 1
[[2,3],[3,4]]
=> [2,2]
=> 1100 => 1
[[2,3],[4,4]]
=> [2,2]
=> 1100 => 1
[[3,3],[4,4]]
=> [2,2]
=> 1100 => 1
[[1,1],[2],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,1],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[2],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4],[2],[3]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4],[2],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,4],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[2,2],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[2,3],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[2,4],[3],[4]]
=> [2,1,1]
=> 10110 => ? = 2
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> 11110 => ? = 2
[[1,1,1],[2,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,1],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,2],[2,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,3],[2,2]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,2],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,3],[2,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,3],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,2,2],[2,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,2,2],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,2,3],[2,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,2,3],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[2,2,2],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[2,2,3],[3,3]]
=> [3,2]
=> 10100 => ? = 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,1,2],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,1,3],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,2,2],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,2,3],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,3,3],[2],[3]]
=> [3,1,1]
=> 100110 => ? = 2
[[1,1],[2,2],[3]]
=> [2,2,1]
=> 11010 => ? = 3
[[1,1],[2,3],[3]]
=> [2,2,1]
=> 11010 => ? = 3
[[1,2],[2,3],[3]]
=> [2,2,1]
=> 11010 => ? = 3
[[1,1,1,1],[2,2]]
=> [4,2]
=> 100100 => ? = 1
[[1,1,1,2],[2,2]]
=> [4,2]
=> 100100 => ? = 1
[[1,1,2,2],[2,2]]
=> [4,2]
=> 100100 => ? = 1
[[1,1,1],[2,2,2]]
=> [3,3]
=> 11000 => ? = 1
[[1],[2],[6]]
=> [1,1,1]
=> 1110 => 2
[[1],[3],[6]]
=> [1,1,1]
=> 1110 => 2
[[1],[4],[6]]
=> [1,1,1]
=> 1110 => 2
[[1],[5],[6]]
=> [1,1,1]
=> 1110 => 2
[[2],[3],[6]]
=> [1,1,1]
=> 1110 => 2
[[2],[4],[6]]
=> [1,1,1]
=> 1110 => 2
[[2],[5],[6]]
=> [1,1,1]
=> 1110 => 2
[[3],[4],[6]]
=> [1,1,1]
=> 1110 => 2
[[3],[5],[6]]
=> [1,1,1]
=> 1110 => 2
[[4],[5],[6]]
=> [1,1,1]
=> 1110 => 2
[[1,1],[2,5]]
=> [2,2]
=> 1100 => 1
[[1,1],[3,5]]
=> [2,2]
=> 1100 => 1
[[1,1],[4,5]]
=> [2,2]
=> 1100 => 1
[[1,1],[5,5]]
=> [2,2]
=> 1100 => 1
[[1,2],[2,5]]
=> [2,2]
=> 1100 => 1
[[1,2],[3,5]]
=> [2,2]
=> 1100 => 1
[[1,3],[2,5]]
=> [2,2]
=> 1100 => 1
[[1,2],[4,5]]
=> [2,2]
=> 1100 => 1
[[1,4],[2,5]]
=> [2,2]
=> 1100 => 1
[[1,2],[5,5]]
=> [2,2]
=> 1100 => 1
[[1,1],[2],[5]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,1],[3],[5]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,1],[4],[5]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[2],[5]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,2],[3],[5]]
=> [2,1,1]
=> 10110 => ? = 2
[[1,3],[2],[5]]
=> [2,1,1]
=> 10110 => ? = 2
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
The following 68 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St001754The number of tolerances of a finite lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ 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. 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. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000080The rank of the poset. 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. St000782The indicator function of whether a given perfect matching is an L & P matching. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. 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. St001625The Möbius invariant of a lattice. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000094The depth of an ordered tree. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. 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. St001645The pebbling number of a connected graph. St001713The difference of the first and last value in the first row of the Gelfand-Tsetlin pattern. St001877Number of indecomposable injective modules with projective dimension 2. St001964The interval resolution global dimension of a poset. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. 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 $K[x]/(x^n)$. St000189The number of elements in the poset. 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. St000529The number of permutations whose descent word is the given binary word. St000180The number of chains 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 $K[x]/(x^n)$. St000100The number of linear extensions of a poset. St001909The number of interval-closed sets of a poset. St000410The tree factorial of an ordered tree. St000634The number of endomorphisms of a poset. St000454The largest eigenvalue of a graph if it is integral. St000422The energy of a graph, if it is integral.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!