searching the database
Your data matches 8 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: St000923
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000923: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 2
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 2
[-1,3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,-3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[2,-1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[-2,1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 2
[3,-2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 2
[-3,-2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,-2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,-2,4,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[-1,-2,-4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[-1,-2,-4,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,3,-2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
Description
The minimal number with no two order isomorphic substrings of this length in a permutation.
For example, the length 3 substrings of the permutation 12435 are 124, 243 and 435, whereas its length 2 substrings are 12, 24, 43 and 35.
No two sequences among 124, 243 and 435 are order isomorphic, but 12 and 24 are, so the statistic on 12435 is 3.
This is inspired by [[St000922]].
Matching statistic: St001207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 4
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,-1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,-1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,-1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,-1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,-4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[3,-4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,-4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-3,-4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,-1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,-1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,-1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,-1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,-3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[4,-3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,-3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[-4,-3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => ? = 4
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => ? = 4
Description
The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn).
Matching statistic: St000744
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St000744: Standard tableaux ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 4 = 3 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 4 = 3 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 4 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 3 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 4 = 3 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 3 = 2 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 3 = 2 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 3 = 2 + 1
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,-1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,-1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,-1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,-1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,-4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[3,-4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,-4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-3,-4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,-1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,-1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,-1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,-1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,-3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[4,-3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,-3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[-4,-3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[1,2,3,4,9],[5,6,7,8,10]]
=> ? = 4 + 1
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 4 + 1
Description
The length of the path to the largest entry in a standard Young tableau.
Matching statistic: St001515
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 3 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,-1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,-1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,-1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,-1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,-4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[3,-4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,-4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-3,-4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,-1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,-1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,-1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,-1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,-3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[4,-3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,-3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[-4,-3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> ? = 4 + 1
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000044
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000044: Perfect matchings ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000044: Perfect matchings ⟶ ℤResult quality: 33% ●values known / values provided: 59%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 5 = 3 + 2
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> 5 = 3 + 2
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 5 = 3 + 2
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 5 = 3 + 2
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 5 = 3 + 2
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> 5 = 3 + 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 4 + 2
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 5 = 3 + 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> 5 = 3 + 2
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 4 = 2 + 2
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 4 = 2 + 2
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 4 = 2 + 2
[2,3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,-3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,-3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,3,4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,3,-4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,-3,4,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,-3,-4,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,-4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[2,-4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,4,1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,4,-1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,-4,1,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-2,-4,-1,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,-1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,-1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,1,4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,1,-4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,-1,4,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,-1,-4,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,-4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[3,-4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,4,2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,4,-2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,-4,2,-1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-3,-4,-2,1] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,-1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,-1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,1,2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,1,-2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,-1,2,-3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,-1,-2,3] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,-3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[4,-3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,3,1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,3,-1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,-3,1,-2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[-4,-3,-1,2] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> ? = 4 + 2
[1,-2,-3,-4,-5] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 4 + 2
Description
The number of vertices of the unicellular map given by a perfect matching.
If the perfect matching of 2n elements is viewed as a fixed point-free involution ϵ This statistic is counting the number of cycles of the permutation γ∘ϵ where γ is the long cycle (1,2,3,…,2n).
'''Example'''
The perfect matching [(1,3),(2,4)] corresponds to the permutation in S4 with disjoint cycle decomposition (1,3)(2,4). Then the permutation (1,2,3,4)∘(1,3)(2,4)=(1,4,3,2) has only one cycle.
Let ϵv(n) is the number of matchings of 2n such that yield v cycles in the process described above. Harer and Zagier [1] gave the following expression for the generating series of the numbers ϵv(n).
\sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k.
Matching statistic: St000782
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 4 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 3 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 3 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 3 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 3 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 3 - 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin ([1,3],[2,4]), insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}
Matching statistic: St001583
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 4 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 3 + 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001722
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 17% ●values known / values provided: 26%●distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 4 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 3 - 1
Description
The number of minimal chains with small intervals between a binary word and the top element.
A valley in a binary word is a subsequence 01, or a trailing 0. A peak is a subsequence 10 or a trailing 1. Let P be the lattice on binary words of length n, where the covering elements of a word are obtained by replacing a valley with a peak. An interval [w_1, w_2] in P is small if w_2 is obtained from w_1 by replacing some valleys with peaks.
This statistic counts the number of chains w = w_1 < \dots < w_d = 1\dots 1 to the top element of minimal length.
For example, there are two such chains for the word 0110:
0110 < 1011 < 1101 < 1110 < 1111
and
0110 < 1010 < 1101 < 1110 < 1111.
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!