Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000923
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
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]].
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
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 $K[x]/(x^n)$.
Matching statistic: St000744
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00033: Dyck paths to two-row standard tableauStandard 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 typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00199: Dyck paths prime Dyck pathDyck 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 typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect 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 $\epsilon$ This statistic is counting the number of cycles of the permutation $\gamma \circ \epsilon$ where $\gamma$ is the long cycle $(1,2,3,\ldots,2n)$. '''Example''' The perfect matching $[(1,3),(2,4)]$ corresponds to the permutation in $S_4$ with disjoint cycle decomposition $(1,3)(2,4)$. Then the permutation $(1,2,3,4)\circ (1,3)(2,4) = (1,4,3,2)$ has only one cycle. Let $\epsilon_v(n)$ 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 $\epsilon_v(n)$. $$ \sum_{v=1}^{n+1} \epsilon_{v}(n) N^v = (2n-1)!! \sum_{k\geq 0}^n \binom{N}{k+1}\binom{n}{k}2^k. $$
Matching statistic: St000782
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect 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 typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
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 typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary 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 $$ 0110 < 1010 < 1101 < 1110 < 1111. $$