Your data matches 168 different statistics following compositions of up to 3 maps.
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Matching statistic: St001091
St001091: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 1
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 2
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 1
[1,1,1,1]
=> 3
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 0
[3,1,1]
=> 1
[2,2,1]
=> 1
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 4
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 0
[4,1,1]
=> 1
[3,3]
=> 1
[3,2,1]
=> 0
[3,1,1,1]
=> 2
[2,2,2]
=> 2
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 3
[1,1,1,1,1,1]
=> 5
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 0
[5,1,1]
=> 1
[4,3]
=> 0
[4,2,1]
=> 0
[4,1,1,1]
=> 2
[3,3,1]
=> 1
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 3
[2,1,1,1,1,1]
=> 4
[1,1,1,1,1,1,1]
=> 6
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 0
[6,1,1]
=> 1
[5,3]
=> 0
[5,2,1]
=> 0
Description
The number of parts in an integer partition whose next smaller part has the same size. In other words, this is the number of distinct parts subtracted from the number of all parts.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
St000052: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 7
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> 4
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
Description
The number of valleys of a Dyck path not on the x-axis. That is, the number of valleys of nonminimal height. This corresponds to the number of -1's in an inclusion of Dyck paths into alternating sign matrices.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00031: Dyck paths to 312-avoiding permutationPermutations
St000356: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,2] => 0
[2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,5,4,3,2,1,7] => 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => 5
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,6,5,4,3,2,1,8] => 0
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,6,4,3,2,1,7] => 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,3,5,2,1,6] => 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,5,4,2,1,6] => 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => 3
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => 4
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => 6
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,7,6,5,4,3,2,1,9] => 0
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,7,5,4,3,2,1,8] => 0
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,4,6,3,2,1,7] => 0
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,6,5,3,2,1,7] => 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,3,2,5,1,6] => 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,4,5,2,1,6] => 0
Description
The number of occurrences of the pattern 13-2. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St001777
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00207: Standard tableaux horizontal strip sizesInteger compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> [[1]]
=> [1] => 0
[2]
=> [[1,2]]
=> [2] => 0
[1,1]
=> [[1],[2]]
=> [1,1] => 1
[3]
=> [[1,2,3]]
=> [3] => 0
[2,1]
=> [[1,3],[2]]
=> [1,2] => 0
[1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[4]
=> [[1,2,3,4]]
=> [4] => 0
[3,1]
=> [[1,3,4],[2]]
=> [1,3] => 0
[2,2]
=> [[1,2],[3,4]]
=> [2,2] => 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 3
[5]
=> [[1,2,3,4,5]]
=> [5] => 0
[4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 0
[3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 0
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 2
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 4
[6]
=> [[1,2,3,4,5,6]]
=> [6] => 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 0
[4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => 0
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 0
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 2
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 3
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 5
[7]
=> [[1,2,3,4,5,6,7]]
=> [7] => 0
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [1,6] => 0
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [2,5] => 0
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [1,1,5] => 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [3,4] => 0
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [1,2,4] => 0
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [1,1,1,4] => 2
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [1,3,3] => 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [2,2,3] => 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [1,1,2,3] => 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [1,1,1,1,3] => 3
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [1,2,2,2] => 2
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [1,1,1,2,2] => 3
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,2] => 4
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1] => 6
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [8] => 0
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [1,7] => 0
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [2,6] => 0
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [1,1,6] => 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [3,5] => 0
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [1,2,5] => 0
Description
The number of weak descents in an integer composition. A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St000160
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 1 = 0 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1 = 0 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 0 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1 = 0 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 1 = 0 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 4 = 3 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1 = 0 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 1 = 0 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 1 = 0 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1 = 0 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 1 = 0 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1 = 0 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2 = 1 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1 = 0 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 1 = 0 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 1 = 0 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2 = 1 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 1 = 0 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 7 = 6 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1 = 0 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 1 = 0 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 1 = 0 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2 = 1 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 1 = 0 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 1 = 0 + 1
Description
The multiplicity of the smallest part of a partition. This counts the number of occurrences of the smallest part $spt(\lambda)$ of a partition $\lambda$. The sum $spt(n) = \sum_{\lambda \vdash n} spt(\lambda)$ satisfies the congruences \begin{align*} spt(5n+4) &\equiv 0\quad \pmod{5}\\\ spt(7n+5) &\equiv 0\quad \pmod{7}\\\ spt(13n+6) &\equiv 0\quad \pmod{13}, \end{align*} analogous to those of the counting function of partitions, see [1] and [2].
Matching statistic: St000475
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 1 = 0 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 2 = 1 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 1 = 0 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 0 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 3 = 2 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 1 = 0 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 1 = 0 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 2 = 1 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 4 = 3 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 1 = 0 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 1 = 0 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 1 = 0 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 2 = 1 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 5 = 4 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 1 = 0 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 1 = 0 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 1 = 0 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 2 = 1 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 2 = 1 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 3 = 2 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 3 = 2 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 6 = 5 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 1 = 0 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 1 = 0 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 1 = 0 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 2 = 1 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 1 = 0 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 1 = 0 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 2 = 1 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 4 = 3 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 3 = 2 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 5 = 4 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 7 = 6 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 1 = 0 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 1 = 0 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 1 = 0 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 2 = 1 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 1 = 0 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 1 = 0 + 1
Description
The number of parts equal to 1 in a partition.
Mp00095: Integer partitions to binary wordBinary words
Mp00136: Binary words rotate back-to-frontBinary words
Mp00224: Binary words runsortBinary words
St000877: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1]
=> 10 => 01 => 01 => 1 = 0 + 1
[2]
=> 100 => 010 => 001 => 2 = 1 + 1
[1,1]
=> 110 => 011 => 011 => 1 = 0 + 1
[3]
=> 1000 => 0100 => 0001 => 3 = 2 + 1
[2,1]
=> 1010 => 0101 => 0101 => 1 = 0 + 1
[1,1,1]
=> 1110 => 0111 => 0111 => 1 = 0 + 1
[4]
=> 10000 => 01000 => 00001 => 4 = 3 + 1
[3,1]
=> 10010 => 01001 => 00101 => 2 = 1 + 1
[2,2]
=> 1100 => 0110 => 0011 => 2 = 1 + 1
[2,1,1]
=> 10110 => 01011 => 01011 => 1 = 0 + 1
[1,1,1,1]
=> 11110 => 01111 => 01111 => 1 = 0 + 1
[5]
=> 100000 => 010000 => 000001 => 5 = 4 + 1
[4,1]
=> 100010 => 010001 => 000101 => 3 = 2 + 1
[3,2]
=> 10100 => 01010 => 00101 => 2 = 1 + 1
[3,1,1]
=> 100110 => 010011 => 001101 => 2 = 1 + 1
[2,2,1]
=> 11010 => 01101 => 01011 => 1 = 0 + 1
[2,1,1,1]
=> 101110 => 010111 => 010111 => 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => 011111 => 011111 => 1 = 0 + 1
[6]
=> 1000000 => 0100000 => 0000001 => 6 = 5 + 1
[5,1]
=> 1000010 => 0100001 => 0000101 => 4 = 3 + 1
[4,2]
=> 100100 => 010010 => 000101 => 3 = 2 + 1
[4,1,1]
=> 1000110 => 0100011 => 0001101 => 3 = 2 + 1
[3,3]
=> 11000 => 01100 => 00011 => 3 = 2 + 1
[3,2,1]
=> 101010 => 010101 => 010101 => 1 = 0 + 1
[3,1,1,1]
=> 1001110 => 0100111 => 0011101 => 2 = 1 + 1
[2,2,2]
=> 11100 => 01110 => 00111 => 2 = 1 + 1
[2,2,1,1]
=> 110110 => 011011 => 011011 => 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => 0101111 => 0101111 => 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => 0111111 => 0111111 => 1 = 0 + 1
[7]
=> 10000000 => 01000000 => 00000001 => 7 = 6 + 1
[6,1]
=> 10000010 => 01000001 => 00000101 => 5 = 4 + 1
[5,2]
=> 1000100 => 0100010 => 0000101 => 4 = 3 + 1
[5,1,1]
=> 10000110 => 01000011 => 00001101 => 4 = 3 + 1
[4,3]
=> 101000 => 010100 => 000101 => 3 = 2 + 1
[4,2,1]
=> 1001010 => 0100101 => 0010101 => 2 = 1 + 1
[4,1,1,1]
=> 10001110 => 01000111 => 00011101 => 3 = 2 + 1
[3,3,1]
=> 110010 => 011001 => 001011 => 2 = 1 + 1
[3,2,2]
=> 101100 => 010110 => 001011 => 2 = 1 + 1
[3,2,1,1]
=> 1010110 => 0101011 => 0101011 => 1 = 0 + 1
[3,1,1,1,1]
=> 10011110 => 01001111 => 00111101 => 2 = 1 + 1
[2,2,2,1]
=> 111010 => 011101 => 010111 => 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => 0110111 => 0110111 => 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => 01011111 => 01011111 => 1 = 0 + 1
[1,1,1,1,1,1,1]
=> 11111110 => 01111111 => 01111111 => 1 = 0 + 1
[8]
=> 100000000 => 010000000 => 000000001 => 8 = 7 + 1
[7,1]
=> 100000010 => 010000001 => 000000101 => 6 = 5 + 1
[6,2]
=> 10000100 => 01000010 => 00000101 => 5 = 4 + 1
[6,1,1]
=> 100000110 => 010000011 => 000001101 => 5 = 4 + 1
[5,3]
=> 1001000 => 0100100 => 0000101 => 4 = 3 + 1
[5,2,1]
=> 10001010 => 01000101 => 00010101 => 3 = 2 + 1
Description
The depth of the binary word interpreted as a path. This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2]. The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000011
Mp00230: Integer partitions parallelogram polyominoDyck paths
Mp00227: Dyck paths Delest-Viennot-inverseDyck paths
Mp00030: Dyck paths zeta mapDyck paths
St000011: Dyck paths ⟶ ℤResult quality: 84% values known / values provided: 84%distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 1 = 0 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3 = 2 + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2 = 1 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 4 + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 4 = 3 + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 3 = 2 + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 1 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 1 + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1 = 0 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 6 = 5 + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 5 = 4 + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 5 = 4 + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 4 = 3 + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,2,2,3,3,4,5} + 1
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,2,2,3,3,3,4,4,4,5,5,6,6} + 1
Description
The number of touch points (or returns) of a Dyck path. This is the number of points, excluding the origin, where the Dyck path has height 0.
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00035: Dyck paths to alternating sign matrixAlternating sign matrices
St000065: Alternating sign matrices ⟶ ℤResult quality: 83% values known / values provided: 83%distinct values known / distinct values provided: 90%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 0
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 0
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 3
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
=> 0
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 0
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 4
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 0
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 0
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 5
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 0
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1],[0,0,0,0,0,1,0]]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> 6
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,-1,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [[0,0,1,0,0],[1,0,-1,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
=> 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0],[1,-1,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 0
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 0
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0],[1,0,0,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1],[0,0,0,0,0,0,1,0]]
=> 0
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> 7
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> 5
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0],[1,0,-1,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,-1,1,0,0],[0,0,0,1,-1,1,0],[0,0,0,0,1,-1,1],[0,0,0,0,0,1,0]]
=> 4
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,-1,1,0,0],[0,1,0,-1,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [[0,0,1,0,0,0],[1,0,0,0,0,0],[0,1,-1,0,1,0],[0,0,1,0,0,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,0,1,0,0,0],[0,1,0,0,0,0,0],[0,0,1,0,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,1}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,1}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,1,2,5}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,1,2,5}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,-1,1,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0,0,0],[0,0,0,0,1,-1,1,0,0,0],[0,0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0,0],[1,0,-1,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,-1,1,0,0,0,0],[0,0,0,1,-1,1,0,0,0],[0,0,0,0,1,-1,1,0,0],[0,0,0,0,0,1,-1,1,0],[0,0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,-1,1,0,0,0,0],[0,1,0,-1,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,-1,1,0,0],[0,0,0,0,1,-1,1,0],[0,0,0,0,0,1,-1,1],[0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,-1,1,0,0,0],[0,0,1,-1,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,1,1,0,0,0,1,0]
=> [[0,1,0,0,0,0,0],[1,-1,1,0,0,0,0],[0,1,0,0,0,0,0],[0,0,0,0,0,1,0],[0,0,0,1,0,0,0],[0,0,0,0,1,0,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,-1,1,0,0,0,0,0],[0,1,-1,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [[1,0,0,0,0,0,0],[0,0,1,0,0,0,0],[0,1,-1,0,1,0,0],[0,0,1,0,0,0,0],[0,0,0,1,0,0,0],[0,0,0,0,0,1,0],[0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,1,0,0,0,0,0,0,0],[1,-1,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0],[0,0,1,0,0,0,0,0],[0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0],[0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0],[0,1,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0],[0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [[0,0,1,0,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,1,0],[0,0,0,0,0,0,0,0,0,1]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0],[0,0,0,1,0,0,0,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0],[0,0,0,0,0,1,0,0,0,0,0],[0,0,0,0,0,0,1,0,0,0,0],[0,0,0,0,0,0,0,1,0,0,0],[0,0,0,0,0,0,0,0,1,0,0],[0,0,0,0,0,0,0,0,0,0,1],[0,0,0,0,0,0,0,0,0,1,0]]
=> ? ∊ {0,0,0,0,0,1,1,1,2,3,4,5,5,6,6,7,9}
Description
The number of entries equal to -1 in an alternating sign matrix. The number of nonzero entries, [[St000890]] is twice this number plus the dimension of the matrix.
Matching statistic: St000319
Mp00095: Integer partitions to binary wordBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00040: Integer compositions to partitionInteger partitions
St000319: Integer partitions ⟶ ℤResult quality: 80% values known / values provided: 81%distinct values known / distinct values provided: 80%
Values
[1]
=> 10 => [1,2] => [2,1]
=> 1 = 0 + 1
[2]
=> 100 => [1,3] => [3,1]
=> 2 = 1 + 1
[1,1]
=> 110 => [1,1,2] => [2,1,1]
=> 1 = 0 + 1
[3]
=> 1000 => [1,4] => [4,1]
=> 3 = 2 + 1
[2,1]
=> 1010 => [1,2,2] => [2,2,1]
=> 1 = 0 + 1
[1,1,1]
=> 1110 => [1,1,1,2] => [2,1,1,1]
=> 1 = 0 + 1
[4]
=> 10000 => [1,5] => [5,1]
=> 4 = 3 + 1
[3,1]
=> 10010 => [1,3,2] => [3,2,1]
=> 2 = 1 + 1
[2,2]
=> 1100 => [1,1,3] => [3,1,1]
=> 2 = 1 + 1
[2,1,1]
=> 10110 => [1,2,1,2] => [2,2,1,1]
=> 1 = 0 + 1
[1,1,1,1]
=> 11110 => [1,1,1,1,2] => [2,1,1,1,1]
=> 1 = 0 + 1
[5]
=> 100000 => [1,6] => [6,1]
=> 5 = 4 + 1
[4,1]
=> 100010 => [1,4,2] => [4,2,1]
=> 3 = 2 + 1
[3,2]
=> 10100 => [1,2,3] => [3,2,1]
=> 2 = 1 + 1
[3,1,1]
=> 100110 => [1,3,1,2] => [3,2,1,1]
=> 2 = 1 + 1
[2,2,1]
=> 11010 => [1,1,2,2] => [2,2,1,1]
=> 1 = 0 + 1
[2,1,1,1]
=> 101110 => [1,2,1,1,2] => [2,2,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1]
=> 111110 => [1,1,1,1,1,2] => [2,1,1,1,1,1]
=> 1 = 0 + 1
[6]
=> 1000000 => [1,7] => [7,1]
=> 6 = 5 + 1
[5,1]
=> 1000010 => [1,5,2] => [5,2,1]
=> 4 = 3 + 1
[4,2]
=> 100100 => [1,3,3] => [3,3,1]
=> 3 = 2 + 1
[4,1,1]
=> 1000110 => [1,4,1,2] => [4,2,1,1]
=> 3 = 2 + 1
[3,3]
=> 11000 => [1,1,4] => [4,1,1]
=> 3 = 2 + 1
[3,2,1]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 1 = 0 + 1
[3,1,1,1]
=> 1001110 => [1,3,1,1,2] => [3,2,1,1,1]
=> 2 = 1 + 1
[2,2,2]
=> 11100 => [1,1,1,3] => [3,1,1,1]
=> 2 = 1 + 1
[2,2,1,1]
=> 110110 => [1,1,2,1,2] => [2,2,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1]
=> 1011110 => [1,2,1,1,1,2] => [2,2,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> 1111110 => [1,1,1,1,1,1,2] => [2,1,1,1,1,1,1]
=> 1 = 0 + 1
[7]
=> 10000000 => [1,8] => [8,1]
=> 7 = 6 + 1
[6,1]
=> 10000010 => [1,6,2] => [6,2,1]
=> 5 = 4 + 1
[5,2]
=> 1000100 => [1,4,3] => [4,3,1]
=> 4 = 3 + 1
[5,1,1]
=> 10000110 => [1,5,1,2] => [5,2,1,1]
=> 4 = 3 + 1
[4,3]
=> 101000 => [1,2,4] => [4,2,1]
=> 3 = 2 + 1
[4,2,1]
=> 1001010 => [1,3,2,2] => [3,2,2,1]
=> 2 = 1 + 1
[4,1,1,1]
=> 10001110 => [1,4,1,1,2] => [4,2,1,1,1]
=> 3 = 2 + 1
[3,3,1]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,2]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 2 = 1 + 1
[3,2,1,1]
=> 1010110 => [1,2,2,1,2] => [2,2,2,1,1]
=> 1 = 0 + 1
[3,1,1,1,1]
=> 10011110 => [1,3,1,1,1,2] => [3,2,1,1,1,1]
=> 2 = 1 + 1
[2,2,2,1]
=> 111010 => [1,1,1,2,2] => [2,2,1,1,1]
=> 1 = 0 + 1
[2,2,1,1,1]
=> 1101110 => [1,1,2,1,1,2] => [2,2,1,1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> 10111110 => [1,2,1,1,1,1,2] => [2,2,1,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> 11111110 => [1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1]
=> 1 = 0 + 1
[8]
=> 100000000 => [1,9] => [9,1]
=> 8 = 7 + 1
[7,1]
=> 100000010 => [1,7,2] => [7,2,1]
=> 6 = 5 + 1
[6,2]
=> 10000100 => [1,5,3] => [5,3,1]
=> 5 = 4 + 1
[6,1,1]
=> 100000110 => [1,6,1,2] => [6,2,1,1]
=> 5 = 4 + 1
[5,3]
=> 1001000 => [1,3,4] => [4,3,1]
=> 4 = 3 + 1
[5,2,1]
=> 10001010 => [1,4,2,2] => [4,2,2,1]
=> 3 = 2 + 1
[9]
=> 1000000000 => [1,10] => [10,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[8,1]
=> 1000000010 => [1,8,2] => [8,2,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[7,1,1]
=> 1000000110 => [1,7,1,2] => [7,2,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[6,1,1,1]
=> 1000001110 => [1,6,1,1,2] => [6,2,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[5,1,1,1,1]
=> 1000011110 => [1,5,1,1,1,2] => [5,2,1,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[4,1,1,1,1,1]
=> 1000111110 => [1,4,1,1,1,1,2] => [4,2,1,1,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[3,1,1,1,1,1,1]
=> 1001111110 => [1,3,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[2,1,1,1,1,1,1,1]
=> 1011111110 => [1,2,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[1,1,1,1,1,1,1,1,1]
=> 1111111110 => [1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,1,2,3,4,5,6,8} + 1
[10]
=> 10000000000 => [1,11] => [11,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[9,1]
=> 10000000010 => [1,9,2] => [9,2,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[8,2]
=> 1000000100 => [1,7,3] => [7,3,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[8,1,1]
=> 10000000110 => [1,8,1,2] => [8,2,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[7,2,1]
=> 1000001010 => [1,6,2,2] => [6,2,2,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[7,1,1,1]
=> 10000001110 => [1,7,1,1,2] => [7,2,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[6,2,1,1]
=> 1000010110 => [1,5,2,1,2] => [5,2,2,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[6,1,1,1,1]
=> 10000011110 => [1,6,1,1,1,2] => [6,2,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[5,2,1,1,1]
=> 1000101110 => [1,4,2,1,1,2] => [4,2,2,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[5,1,1,1,1,1]
=> 10000111110 => [1,5,1,1,1,1,2] => [5,2,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[4,2,1,1,1,1]
=> 1001011110 => [1,3,2,1,1,1,2] => [3,2,2,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[4,1,1,1,1,1,1]
=> 10001111110 => [1,4,1,1,1,1,1,2] => [4,2,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[3,2,1,1,1,1,1]
=> 1010111110 => [1,2,2,1,1,1,1,2] => [2,2,2,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[3,1,1,1,1,1,1,1]
=> 10011111110 => [1,3,1,1,1,1,1,1,2] => [3,2,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[2,2,1,1,1,1,1,1]
=> 1101111110 => [1,1,2,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[2,1,1,1,1,1,1,1,1]
=> 10111111110 => [1,2,1,1,1,1,1,1,1,2] => [2,2,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
[1,1,1,1,1,1,1,1,1,1]
=> 11111111110 => [1,1,1,1,1,1,1,1,1,1,2] => [2,1,1,1,1,1,1,1,1,1,1]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,4,4,5,6,6,7,9} + 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
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St000320The dinv adjustment of an integer partition. St000204The number of internal nodes of a binary tree. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000359The number of occurrences of the pattern 23-1. St000214The number of adjacencies of a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000028The number of stack-sorts needed to sort a permutation. St000711The number of big exceedences of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000358The number of occurrences of the pattern 31-2. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000039The number of crossings of a permutation. St000317The cycle descent number of a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000021The number of descents of a permutation. St000314The number of left-to-right-maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001498The normalised height of a Nakayama algebra with magnitude 1. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000444The length of the maximal rise of a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000445The number of rises of length 1 of a Dyck path. St001176The size of a partition minus its first part. St000013The height of a Dyck path. St000654The first descent of a permutation. St000074The number of special entries. St000931The number of occurrences of the pattern UUU in a Dyck path. St000674The number of hills of a Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000996The number of exclusive left-to-right maxima of a permutation. St000740The last entry of a permutation. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000215The number of adjacencies of a permutation, zero appended. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St000193The row of the unique '1' in the first column of the alternating sign matrix. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St000365The number of double ascents of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000989The number of final rises of a permutation. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000051The size of the left subtree of a binary tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000710The number of big deficiencies of a permutation. St000990The first ascent of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000061The number of nodes on the left branch of a binary tree. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000025The number of initial rises of a Dyck path. St000738The first entry in the last row of a standard tableau. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St000686The finitistic dominant dimension of a Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001438The number of missing boxes of a skew partition. St001435The number of missing boxes in the first row. St001868The number of alignments of type NE of a signed permutation. St001487The number of inner corners of a skew partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001867The number of alignments of type EN of a signed permutation. St000259The diameter of a connected graph. St001730The number of times the path corresponding to a binary word crosses the base line. St000454The largest eigenvalue of a graph if it is integral. St000899The maximal number of repetitions of an integer composition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St000326The position of the first one in a binary word after appending a 1 at the end. St000383The last part of an integer composition. St000406The number of occurrences of the pattern 3241 in a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000873The aix statistic of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001847The number of occurrences of the pattern 1432 in a permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000120The number of left tunnels of a Dyck path. St000657The smallest part of an integer composition. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001864The number of excedances of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001905The number of preferred parking spots in a parking function less than the index of the car. St001937The size of the center of a parking function.