Your data matches 179 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000142
Mapping
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 1
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 1
[5,1]
 => 0
[4,2]
 => 2
[4,1,1]
 => 1
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 3
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 1
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 1
[4,2,1]
 => 2
[4,1,1,1]
 => 1
[3,3,1]
 => 0
[3,2,2]
 => 2
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 3
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 1
[7,1]
 => 0
[6,2]
 => 2
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 1
Description
The number of even parts of a partition.
Matching statistic: St000143
(load all 2 compositions to match this statistic)
Mapping
St000143: 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]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 2
[2,1,1]
 => 1
[1,1,1,1]
 => 1
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 1
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 3
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 2
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 1
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 1
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 1
[3,3,1]
 => 3
[3,2,2]
 => 2
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 2
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 1
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 0
Description
The largest repeated part of a partition. If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
Matching statistic: St000149
(load all 4 compositions to match this statistic)
Mapping
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 0
[4]
 => 2
[3,1]
 => 1
[2,2]
 => 1
[2,1,1]
 => 0
[1,1,1,1]
 => 0
[5]
 => 2
[4,1]
 => 1
[3,2]
 => 1
[3,1,1]
 => 1
[2,2,1]
 => 0
[2,1,1,1]
 => 0
[1,1,1,1,1]
 => 0
[6]
 => 3
[5,1]
 => 2
[4,2]
 => 2
[4,1,1]
 => 1
[3,3]
 => 1
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 1
[2,2,1,1]
 => 0
[2,1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => 0
[7]
 => 3
[6,1]
 => 2
[5,2]
 => 2
[5,1,1]
 => 2
[4,3]
 => 1
[4,2,1]
 => 1
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 0
[3,1,1,1,1]
 => 1
[2,2,2,1]
 => 0
[2,2,1,1,1]
 => 0
[2,1,1,1,1,1]
 => 0
[1,1,1,1,1,1,1]
 => 0
[8]
 => 4
[7,1]
 => 3
[6,2]
 => 3
[6,1,1]
 => 2
[5,3]
 => 2
[5,2,1]
 => 1
Description
The number of cells of the partition whose leg is zero and arm is odd. This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
Mapping
St000150: 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]
 => 1
[4]
 => 0
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 2
[5]
 => 0
[4,1]
 => 0
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 2
[6]
 => 0
[5,1]
 => 0
[4,2]
 => 0
[4,1,1]
 => 1
[3,3]
 => 1
[3,2,1]
 => 0
[3,1,1,1]
 => 1
[2,2,2]
 => 1
[2,2,1,1]
 => 2
[2,1,1,1,1]
 => 2
[1,1,1,1,1,1]
 => 3
[7]
 => 0
[6,1]
 => 0
[5,2]
 => 0
[5,1,1]
 => 1
[4,3]
 => 0
[4,2,1]
 => 0
[4,1,1,1]
 => 1
[3,3,1]
 => 1
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 2
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 2
[2,1,1,1,1,1]
 => 2
[1,1,1,1,1,1,1]
 => 3
[8]
 => 0
[7,1]
 => 0
[6,2]
 => 0
[6,1,1]
 => 1
[5,3]
 => 0
[5,2,1]
 => 0
Description
The floored half-sum of the multiplicities of a partition. This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St001587
(load all 3 compositions to match this statistic)
Mapping
St001587: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 1
[1,1]
 => 0
[3]
 => 0
[2,1]
 => 1
[1,1,1]
 => 0
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 1
[1,1,1,1]
 => 0
[5]
 => 0
[4,1]
 => 2
[3,2]
 => 1
[3,1,1]
 => 0
[2,2,1]
 => 1
[2,1,1,1]
 => 1
[1,1,1,1,1]
 => 0
[6]
 => 3
[5,1]
 => 0
[4,2]
 => 2
[4,1,1]
 => 2
[3,3]
 => 0
[3,2,1]
 => 1
[3,1,1,1]
 => 0
[2,2,2]
 => 1
[2,2,1,1]
 => 1
[2,1,1,1,1]
 => 1
[1,1,1,1,1,1]
 => 0
[7]
 => 0
[6,1]
 => 3
[5,2]
 => 1
[5,1,1]
 => 0
[4,3]
 => 2
[4,2,1]
 => 2
[4,1,1,1]
 => 2
[3,3,1]
 => 0
[3,2,2]
 => 1
[3,2,1,1]
 => 1
[3,1,1,1,1]
 => 0
[2,2,2,1]
 => 1
[2,2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => 0
[8]
 => 4
[7,1]
 => 0
[6,2]
 => 3
[6,1,1]
 => 3
[5,3]
 => 0
[5,2,1]
 => 1
Description
Half of the largest even part of an integer partition. The largest even part is recorded by [[St000995]].
Matching statistic: St000992
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000992: 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]
 => 1
[3]
 => []
 => 0
[2,1]
 => [1]
 => 1
[1,1,1]
 => [1,1]
 => 0
[4]
 => []
 => 0
[3,1]
 => [1]
 => 1
[2,2]
 => [2]
 => 2
[2,1,1]
 => [1,1]
 => 0
[1,1,1,1]
 => [1,1,1]
 => 1
[5]
 => []
 => 0
[4,1]
 => [1]
 => 1
[3,2]
 => [2]
 => 2
[3,1,1]
 => [1,1]
 => 0
[2,2,1]
 => [2,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => 0
[6]
 => []
 => 0
[5,1]
 => [1]
 => 1
[4,2]
 => [2]
 => 2
[4,1,1]
 => [1,1]
 => 0
[3,3]
 => [3]
 => 3
[3,2,1]
 => [2,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => 1
[2,2,2]
 => [2,2]
 => 0
[2,2,1,1]
 => [2,1,1]
 => 2
[2,1,1,1,1]
 => [1,1,1,1]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[7]
 => []
 => 0
[6,1]
 => [1]
 => 1
[5,2]
 => [2]
 => 2
[5,1,1]
 => [1,1]
 => 0
[4,3]
 => [3]
 => 3
[4,2,1]
 => [2,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => 1
[3,3,1]
 => [3,1]
 => 2
[3,2,2]
 => [2,2]
 => 0
[3,2,1,1]
 => [2,1,1]
 => 2
[3,1,1,1,1]
 => [1,1,1,1]
 => 0
[2,2,2,1]
 => [2,2,1]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 0
[8]
 => []
 => 0
[7,1]
 => [1]
 => 1
[6,2]
 => [2]
 => 2
[6,1,1]
 => [1,1]
 => 0
[5,3]
 => [3]
 => 3
[5,2,1]
 => [2,1]
 => 1
Description
The alternating sum of the parts of an integer partition. For a partition $\lambda = (\lambda_1,\ldots,\lambda_k)$, this is $\lambda_1 - \lambda_2 + \cdots \pm \lambda_k$.
Matching statistic: St000148
(load all 2 compositions to match this statistic)
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000148: 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]
 => [1]
 => 1
[3]
 => []
 => []
 => 0
[2,1]
 => [1]
 => [1]
 => 1
[1,1,1]
 => [1,1]
 => [2]
 => 0
[4]
 => []
 => []
 => 0
[3,1]
 => [1]
 => [1]
 => 1
[2,2]
 => [2]
 => [1,1]
 => 2
[2,1,1]
 => [1,1]
 => [2]
 => 0
[1,1,1,1]
 => [1,1,1]
 => [3]
 => 1
[5]
 => []
 => []
 => 0
[4,1]
 => [1]
 => [1]
 => 1
[3,2]
 => [2]
 => [1,1]
 => 2
[3,1,1]
 => [1,1]
 => [2]
 => 0
[2,2,1]
 => [2,1]
 => [2,1]
 => 1
[2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 0
[6]
 => []
 => []
 => 0
[5,1]
 => [1]
 => [1]
 => 1
[4,2]
 => [2]
 => [1,1]
 => 2
[4,1,1]
 => [1,1]
 => [2]
 => 0
[3,3]
 => [3]
 => [1,1,1]
 => 3
[3,2,1]
 => [2,1]
 => [2,1]
 => 1
[3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
[2,2,2]
 => [2,2]
 => [2,2]
 => 0
[2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 2
[2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1
[7]
 => []
 => []
 => 0
[6,1]
 => [1]
 => [1]
 => 1
[5,2]
 => [2]
 => [1,1]
 => 2
[5,1,1]
 => [1,1]
 => [2]
 => 0
[4,3]
 => [3]
 => [1,1,1]
 => 3
[4,2,1]
 => [2,1]
 => [2,1]
 => 1
[4,1,1,1]
 => [1,1,1]
 => [3]
 => 1
[3,3,1]
 => [3,1]
 => [2,1,1]
 => 2
[3,2,2]
 => [2,2]
 => [2,2]
 => 0
[3,2,1,1]
 => [2,1,1]
 => [3,1]
 => 2
[3,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 0
[2,2,2,1]
 => [2,2,1]
 => [3,2]
 => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [5]
 => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [6]
 => 0
[8]
 => []
 => []
 => 0
[7,1]
 => [1]
 => [1]
 => 1
[6,2]
 => [2]
 => [1,1]
 => 2
[6,1,1]
 => [1,1]
 => [2]
 => 0
[5,3]
 => [3]
 => [1,1,1]
 => 3
[5,2,1]
 => [2,1]
 => [2,1]
 => 1
Description
The number of odd parts of a partition.
Matching statistic: St000022
Mapping
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => []
 => []
 => [] => 0
[2]
 => []
 => []
 => [] => 0
[1,1]
 => [1]
 => [[1]]
 => [1] => 1
[3]
 => []
 => []
 => [] => 0
[2,1]
 => [1]
 => [[1]]
 => [1] => 1
[1,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[4]
 => []
 => []
 => [] => 0
[3,1]
 => [1]
 => [[1]]
 => [1] => 1
[2,2]
 => [2]
 => [[1,2]]
 => [1,2] => 2
[2,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[1,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[5]
 => []
 => []
 => [] => 0
[4,1]
 => [1]
 => [[1]]
 => [1] => 1
[3,2]
 => [2]
 => [[1,2]]
 => [1,2] => 2
[3,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[2,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[2,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[1,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[6]
 => []
 => []
 => [] => 0
[5,1]
 => [1]
 => [[1]]
 => [1] => 1
[4,2]
 => [2]
 => [[1,2]]
 => [1,2] => 2
[4,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[3,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 3
[3,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[3,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[2,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[2,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[2,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[1,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
[7]
 => []
 => []
 => [] => 0
[6,1]
 => [1]
 => [[1]]
 => [1] => 1
[5,2]
 => [2]
 => [[1,2]]
 => [1,2] => 2
[5,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[4,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 3
[4,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
[4,1,1,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [3,2,1] => 1
[3,3,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [2,1,3,4] => 2
[3,2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[3,2,1,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [3,2,1,4] => 2
[3,1,1,1,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [4,3,2,1] => 0
[2,2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 1
[2,2,1,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 1
[2,1,1,1,1,1]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
[1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [6,5,4,3,2,1] => 0
[8]
 => []
 => []
 => [] => 0
[7,1]
 => [1]
 => [[1]]
 => [1] => 1
[6,2]
 => [2]
 => [[1,2]]
 => [1,2] => 2
[6,1,1]
 => [1,1]
 => [[1],[2]]
 => [2,1] => 0
[5,3]
 => [3]
 => [[1,2,3]]
 => [1,2,3] => 3
[5,2,1]
 => [2,1]
 => [[1,3],[2]]
 => [2,1,3] => 1
Description
The number of fixed points of a permutation.
Matching statistic: St000392
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
Mp00224: Binary words —runsort⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => 0 => 0 => 0
[2]
 => 0 => 1 => 1 => 1
[1,1]
 => 11 => 00 => 00 => 0
[3]
 => 1 => 0 => 0 => 0
[2,1]
 => 01 => 10 => 01 => 1
[1,1,1]
 => 111 => 000 => 000 => 0
[4]
 => 0 => 1 => 1 => 1
[3,1]
 => 11 => 00 => 00 => 0
[2,2]
 => 00 => 11 => 11 => 2
[2,1,1]
 => 011 => 100 => 001 => 1
[1,1,1,1]
 => 1111 => 0000 => 0000 => 0
[5]
 => 1 => 0 => 0 => 0
[4,1]
 => 01 => 10 => 01 => 1
[3,2]
 => 10 => 01 => 01 => 1
[3,1,1]
 => 111 => 000 => 000 => 0
[2,2,1]
 => 001 => 110 => 011 => 2
[2,1,1,1]
 => 0111 => 1000 => 0001 => 1
[1,1,1,1,1]
 => 11111 => 00000 => 00000 => 0
[6]
 => 0 => 1 => 1 => 1
[5,1]
 => 11 => 00 => 00 => 0
[4,2]
 => 00 => 11 => 11 => 2
[4,1,1]
 => 011 => 100 => 001 => 1
[3,3]
 => 11 => 00 => 00 => 0
[3,2,1]
 => 101 => 010 => 001 => 1
[3,1,1,1]
 => 1111 => 0000 => 0000 => 0
[2,2,2]
 => 000 => 111 => 111 => 3
[2,2,1,1]
 => 0011 => 1100 => 0011 => 2
[2,1,1,1,1]
 => 01111 => 10000 => 00001 => 1
[1,1,1,1,1,1]
 => 111111 => 000000 => 000000 => 0
[7]
 => 1 => 0 => 0 => 0
[6,1]
 => 01 => 10 => 01 => 1
[5,2]
 => 10 => 01 => 01 => 1
[5,1,1]
 => 111 => 000 => 000 => 0
[4,3]
 => 01 => 10 => 01 => 1
[4,2,1]
 => 001 => 110 => 011 => 2
[4,1,1,1]
 => 0111 => 1000 => 0001 => 1
[3,3,1]
 => 111 => 000 => 000 => 0
[3,2,2]
 => 100 => 011 => 011 => 2
[3,2,1,1]
 => 1011 => 0100 => 0001 => 1
[3,1,1,1,1]
 => 11111 => 00000 => 00000 => 0
[2,2,2,1]
 => 0001 => 1110 => 0111 => 3
[2,2,1,1,1]
 => 00111 => 11000 => 00011 => 2
[2,1,1,1,1,1]
 => 011111 => 100000 => 000001 => 1
[1,1,1,1,1,1,1]
 => 1111111 => 0000000 => 0000000 => 0
[8]
 => 0 => 1 => 1 => 1
[7,1]
 => 11 => 00 => 00 => 0
[6,2]
 => 00 => 11 => 11 => 2
[6,1,1]
 => 011 => 100 => 001 => 1
[5,3]
 => 11 => 00 => 00 => 0
[5,2,1]
 => 101 => 010 => 001 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001389
Mapping
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 1 => [1,1] => [1,1]
 => 1 = 0 + 1
[2]
 => 0 => [2] => [2]
 => 2 = 1 + 1
[1,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[3]
 => 1 => [1,1] => [1,1]
 => 1 = 0 + 1
[2,1]
 => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[1,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1 = 0 + 1
[4]
 => 0 => [2] => [2]
 => 2 = 1 + 1
[3,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[2,2]
 => 00 => [3] => [3]
 => 3 = 2 + 1
[2,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2 = 1 + 1
[1,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[5]
 => 1 => [1,1] => [1,1]
 => 1 = 0 + 1
[4,1]
 => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[3,2]
 => 10 => [1,2] => [2,1]
 => 2 = 1 + 1
[3,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1 = 0 + 1
[2,2,1]
 => 001 => [3,1] => [3,1]
 => 3 = 2 + 1
[2,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[6]
 => 0 => [2] => [2]
 => 2 = 1 + 1
[5,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[4,2]
 => 00 => [3] => [3]
 => 3 = 2 + 1
[4,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2 = 1 + 1
[3,3]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[3,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 2 = 1 + 1
[3,1,1,1]
 => 1111 => [1,1,1,1,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[2,2,2]
 => 000 => [4] => [4]
 => 4 = 3 + 1
[2,2,1,1]
 => 0011 => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[2,1,1,1,1]
 => 01111 => [2,1,1,1,1] => [2,1,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1,1]
 => 111111 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1]
 => 1 = 0 + 1
[7]
 => 1 => [1,1] => [1,1]
 => 1 = 0 + 1
[6,1]
 => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[5,2]
 => 10 => [1,2] => [2,1]
 => 2 = 1 + 1
[5,1,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1 = 0 + 1
[4,3]
 => 01 => [2,1] => [2,1]
 => 2 = 1 + 1
[4,2,1]
 => 001 => [3,1] => [3,1]
 => 3 = 2 + 1
[4,1,1,1]
 => 0111 => [2,1,1,1] => [2,1,1,1]
 => 2 = 1 + 1
[3,3,1]
 => 111 => [1,1,1,1] => [1,1,1,1]
 => 1 = 0 + 1
[3,2,2]
 => 100 => [1,3] => [3,1]
 => 3 = 2 + 1
[3,2,1,1]
 => 1011 => [1,2,1,1] => [2,1,1,1]
 => 2 = 1 + 1
[3,1,1,1,1]
 => 11111 => [1,1,1,1,1,1] => [1,1,1,1,1,1]
 => 1 = 0 + 1
[2,2,2,1]
 => 0001 => [4,1] => [4,1]
 => 4 = 3 + 1
[2,2,1,1,1]
 => 00111 => [3,1,1,1] => [3,1,1,1]
 => 3 = 2 + 1
[2,1,1,1,1,1]
 => 011111 => [2,1,1,1,1,1] => [2,1,1,1,1,1]
 => 2 = 1 + 1
[1,1,1,1,1,1,1]
 => 1111111 => [1,1,1,1,1,1,1,1] => [1,1,1,1,1,1,1,1]
 => 1 = 0 + 1
[8]
 => 0 => [2] => [2]
 => 2 = 1 + 1
[7,1]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[6,2]
 => 00 => [3] => [3]
 => 3 = 2 + 1
[6,1,1]
 => 011 => [2,1,1] => [2,1,1]
 => 2 = 1 + 1
[5,3]
 => 11 => [1,1,1] => [1,1,1]
 => 1 = 0 + 1
[5,2,1]
 => 101 => [1,2,1] => [2,1,1]
 => 2 = 1 + 1
Description
The number of partitions of the same length below the given integer partition. For a partition $\lambda_1 \geq \dots \lambda_k > 0$, this number is $$ \det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$$
The following 169 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000877The depth of the binary word interpreted as a path. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St000326The position of the first one in a binary word after appending a 1 at the end. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001584The area statistic between a Dyck path and its bounce path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000895The number of ones on the main diagonal of an alternating sign matrix. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000864The number of circled entries of the shifted recording tableau of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001843The Z-index of a set partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000065The number of entries equal to -1 in an alternating sign matrix. St000237The number of small exceedances. St000461The rix statistic 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. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000359The number of occurrences of the pattern 23-1. St001394The genus of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000105The number of blocks in the set partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St000039The number of crossings of a permutation. St000355The number of occurrences of the pattern 21-3. St000711The number of big exceedences of a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000335The difference of lower and upper interactions. St000678The number of up steps after the last double rise of a Dyck path. St000990The first ascent of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001530The depth of a Dyck path. St000475The number of parts equal to 1 in a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001498The normalised height of a Nakayama algebra with magnitude 1. St001557The number of inversions of the second entry of a permutation. St000153The number of adjacent cycles of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000654The first descent of a permutation. St000314The number of left-to-right-maxima of a permutation. St001597The Frobenius rank of a skew partition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001114The number of odd descents of a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000681The Grundy value of Chomp on Ferrers diagrams. St000933The number of multipartitions of sizes given by an integer partition. St000993The multiplicity of the largest part of an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001115The number of even descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001052The length of the exterior of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000338The number of pixed points of a permutation. St000516The number of stretching pairs of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000732The number of double deficiencies of a permutation. St000989The number of final rises of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St000259The diameter of a connected graph. St000260The radius of a connected graph. St001964The interval resolution global dimension of a poset. St000741The Colin de Verdière graph invariant. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001487The number of inner corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001488The number of corners of a skew partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000768The number of peaks in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000884The number of isolated descents of a permutation. St000647The number of big descents of a permutation. St001423The number of distinct cubes in a binary word. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000455The second largest eigenvalue of a graph if it is integral. St000761The number of ascents in an integer composition. St000807The sum of the heights of the valleys of the associated bargraph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St000383The last part of an integer composition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000805The number of peaks of the associated bargraph. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001569The maximal modular displacement of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St000735The last entry on the main diagonal of a standard tableau.