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Your data matches 207 different statistics following compositions of up to 3 maps.
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Matching statistic: St001435
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001435: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> 0
[[2],[]]
=> 0
[[1,1],[]]
=> 0
[[2,1],[1]]
=> 1
[[3],[]]
=> 0
[[2,1],[]]
=> 0
[[3,1],[1]]
=> 1
[[2,2],[1]]
=> 1
[[3,2],[2]]
=> 2
[[1,1,1],[]]
=> 0
[[2,2,1],[1,1]]
=> 1
[[2,1,1],[1]]
=> 1
[[3,2,1],[2,1]]
=> 2
[[4],[]]
=> 0
[[3,1],[]]
=> 0
[[4,1],[1]]
=> 1
[[2,2],[]]
=> 0
[[3,2],[1]]
=> 1
[[4,2],[2]]
=> 2
[[2,1,1],[]]
=> 0
[[3,2,1],[1,1]]
=> 1
[[3,1,1],[1]]
=> 1
[[4,2,1],[2,1]]
=> 2
[[3,3],[2]]
=> 2
[[4,3],[3]]
=> 3
[[2,2,1],[1]]
=> 1
[[3,3,1],[2,1]]
=> 2
[[3,2,1],[2]]
=> 2
[[4,3,1],[3,1]]
=> 3
[[2,2,2],[1,1]]
=> 1
[[3,3,2],[2,2]]
=> 2
[[3,2,2],[2,1]]
=> 2
[[4,3,2],[3,2]]
=> 3
[[1,1,1,1],[]]
=> 0
[[2,2,2,1],[1,1,1]]
=> 1
[[2,2,1,1],[1,1]]
=> 1
[[3,3,2,1],[2,2,1]]
=> 2
[[2,1,1,1],[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> 2
[[3,2,1,1],[2,1]]
=> 2
[[4,3,2,1],[3,2,1]]
=> 3
[[5],[]]
=> 0
[[4,1],[]]
=> 0
[[5,1],[1]]
=> 1
[[3,2],[]]
=> 0
[[4,2],[1]]
=> 1
[[5,2],[2]]
=> 2
[[3,1,1],[]]
=> 0
[[4,2,1],[1,1]]
=> 1
[[4,1,1],[1]]
=> 1
Description
The number of missing boxes in the first row.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> 0
[[2],[]]
=> []
=> 0
[[1,1],[]]
=> []
=> 0
[[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> []
=> 0
[[2,1],[]]
=> []
=> 0
[[3,1],[1]]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> 1
[[1,1,1],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> 2
[[2,1,1],[1]]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> []
=> 0
[[3,1],[]]
=> []
=> 0
[[4,1],[1]]
=> [1]
=> 1
[[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> 1
[[2,1,1],[]]
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> 2
[[3,1,1],[1]]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> 1
[[4,3],[3]]
=> [3]
=> 1
[[2,2,1],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> 2
[[1,1,1,1],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> 0
[[4,1],[]]
=> []
=> 0
[[5,1],[1]]
=> [1]
=> 1
[[3,2],[]]
=> []
=> 0
[[4,2],[1]]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> 1
[[3,1,1],[]]
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> 2
[[4,1,1],[1]]
=> [1]
=> 1
Description
The length of the partition.
Matching statistic: St000147
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> 0
[[2],[]]
=> []
=> 0
[[1,1],[]]
=> []
=> 0
[[2,1],[1]]
=> [1]
=> 1
[[3],[]]
=> []
=> 0
[[2,1],[]]
=> []
=> 0
[[3,1],[1]]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> 2
[[1,1,1],[]]
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> 1
[[2,1,1],[1]]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> 2
[[4],[]]
=> []
=> 0
[[3,1],[]]
=> []
=> 0
[[4,1],[1]]
=> [1]
=> 1
[[2,2],[]]
=> []
=> 0
[[3,2],[1]]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> 2
[[2,1,1],[]]
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> 1
[[3,1,1],[1]]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> 2
[[3,3],[2]]
=> [2]
=> 2
[[4,3],[3]]
=> [3]
=> 3
[[2,2,1],[1]]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> 2
[[3,2,1],[2]]
=> [2]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> 3
[[1,1,1,1],[]]
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> 3
[[5],[]]
=> []
=> 0
[[4,1],[]]
=> []
=> 0
[[5,1],[1]]
=> [1]
=> 1
[[3,2],[]]
=> []
=> 0
[[4,2],[1]]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> 2
[[3,1,1],[]]
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> 1
[[4,1,1],[1]]
=> [1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000225
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> [1]
=> 0
[[2],[]]
=> [2]
=> 0
[[1,1],[]]
=> [1,1]
=> 0
[[2,1],[1]]
=> [2,1]
=> 1
[[3],[]]
=> [3]
=> 0
[[2,1],[]]
=> [2,1]
=> 1
[[3,1],[1]]
=> [3,1]
=> 2
[[2,2],[1]]
=> [2,2]
=> 0
[[3,2],[2]]
=> [3,2]
=> 1
[[1,1,1],[]]
=> [1,1,1]
=> 0
[[2,2,1],[1,1]]
=> [2,2,1]
=> 1
[[2,1,1],[1]]
=> [2,1,1]
=> 1
[[3,2,1],[2,1]]
=> [3,2,1]
=> 2
[[4],[]]
=> [4]
=> 0
[[3,1],[]]
=> [3,1]
=> 2
[[4,1],[1]]
=> [4,1]
=> 3
[[2,2],[]]
=> [2,2]
=> 0
[[3,2],[1]]
=> [3,2]
=> 1
[[4,2],[2]]
=> [4,2]
=> 2
[[2,1,1],[]]
=> [2,1,1]
=> 1
[[3,2,1],[1,1]]
=> [3,2,1]
=> 2
[[3,1,1],[1]]
=> [3,1,1]
=> 2
[[4,2,1],[2,1]]
=> [4,2,1]
=> 3
[[3,3],[2]]
=> [3,3]
=> 0
[[4,3],[3]]
=> [4,3]
=> 1
[[2,2,1],[1]]
=> [2,2,1]
=> 1
[[3,3,1],[2,1]]
=> [3,3,1]
=> 2
[[3,2,1],[2]]
=> [3,2,1]
=> 2
[[4,3,1],[3,1]]
=> [4,3,1]
=> 3
[[2,2,2],[1,1]]
=> [2,2,2]
=> 0
[[3,3,2],[2,2]]
=> [3,3,2]
=> 1
[[3,2,2],[2,1]]
=> [3,2,2]
=> 1
[[4,3,2],[3,2]]
=> [4,3,2]
=> 2
[[1,1,1,1],[]]
=> [1,1,1,1]
=> 0
[[2,2,2,1],[1,1,1]]
=> [2,2,2,1]
=> 1
[[2,2,1,1],[1,1]]
=> [2,2,1,1]
=> 1
[[3,3,2,1],[2,2,1]]
=> [3,3,2,1]
=> 2
[[2,1,1,1],[1]]
=> [2,1,1,1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [3,2,2,1]
=> 2
[[3,2,1,1],[2,1]]
=> [3,2,1,1]
=> 2
[[4,3,2,1],[3,2,1]]
=> [4,3,2,1]
=> 3
[[5],[]]
=> [5]
=> 0
[[4,1],[]]
=> [4,1]
=> 3
[[5,1],[1]]
=> [5,1]
=> 4
[[3,2],[]]
=> [3,2]
=> 1
[[4,2],[1]]
=> [4,2]
=> 2
[[5,2],[2]]
=> [5,2]
=> 3
[[3,1,1],[]]
=> [3,1,1]
=> 2
[[4,2,1],[1,1]]
=> [4,2,1]
=> 3
[[4,1,1],[1]]
=> [4,1,1]
=> 3
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000013
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 3
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> 1
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> 1
Description
The height of a Dyck path.
The height of a Dyck path $D$ of semilength $n$ is defined as the maximal height of a peak of $D$. The height of $D$ at position $i$ is the number of up-steps minus the number of down-steps before position $i$.
Matching statistic: St000378
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[2]]
=> [2]
=> [1,1]
=> 1
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[2,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1]
=> 1
[[4,2],[2]]
=> [2]
=> [1,1]
=> 1
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,1,1],[1]]
=> [1]
=> [1]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[3,3],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3],[3]]
=> [3]
=> [1,1,1]
=> 1
[[2,2,1],[1]]
=> [1]
=> [1]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,1]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [2,1,1]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [4]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [5]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [2,2,1]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [2,2]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [3]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [5,1]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1]
=> 1
[[5,2],[2]]
=> [2]
=> [1,1]
=> 1
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2
[[4,1,1],[1]]
=> [1]
=> [1]
=> 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000676
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[1,1,1],[]]
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4],[]]
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[[5],[]]
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> 1
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000007
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> [] => 0
[[2],[]]
=> []
=> []
=> [] => 0
[[1,1],[]]
=> []
=> []
=> [] => 0
[[2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3],[]]
=> []
=> []
=> [] => 0
[[2,1],[]]
=> []
=> []
=> [] => 0
[[3,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[1,1,1],[]]
=> []
=> []
=> [] => 0
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4],[]]
=> []
=> []
=> [] => 0
[[3,1],[]]
=> []
=> []
=> [] => 0
[[4,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[2,2],[]]
=> []
=> []
=> [] => 0
[[3,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[2,1,1],[]]
=> []
=> []
=> [] => 0
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[4,3,1],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[[1,1,1,1],[]]
=> []
=> []
=> [] => 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 3
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 3
[[5],[]]
=> []
=> []
=> [] => 0
[[4,1],[]]
=> []
=> []
=> [] => 0
[[5,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[3,2],[]]
=> []
=> []
=> [] => 0
[[4,2],[1]]
=> [1]
=> [[1]]
=> [1] => 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[[3,1,1],[]]
=> []
=> []
=> [] => 0
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> [1] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000025
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[2,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[[2,2,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[3,2,1],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[4,3,1],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[[2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[[2,1,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[[3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[4,3,2,1],[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]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
[[5,2],[2]]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[[4,1,1],[1]]
=> [1]
=> [1,0]
=> [1,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1],[]]
=> []
=> []
=> []
=> 0
[[2],[]]
=> []
=> []
=> []
=> 0
[[1,1],[]]
=> []
=> []
=> []
=> 0
[[2,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3],[]]
=> []
=> []
=> []
=> 0
[[2,1],[]]
=> []
=> []
=> []
=> 0
[[3,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[2,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[2,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4],[]]
=> []
=> []
=> []
=> 0
[[3,1],[]]
=> []
=> []
=> []
=> 0
[[4,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[2,2],[]]
=> []
=> []
=> []
=> 0
[[3,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[4,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[2,1,1],[]]
=> []
=> []
=> []
=> 0
[[3,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[4,2,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[3,3],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[4,3],[3]]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1
[[2,2,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,3,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[3,2,1],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[4,3,1],[3,1]]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,3,2],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[[3,2,2],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4,3,2],[3,2]]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
[[1,1,1,1],[]]
=> []
=> []
=> []
=> 0
[[2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3
[[2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[3,3,2,1],[2,2,1]]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 3
[[2,1,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2,2,1],[2,1,1]]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 3
[[3,2,1,1],[2,1]]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 2
[[4,3,2,1],[3,2,1]]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 3
[[5],[]]
=> []
=> []
=> []
=> 0
[[4,1],[]]
=> []
=> []
=> []
=> 0
[[5,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[3,2],[]]
=> []
=> []
=> []
=> 0
[[4,2],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
[[5,2],[2]]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[[3,1,1],[]]
=> []
=> []
=> []
=> 0
[[4,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2
[[4,1,1],[1]]
=> [1]
=> [[1]]
=> [[1]]
=> 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
The following 197 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000740The last entry of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000144The pyramid weight of the Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000288The number of ones in a binary word. St000733The row containing the largest entry of a standard tableau. St000734The last entry in the first row of a standard tableau. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St000141The maximum drop size of a permutation. St000167The number of leaves of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000470The number of runs in a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000653The last descent of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000691The number of changes of a binary word. St000840The number of closers smaller than the largest opener in a perfect matching. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows:
St001480The number of simple summands of the module J^2/J^3. St001809The index of the step at the first peak of maximal height in a Dyck path. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000166The depth minus 1 of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000291The number of descents of a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000390The number of runs of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000542The number of left-to-right-minima of a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St000991The number of right-to-left minima of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a 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. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St000738The first entry in the last row of a standard tableau. St001462The number of factors of a standard tableaux under concatenation. St000383The last part of an integer composition. St000006The dinv of a Dyck path. St000505The biggest entry in the block containing the 1. St000746The number of pairs with odd minimum in a perfect matching. St000971The smallest closer of a set partition. St000702The number of weak deficiencies of a permutation. St001589The nesting number of a perfect matching. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000489The number of cycles of a permutation of length at most 3. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St001246The maximal difference between two consecutive entries of a permutation. St000488The number of cycles of a permutation of length at most 2. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001062The maximal size of a block of a set partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001488The number of corners of a skew partition. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000259The diameter of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001839The number of excedances of a set partition. St001568The smallest positive integer that does not appear twice in the partition. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001060The distinguishing index of a graph. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001645The pebbling number of a connected graph. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000681The Grundy value of Chomp on Ferrers diagrams. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000993The multiplicity of the largest part of an integer partition. St001778The largest greatest common divisor of an element and its image in 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. St000454The largest eigenvalue of a graph if it is integral. 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 $eA$. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000264The girth of a graph, which is not a tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St001556The number of inversions of the third entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. 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. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001875The number of simple modules with projective dimension at most 1.
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