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Your data matches 448 different statistics following compositions of up to 3 maps.
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Matching statistic: St000105
(load all 927 compositions to match this statistic)
Mapping
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => 1 = 0 + 1
{{1,2}}
 => 1 = 0 + 1
{{1},{2}}
 => 2 = 1 + 1
{{1,2,3}}
 => 1 = 0 + 1
{{1,2},{3}}
 => 2 = 1 + 1
{{1,3},{2}}
 => 2 = 1 + 1
{{1},{2,3}}
 => 2 = 1 + 1
{{1},{2},{3}}
 => 3 = 2 + 1
{{1,2,3,4}}
 => 1 = 0 + 1
{{1,2,3},{4}}
 => 2 = 1 + 1
{{1,2,4},{3}}
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => 3 = 2 + 1
{{1,3,4},{2}}
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => 3 = 2 + 1
{{1},{2,3,4}}
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => 3 = 2 + 1
{{1},{2},{3,4}}
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => 4 = 3 + 1
{{1},{2,3,4,5}}
 => 2 = 1 + 1
{{1},{2,3,4},{5}}
 => 3 = 2 + 1
{{1},{2,3,5},{4}}
 => 3 = 2 + 1
{{1},{2,3},{4},{5}}
 => 4 = 3 + 1
{{1,4,5},{2},{3}}
 => 3 = 2 + 1
{{1,4},{2},{3},{5}}
 => 4 = 3 + 1
{{1},{2,4,5},{3}}
 => 3 = 2 + 1
{{1},{2,4},{3},{5}}
 => 4 = 3 + 1
{{1},{2},{3,4,5}}
 => 3 = 2 + 1
{{1},{2},{3,4},{5}}
 => 4 = 3 + 1
{{1,5},{2},{3},{4}}
 => 4 = 3 + 1
{{1},{2,5},{3},{4}}
 => 4 = 3 + 1
{{1},{2},{3,5},{4}}
 => 4 = 3 + 1
{{1},{2},{3},{4,5}}
 => 4 = 3 + 1
{{1},{2},{3},{4},{5}}
 => 5 = 4 + 1
{{1,2,3,4,5,6}}
 => 1 = 0 + 1
{{1,2,3,4,5},{6}}
 => 2 = 1 + 1
{{1,2,3,4,6},{5}}
 => 2 = 1 + 1
Description
The number of blocks in the set partition. The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
Matching statistic: St000157
(load all 50 compositions to match this statistic)
Mapping
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [[1]]
 => 0
{{1,2}}
 => [[1,2]]
 => 0
{{1},{2}}
 => [[1],[2]]
 => 1
{{1,2,3}}
 => [[1,2,3]]
 => 0
{{1,2},{3}}
 => [[1,2],[3]]
 => 1
{{1,3},{2}}
 => [[1,3],[2]]
 => 1
{{1},{2,3}}
 => [[1,3],[2]]
 => 1
{{1},{2},{3}}
 => [[1],[2],[3]]
 => 2
{{1,2,3,4}}
 => [[1,2,3,4]]
 => 0
{{1,2,3},{4}}
 => [[1,2,3],[4]]
 => 1
{{1,2,4},{3}}
 => [[1,2,4],[3]]
 => 1
{{1,2},{3},{4}}
 => [[1,2],[3],[4]]
 => 2
{{1,3,4},{2}}
 => [[1,3,4],[2]]
 => 1
{{1,3},{2},{4}}
 => [[1,3],[2],[4]]
 => 2
{{1},{2,3,4}}
 => [[1,3,4],[2]]
 => 1
{{1},{2,3},{4}}
 => [[1,3],[2],[4]]
 => 2
{{1,4},{2},{3}}
 => [[1,4],[2],[3]]
 => 2
{{1},{2,4},{3}}
 => [[1,4],[2],[3]]
 => 2
{{1},{2},{3,4}}
 => [[1,4],[2],[3]]
 => 2
{{1},{2},{3},{4}}
 => [[1],[2],[3],[4]]
 => 3
{{1,2,3,4,5}}
 => [[1,2,3,4,5]]
 => 0
{{1,2,3,4},{5}}
 => [[1,2,3,4],[5]]
 => 1
{{1,2,3,5},{4}}
 => [[1,2,3,5],[4]]
 => 1
{{1,2,3},{4},{5}}
 => [[1,2,3],[4],[5]]
 => 2
{{1,2,4,5},{3}}
 => [[1,2,4,5],[3]]
 => 1
{{1,2,4},{3},{5}}
 => [[1,2,4],[3],[5]]
 => 2
{{1,2,5},{3},{4}}
 => [[1,2,5],[3],[4]]
 => 2
{{1,2},{3},{4},{5}}
 => [[1,2],[3],[4],[5]]
 => 3
{{1,3,4,5},{2}}
 => [[1,3,4,5],[2]]
 => 1
{{1,3,4},{2},{5}}
 => [[1,3,4],[2],[5]]
 => 2
{{1,3,5},{2},{4}}
 => [[1,3,5],[2],[4]]
 => 2
{{1,3},{2},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 3
{{1},{2,3,4,5}}
 => [[1,3,4,5],[2]]
 => 1
{{1},{2,3,4},{5}}
 => [[1,3,4],[2],[5]]
 => 2
{{1},{2,3,5},{4}}
 => [[1,3,5],[2],[4]]
 => 2
{{1},{2,3},{4},{5}}
 => [[1,3],[2],[4],[5]]
 => 3
{{1,4,5},{2},{3}}
 => [[1,4,5],[2],[3]]
 => 2
{{1,4},{2},{3},{5}}
 => [[1,4],[2],[3],[5]]
 => 3
{{1},{2,4,5},{3}}
 => [[1,4,5],[2],[3]]
 => 2
{{1},{2,4},{3},{5}}
 => [[1,4],[2],[3],[5]]
 => 3
{{1},{2},{3,4,5}}
 => [[1,4,5],[2],[3]]
 => 2
{{1},{2},{3,4},{5}}
 => [[1,4],[2],[3],[5]]
 => 3
{{1,5},{2},{3},{4}}
 => [[1,5],[2],[3],[4]]
 => 3
{{1},{2,5},{3},{4}}
 => [[1,5],[2],[3],[4]]
 => 3
{{1},{2},{3,5},{4}}
 => [[1,5],[2],[3],[4]]
 => 3
{{1},{2},{3},{4,5}}
 => [[1,5],[2],[3],[4]]
 => 3
{{1},{2},{3},{4},{5}}
 => [[1],[2],[3],[4],[5]]
 => 4
{{1,2,3,4,5,6}}
 => [[1,2,3,4,5,6]]
 => 0
{{1,2,3,4,5},{6}}
 => [[1,2,3,4,5],[6]]
 => 1
{{1,2,3,4,6},{5}}
 => [[1,2,3,4,6],[5]]
 => 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
(load all 156 compositions to match this statistic)
Mapping
Mp00221: Set partitions —conjugate⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1},{2}}
 => 0
{{1},{2}}
 => {{1,2}}
 => 1
{{1,2,3}}
 => {{1},{2},{3}}
 => 0
{{1,2},{3}}
 => {{1,2},{3}}
 => 1
{{1,3},{2}}
 => {{1},{2,3}}
 => 1
{{1},{2,3}}
 => {{1,3},{2}}
 => 1
{{1},{2},{3}}
 => {{1,2,3}}
 => 2
{{1,2,3,4}}
 => {{1},{2},{3},{4}}
 => 0
{{1,2,3},{4}}
 => {{1,2},{3},{4}}
 => 1
{{1,2,4},{3}}
 => {{1},{2,3},{4}}
 => 1
{{1,2},{3},{4}}
 => {{1,2,3},{4}}
 => 2
{{1,3,4},{2}}
 => {{1},{2},{3,4}}
 => 1
{{1,3},{2},{4}}
 => {{1,2},{3,4}}
 => 2
{{1},{2,3,4}}
 => {{1,4},{2},{3}}
 => 1
{{1},{2,3},{4}}
 => {{1,2,4},{3}}
 => 2
{{1,4},{2},{3}}
 => {{1},{2,3,4}}
 => 2
{{1},{2,4},{3}}
 => {{1,4},{2,3}}
 => 2
{{1},{2},{3,4}}
 => {{1,3,4},{2}}
 => 2
{{1},{2},{3},{4}}
 => {{1,2,3,4}}
 => 3
{{1,2,3,4,5}}
 => {{1},{2},{3},{4},{5}}
 => 0
{{1,2,3,4},{5}}
 => {{1,2},{3},{4},{5}}
 => 1
{{1,2,3,5},{4}}
 => {{1},{2,3},{4},{5}}
 => 1
{{1,2,3},{4},{5}}
 => {{1,2,3},{4},{5}}
 => 2
{{1,2,4,5},{3}}
 => {{1},{2},{3,4},{5}}
 => 1
{{1,2,4},{3},{5}}
 => {{1,2},{3,4},{5}}
 => 2
{{1,2,5},{3},{4}}
 => {{1},{2,3,4},{5}}
 => 2
{{1,2},{3},{4},{5}}
 => {{1,2,3,4},{5}}
 => 3
{{1,3,4,5},{2}}
 => {{1},{2},{3},{4,5}}
 => 1
{{1,3,4},{2},{5}}
 => {{1,2},{3},{4,5}}
 => 2
{{1,3,5},{2},{4}}
 => {{1},{2,3},{4,5}}
 => 2
{{1,3},{2},{4},{5}}
 => {{1,2,3},{4,5}}
 => 3
{{1},{2,3,4,5}}
 => {{1,5},{2},{3},{4}}
 => 1
{{1},{2,3,4},{5}}
 => {{1,2,5},{3},{4}}
 => 2
{{1},{2,3,5},{4}}
 => {{1,5},{2,3},{4}}
 => 2
{{1},{2,3},{4},{5}}
 => {{1,2,3,5},{4}}
 => 3
{{1,4,5},{2},{3}}
 => {{1},{2},{3,4,5}}
 => 2
{{1,4},{2},{3},{5}}
 => {{1,2},{3,4,5}}
 => 3
{{1},{2,4,5},{3}}
 => {{1,5},{2},{3,4}}
 => 2
{{1},{2,4},{3},{5}}
 => {{1,2,5},{3,4}}
 => 3
{{1},{2},{3,4,5}}
 => {{1,4,5},{2},{3}}
 => 2
{{1},{2},{3,4},{5}}
 => {{1,2,4,5},{3}}
 => 3
{{1,5},{2},{3},{4}}
 => {{1},{2,3,4,5}}
 => 3
{{1},{2,5},{3},{4}}
 => {{1,5},{2,3,4}}
 => 3
{{1},{2},{3,5},{4}}
 => {{1,4,5},{2,3}}
 => 3
{{1},{2},{3},{4,5}}
 => {{1,3,4,5},{2}}
 => 3
{{1},{2},{3},{4},{5}}
 => {{1,2,3,4,5}}
 => 4
{{1,2,3,4,5,6}}
 => {{1},{2},{3},{4},{5},{6}}
 => 0
{{1,2,3,4,5},{6}}
 => {{1,2},{3},{4},{5},{6}}
 => 1
{{1,2,3,4,6},{5}}
 => {{1},{2,3},{4},{5},{6}}
 => 1
Description
The rank of the set partition. This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St001176
(load all 5 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => 0
{{1,2}}
 => [2]
 => 0
{{1},{2}}
 => [1,1]
 => 1
{{1,2,3}}
 => [3]
 => 0
{{1,2},{3}}
 => [2,1]
 => 1
{{1,3},{2}}
 => [2,1]
 => 1
{{1},{2,3}}
 => [2,1]
 => 1
{{1},{2},{3}}
 => [1,1,1]
 => 2
{{1,2,3,4}}
 => [4]
 => 0
{{1,2,3},{4}}
 => [3,1]
 => 1
{{1,2,4},{3}}
 => [3,1]
 => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => 2
{{1,3,4},{2}}
 => [3,1]
 => 1
{{1,3},{2},{4}}
 => [2,1,1]
 => 2
{{1},{2,3,4}}
 => [3,1]
 => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => 2
{{1,4},{2},{3}}
 => [2,1,1]
 => 2
{{1},{2,4},{3}}
 => [2,1,1]
 => 2
{{1},{2},{3,4}}
 => [2,1,1]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 3
{{1,2,3,4,5}}
 => [5]
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => 2
{{1,2,4,5},{3}}
 => [4,1]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => 3
{{1,3,4,5},{2}}
 => [4,1]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => 3
{{1},{2,3,4,5}}
 => [4,1]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => 2
{{1},{2,3,5},{4}}
 => [3,1,1]
 => 2
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => 3
{{1},{2,4,5},{3}}
 => [3,1,1]
 => 2
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => 3
{{1},{2},{3,4,5}}
 => [3,1,1]
 => 2
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => 3
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => 3
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 3
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 4
{{1,2,3,4,5,6}}
 => [6]
 => 0
{{1,2,3,4,5},{6}}
 => [5,1]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => 1
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000010
(load all 63 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => 1 = 0 + 1
{{1,2}}
 => [2]
 => 1 = 0 + 1
{{1},{2}}
 => [1,1]
 => 2 = 1 + 1
{{1,2,3}}
 => [3]
 => 1 = 0 + 1
{{1,2},{3}}
 => [2,1]
 => 2 = 1 + 1
{{1,3},{2}}
 => [2,1]
 => 2 = 1 + 1
{{1},{2,3}}
 => [2,1]
 => 2 = 1 + 1
{{1},{2},{3}}
 => [1,1,1]
 => 3 = 2 + 1
{{1,2,3,4}}
 => [4]
 => 1 = 0 + 1
{{1,2,3},{4}}
 => [3,1]
 => 2 = 1 + 1
{{1,2,4},{3}}
 => [3,1]
 => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,1]
 => 3 = 2 + 1
{{1,3,4},{2}}
 => [3,1]
 => 2 = 1 + 1
{{1,3},{2},{4}}
 => [2,1,1]
 => 3 = 2 + 1
{{1},{2,3,4}}
 => [3,1]
 => 2 = 1 + 1
{{1},{2,3},{4}}
 => [2,1,1]
 => 3 = 2 + 1
{{1,4},{2},{3}}
 => [2,1,1]
 => 3 = 2 + 1
{{1},{2,4},{3}}
 => [2,1,1]
 => 3 = 2 + 1
{{1},{2},{3,4}}
 => [2,1,1]
 => 3 = 2 + 1
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => 4 = 3 + 1
{{1,2,3,4,5}}
 => [5]
 => 1 = 0 + 1
{{1,2,3,4},{5}}
 => [4,1]
 => 2 = 1 + 1
{{1,2,3,5},{4}}
 => [4,1]
 => 2 = 1 + 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,2,4,5},{3}}
 => [4,1]
 => 2 = 1 + 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,2,5},{3},{4}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1,3,4,5},{2}}
 => [4,1]
 => 2 = 1 + 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,3,5},{2},{4}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2,3,4,5}}
 => [4,1]
 => 2 = 1 + 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => 3 = 2 + 1
{{1},{2,3,5},{4}}
 => [3,1,1]
 => 3 = 2 + 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1,4,5},{2},{3}}
 => [3,1,1]
 => 3 = 2 + 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2,4,5},{3}}
 => [3,1,1]
 => 3 = 2 + 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2},{3,4,5}}
 => [3,1,1]
 => 3 = 2 + 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => 4 = 3 + 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => 5 = 4 + 1
{{1,2,3,4,5,6}}
 => [6]
 => 1 = 0 + 1
{{1,2,3,4,5},{6}}
 => [5,1]
 => 2 = 1 + 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000012
(load all 4 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => [1,0]
 => 0
{{1,2}}
 => [2]
 => [1,0,1,0]
 => 0
{{1},{2}}
 => [1,1]
 => [1,1,0,0]
 => 1
{{1,2,3}}
 => [3]
 => [1,0,1,0,1,0]
 => 0
{{1,2},{3}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1,3},{2}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1},{2,3}}
 => [2,1]
 => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1]
 => [1,1,0,1,0,0]
 => 2
{{1,2,3,4}}
 => [4]
 => [1,0,1,0,1,0,1,0]
 => 0
{{1,2,3},{4}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1,3,4},{2}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1},{2,3,4}}
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => 3
{{1,2,3,4,5}}
 => [5]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,2,4,5},{3}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1,3,4,5},{2}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2,3,4,5}}
 => [4,1]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
{{1,2,3,4,5,6}}
 => [6]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 0
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => 1
Description
The area of a Dyck path. This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic. 1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$. 2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$ 3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000053
(load all 20 compositions to match this statistic)
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => 0
{{1,2}}
 => [2] => [1,1,0,0]
 => 0
{{1},{2}}
 => [1,1] => [1,0,1,0]
 => 1
{{1,2,3}}
 => [3] => [1,1,1,0,0,0]
 => 0
{{1,2},{3}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1,3},{2}}
 => [2,1] => [1,1,0,0,1,0]
 => 1
{{1},{2,3}}
 => [1,2] => [1,0,1,1,0,0]
 => 1
{{1},{2},{3}}
 => [1,1,1] => [1,0,1,0,1,0]
 => 2
{{1,2,3,4}}
 => [4] => [1,1,1,1,0,0,0,0]
 => 0
{{1,2,3},{4}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,2,4},{3}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1,3,4},{2}}
 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
{{1,3},{2},{4}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,3,4}}
 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1,4},{2},{3}}
 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
{{1},{2,4},{3}}
 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
{{1},{2},{3,4}}
 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,3,5},{4}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,4,5},{3}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1,3,4,5},{2}}
 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1},{2,3,4,5}}
 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1},{2,3,4},{5}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
{{1},{2,3,5},{4}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1},{2,4,5},{3}}
 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
{{1},{2,4},{3},{5}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1},{2},{3,4,5}}
 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
{{1},{2,5},{3},{4}}
 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3
{{1},{2},{3,5},{4}}
 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 4
{{1,2,3,4,5,6}}
 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 0
{{1,2,3,4,5},{6}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 1
Description
The number of valleys of the Dyck path.
Matching statistic: St000148
(load all 3 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000148: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => []
 => 0
{{1,2}}
 => [2]
 => []
 => 0
{{1},{2}}
 => [1,1]
 => [1]
 => 1
{{1,2,3}}
 => [3]
 => []
 => 0
{{1,2},{3}}
 => [2,1]
 => [1]
 => 1
{{1,3},{2}}
 => [2,1]
 => [1]
 => 1
{{1},{2,3}}
 => [2,1]
 => [1]
 => 1
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => 2
{{1,2,3,4}}
 => [4]
 => []
 => 0
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => 1
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => 1
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => 3
{{1,2,3,4,5}}
 => [5]
 => []
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
{{1,2,3,4,5,6}}
 => [6]
 => []
 => 0
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => 1
Description
The number of odd parts of a partition.
Matching statistic: St000160
(load all 3 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000160: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => []
 => 0
{{1,2}}
 => [2]
 => []
 => 0
{{1},{2}}
 => [1,1]
 => [1]
 => 1
{{1,2,3}}
 => [3]
 => []
 => 0
{{1,2},{3}}
 => [2,1]
 => [1]
 => 1
{{1,3},{2}}
 => [2,1]
 => [1]
 => 1
{{1},{2,3}}
 => [2,1]
 => [1]
 => 1
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => 2
{{1,2,3,4}}
 => [4]
 => []
 => 0
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => 1
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => 1
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => 3
{{1,2,3,4,5}}
 => [5]
 => []
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
{{1,2,3,4,5,6}}
 => [6]
 => []
 => 0
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => 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: St000228
(load all 8 compositions to match this statistic)
Mapping
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1]
 => []
 => 0
{{1,2}}
 => [2]
 => []
 => 0
{{1},{2}}
 => [1,1]
 => [1]
 => 1
{{1,2,3}}
 => [3]
 => []
 => 0
{{1,2},{3}}
 => [2,1]
 => [1]
 => 1
{{1,3},{2}}
 => [2,1]
 => [1]
 => 1
{{1},{2,3}}
 => [2,1]
 => [1]
 => 1
{{1},{2},{3}}
 => [1,1,1]
 => [1,1]
 => 2
{{1,2,3,4}}
 => [4]
 => []
 => 0
{{1,2,3},{4}}
 => [3,1]
 => [1]
 => 1
{{1,2,4},{3}}
 => [3,1]
 => [1]
 => 1
{{1,2},{3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,3,4},{2}}
 => [3,1]
 => [1]
 => 1
{{1,3},{2},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,3,4}}
 => [3,1]
 => [1]
 => 1
{{1},{2,3},{4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4}}
 => [2,1,1]
 => [1,1]
 => 2
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => 3
{{1,2,3,4,5}}
 => [5]
 => []
 => 0
{{1,2,3,4},{5}}
 => [4,1]
 => [1]
 => 1
{{1,2,3,5},{4}}
 => [4,1]
 => [1]
 => 1
{{1,2,3},{4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,4,5},{3}}
 => [4,1]
 => [1]
 => 1
{{1,2,4},{3},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2,5},{3},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,3,4,5},{2}}
 => [4,1]
 => [1]
 => 1
{{1,3,4},{2},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3,5},{2},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,3,4,5}}
 => [4,1]
 => [1]
 => 1
{{1},{2,3,4},{5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3,5},{4}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,4,5},{2},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,4,5},{3}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,4,5}}
 => [3,1,1]
 => [1,1]
 => 2
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
{{1,2,3,4,5,6}}
 => [6]
 => []
 => 0
{{1,2,3,4,5},{6}}
 => [5,1]
 => [1]
 => 1
{{1,2,3,4,6},{5}}
 => [5,1]
 => [1]
 => 1
Description
The size of a partition. This statistic is the constant statistic of the level sets.
The following 438 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000475The number of parts equal to 1 in a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000867The sum of the hook lengths in the first row of an integer partition. St001127The sum of the squares of the parts of a partition. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000011The number of touch points (or returns) of a Dyck path. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000063The number of linear extensions of a certain poset defined for an integer partition. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000108The number of partitions contained in the given partition. St000147The largest part of an integer partition. St000288The number of ones in a binary word. St000378The diagonal inversion number of an integer partition. St000507The number of ascents of a standard tableau. St000532The total number of rook placements on a Ferrers board. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. 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: St001372The length of a longest cyclic run of ones of a binary word. St001389The number of partitions of the same length below the given integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001494The Alon-Tarsi number of a graph. St001581The achromatic number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000013The height of a Dyck path. St000019The cardinality of the support of a permutation. St000024The number of double up and double down steps of a Dyck path. St000041The number of nestings of a perfect matching. St000141The maximum drop size of a permutation. St000214The number of adjacencies of a permutation. St000234The number of global ascents of a permutation. St000272The treewidth of a graph. St000293The number of inversions of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000362The size of a minimal vertex cover of a graph. St000377The dinv defect of an integer partition. St000392The length of the longest run of ones in a binary word. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000662The staircase size of the code of a permutation. St000778The metric dimension of a graph. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000921The number of internal inversions of a binary word. St000992The alternating sum of the parts of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St001034The area of the parallelogram polyomino associated with the Dyck path. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001090The number of pop-stack-sorts needed to sort a permutation. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero 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. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001777The number of weak descents in an integer composition. St000007The number of saliances of the permutation. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000054The first entry of the permutation. St000058The order of a permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000167The number of leaves of an ordered tree. St000172The Grundy number of a graph. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000393The number of strictly increasing runs in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000808The number of up steps of the associated bargraph. St000839The largest opener of a set partition. St000883The number of longest increasing subsequences of a permutation. St000982The length of the longest constant subword. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001050The number of terminal closers of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001809The index of the step at the first peak of maximal height in a Dyck path. St001814The number of partitions interlacing the given partition. St001883The mutual visibility number of a graph. St000439The position of the first down step of a Dyck path. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000806The semiperimeter of the associated bargraph. St000925The number of topologically connected components of a set partition. St000984The number of boxes below precisely one peak. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000209Maximum difference of elements in cycles. St000297The number of leading ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000442The maximal area to the right of an up step of a Dyck path. St000446The disorder of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000503The maximal difference between two elements in a common block. St000728The dimension of a set partition. St000730The maximal arc length of a set partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000947The major index east count of a Dyck path. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001726The number of visible inversions of a permutation. St000144The pyramid weight of the Dyck path. St000420The number of Dyck paths that are weakly above a Dyck path. St000444The length of the maximal rise of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001808The box weight or horizontal decoration of a Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001190Number of simple modules with projective dimension at most 4 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 St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000809The reduced reflection length of the permutation. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St000485The length of the longest cycle of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001933The largest multiplicity of a part in an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St000290The major index of a binary word. St000296The length of the symmetric border of a binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000627The exponent of a binary word. St000667The greatest common divisor of the parts of the partition. St000878The number of ones minus the number of zeros of a binary word. St000922The minimal number such that all substrings of this length are unique. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001884The number of borders of a binary word. St000145The Dyson rank of a partition. St000294The number of distinct factors of a binary word. St000295The length of the border of a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000518The number of distinct subsequences in a binary word. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St001091The number of parts in an integer partition whose next smaller part has the same size. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001658The total number of rook placements on a Ferrers board. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001955The number of natural descents for set-valued two row standard Young tableaux. St000438The position of the last up step in a Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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. 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. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. 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. St001959The product of the heights of the peaks of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000308The height of the tree associated to a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. 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. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001213The number of indecomposable modules in the corresponding Nakayama algebra that have vanishing first Ext-group with the regular module. St001259The vector space dimension of the double dual of D(A) in the corresponding Nakayama algebra. St000006The dinv of a Dyck path. St000245The number of ascents of a permutation. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001962The proper pathwidth of a graph. St000822The Hadwiger number of the graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000306The bounce count of a Dyck path. St000741The Colin de Verdière graph invariant. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000339The maf index of a permutation. St000653The last descent of a permutation. St001497The position of the largest weak excedence 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. St000993The multiplicity of the largest part of an integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St000477The weight of a partition according to Alladi. St000770The major index of an integer partition when read from bottom to top. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000744The length of the path to the largest entry in a standard Young tableau. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000932The number of occurrences of the pattern UDU in a Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001812The biclique partition number of a graph. St000672The number of minimal elements in Bruhat order not less than the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000153The number of adjacent cycles of a permutation. St001644The dimension of a graph. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. 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. St000161The sum of the sizes of the right subtrees of a binary tree. St001298The number of repeated entries in the Lehmer code of a permutation. St000654The first descent of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St000937The number of positive values of the symmetric group character corresponding to the 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 $eA$. St000553The number of blocks of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000783The side length of the largest staircase partition fitting into a partition. St000018The number of inversions of a permutation. St000237The number of small exceedances. St000159The number of distinct parts of the integer partition. St000028The number of stack-sorts needed to sort a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000292The number of ascents of a binary word. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000843The decomposition number of a perfect matching. St001432The order dimension of the partition. St001427The number of descents of a signed permutation. St000246The number of non-inversions of a permutation. St000651The maximal size of a rise in a permutation. St000740The last entry of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000542The number of left-to-right-minima of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000501The size of the first part in the decomposition of a permutation. St000021The number of descents of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000325The width of the tree associated to a permutation. St000990The first ascent of a permutation. 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000005The bounce statistic of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000057The Shynar inversion number of a standard tableau. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000171The degree of the graph. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000316The number of non-left-to-right-maxima of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. 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. St001357The maximal degree of a regular spanning subgraph of a graph. St001391The disjunction number of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St000056The decomposition (or block) number of a permutation. St000084The number of subtrees. St000087The number of induced subgraphs. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000286The number of connected components of the complement of a graph. St000328The maximum number of child nodes in a tree. St000363The number of minimal vertex covers of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000926The clique-coclique number of a graph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000991The number of right-to-left minima of a permutation. St001110The 3-dynamic chromatic number of a graph. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001316The domatic number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001530The depth of a Dyck path. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000216The absolute length of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000061The number of nodes on the left branch of a binary tree. St000082The number of elements smaller than a binary tree in Tamari order. St001346The number of parking functions that give the same permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001430The number of positive entries in a signed permutation. St000317The cycle descent number of a permutation. St000358The number of occurrences of the pattern 31-2. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St001668The number of points of the poset minus the width of the poset. St001948The number of augmented double ascents of a permutation. St001863The number of weak excedances of a signed permutation. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. 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. St001773The number of minimal elements in Bruhat order not less than the signed permutation.