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Your data matches 278 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ 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] => [1]
=> 1
[2,1] => [2]
=> 1
[1,3,2] => [2,1]
=> 2
[2,3,1] => [2,1]
=> 2
[3,2,1] => [3]
=> 1
[1,2,4,3] => [2,1,1]
=> 3
[1,3,4,2] => [2,1,1]
=> 3
[1,4,3,2] => [3,1]
=> 2
[2,1,4,3] => [2,2]
=> 2
[2,3,4,1] => [2,1,1]
=> 3
[2,4,3,1] => [3,1]
=> 2
[3,1,4,2] => [2,2]
=> 2
[3,2,4,1] => [3,1]
=> 2
[3,4,2,1] => [3,1]
=> 2
[4,1,3,2] => [3,1]
=> 2
[4,2,3,1] => [3,1]
=> 2
[4,3,2,1] => [4]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> 4
[1,2,4,5,3] => [2,1,1,1]
=> 4
[1,2,5,4,3] => [3,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> 4
[1,3,5,4,2] => [3,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> 3
[1,5,3,4,2] => [3,1,1]
=> 3
[1,5,4,3,2] => [4,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> 3
[2,1,4,5,3] => [2,2,1]
=> 3
[2,1,5,4,3] => [3,2]
=> 2
[2,3,1,5,4] => [2,2,1]
=> 3
[2,3,4,5,1] => [2,1,1,1]
=> 4
[2,3,5,4,1] => [3,1,1]
=> 3
[2,4,1,5,3] => [2,2,1]
=> 3
[2,4,3,5,1] => [3,1,1]
=> 3
[2,4,5,3,1] => [3,1,1]
=> 3
[2,5,1,4,3] => [3,1,1]
=> 3
[2,5,3,4,1] => [3,1,1]
=> 3
[2,5,4,3,1] => [4,1]
=> 2
[3,1,2,5,4] => [2,2,1]
=> 3
[3,1,4,5,2] => [2,2,1]
=> 3
[3,1,5,4,2] => [3,2]
=> 2
[3,2,1,5,4] => [3,2]
=> 2
[3,2,4,5,1] => [3,1,1]
=> 3
[3,2,5,4,1] => [3,2]
=> 2
[3,4,1,5,2] => [2,2,1]
=> 3
[3,4,2,5,1] => [3,1,1]
=> 3
[3,4,5,2,1] => [3,1,1]
=> 3
Description
The length of the partition.
Matching statistic: St000507
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> 1
[2,1] => [[1],[2]]
=> 1
[1,3,2] => [[1,2],[3]]
=> 2
[2,3,1] => [[1,2],[3]]
=> 2
[3,2,1] => [[1],[2],[3]]
=> 1
[1,2,4,3] => [[1,2,3],[4]]
=> 3
[1,3,4,2] => [[1,2,3],[4]]
=> 3
[1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> 3
[2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> 4
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> 4
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 3
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> 3
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> 4
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 3
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> 3
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> 3
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 3
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> 3
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 3
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> 3
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> 3
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 2
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> 3
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 3
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> 3
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 3
[2,4,5,3,1] => [[1,2,3],[4],[5]]
=> 3
[2,5,1,4,3] => [[1,2],[3,4],[5]]
=> 3
[2,5,3,4,1] => [[1,2,4],[3],[5]]
=> 3
[2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 2
[3,1,2,5,4] => [[1,3,4],[2,5]]
=> 3
[3,1,4,5,2] => [[1,3,4],[2,5]]
=> 3
[3,1,5,4,2] => [[1,3],[2,4],[5]]
=> 2
[3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 2
[3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 3
[3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 2
[3,4,1,5,2] => [[1,2,4],[3,5]]
=> 3
[3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 3
[3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 3
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).
Matching statistic: St000093
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000105
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => {{1}}
=> 1
[2,1] => [2,1] => {{1,2}}
=> 1
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,1] => [2,3,1] => {{1,2,3}}
=> 1
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[2,4,3,1] => [3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,1,4,2] => [4,3,1,2] => {{1,4},{2,3}}
=> 2
[3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[3,4,2,1] => [2,4,3,1] => {{1,2,4},{3}}
=> 2
[4,1,3,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[4,2,3,1] => [3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,3,2,1] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,2,3,5,4] => [1,2,3,5,4] => {{1},{2},{3},{4,5}}
=> 4
[1,2,4,5,3] => [1,2,5,4,3] => {{1},{2},{3,5},{4}}
=> 4
[1,2,5,4,3] => [1,2,4,5,3] => {{1},{2},{3,4,5}}
=> 3
[1,3,2,5,4] => [1,3,2,5,4] => {{1},{2,3},{4,5}}
=> 3
[1,3,4,5,2] => [1,5,3,4,2] => {{1},{2,5},{3},{4}}
=> 4
[1,3,5,4,2] => [1,4,3,5,2] => {{1},{2,4,5},{3}}
=> 3
[1,4,2,5,3] => [1,5,4,2,3] => {{1},{2,5},{3,4}}
=> 3
[1,4,3,5,2] => [1,5,4,3,2] => {{1},{2,5},{3,4}}
=> 3
[1,4,5,3,2] => [1,3,5,4,2] => {{1},{2,3,5},{4}}
=> 3
[1,5,2,4,3] => [1,4,5,2,3] => {{1},{2,4},{3,5}}
=> 3
[1,5,3,4,2] => [1,4,5,3,2] => {{1},{2,4},{3,5}}
=> 3
[1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[2,1,3,5,4] => [2,1,3,5,4] => {{1,2},{3},{4,5}}
=> 3
[2,1,4,5,3] => [2,1,5,4,3] => {{1,2},{3,5},{4}}
=> 3
[2,1,5,4,3] => [2,1,4,5,3] => {{1,2},{3,4,5}}
=> 2
[2,3,1,5,4] => [3,2,1,5,4] => {{1,3},{2},{4,5}}
=> 3
[2,3,4,5,1] => [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 4
[2,3,5,4,1] => [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 3
[2,4,1,5,3] => [5,2,4,1,3] => {{1,5},{2},{3,4}}
=> 3
[2,4,3,5,1] => [5,2,4,3,1] => {{1,5},{2},{3,4}}
=> 3
[2,4,5,3,1] => [3,2,5,4,1] => {{1,3,5},{2},{4}}
=> 3
[2,5,1,4,3] => [4,2,5,1,3] => {{1,4},{2},{3,5}}
=> 3
[2,5,3,4,1] => [4,2,5,3,1] => {{1,4},{2},{3,5}}
=> 3
[2,5,4,3,1] => [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 2
[3,1,2,5,4] => [3,1,2,5,4] => {{1,3},{2},{4,5}}
=> 3
[3,1,4,5,2] => [5,3,1,4,2] => {{1,5},{2,3},{4}}
=> 3
[3,1,5,4,2] => [4,3,1,5,2] => {{1,4,5},{2,3}}
=> 2
[3,2,1,5,4] => [2,3,1,5,4] => {{1,2,3},{4,5}}
=> 2
[3,2,4,5,1] => [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 3
[3,2,5,4,1] => [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 2
[3,4,1,5,2] => [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> 3
[3,4,2,5,1] => [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 3
[3,4,5,2,1] => [2,5,3,4,1] => {{1,2,5},{3},{4}}
=> 3
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: St000147
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1
[2,1] => [2]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> 2
[2,3,1] => [2,1]
=> [2,1]
=> 2
[3,2,1] => [3]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 3
[1,3,4,2] => [2,1,1]
=> [3,1]
=> 3
[1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,4,1] => [2,1,1]
=> [3,1]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> [2,2]
=> 2
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 4
[1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [3,2]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> 3
[1,5,3,4,2] => [3,1,1]
=> [3,1,1]
=> 3
[1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [3,2]
=> 3
[2,1,4,5,3] => [2,2,1]
=> [3,2]
=> 3
[2,1,5,4,3] => [3,2]
=> [2,2,1]
=> 2
[2,3,1,5,4] => [2,2,1]
=> [3,2]
=> 3
[2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> 4
[2,3,5,4,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,4,1,5,3] => [2,2,1]
=> [3,2]
=> 3
[2,4,3,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,4,5,3,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,5,1,4,3] => [3,1,1]
=> [3,1,1]
=> 3
[2,5,3,4,1] => [3,1,1]
=> [3,1,1]
=> 3
[2,5,4,3,1] => [4,1]
=> [2,1,1,1]
=> 2
[3,1,2,5,4] => [2,2,1]
=> [3,2]
=> 3
[3,1,4,5,2] => [2,2,1]
=> [3,2]
=> 3
[3,1,5,4,2] => [3,2]
=> [2,2,1]
=> 2
[3,2,1,5,4] => [3,2]
=> [2,2,1]
=> 2
[3,2,4,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[3,2,5,4,1] => [3,2]
=> [2,2,1]
=> 2
[3,4,1,5,2] => [2,2,1]
=> [3,2]
=> 3
[3,4,2,5,1] => [3,1,1]
=> [3,1,1]
=> 3
[3,4,5,2,1] => [3,1,1]
=> [3,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 149 compositions to match this statistic)
(load all 149 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1
[2,1] => [2]
=> 100 => 1
[1,3,2] => [2,1]
=> 1010 => 2
[2,3,1] => [2,1]
=> 1010 => 2
[3,2,1] => [3]
=> 1000 => 1
[1,2,4,3] => [2,1,1]
=> 10110 => 3
[1,3,4,2] => [2,1,1]
=> 10110 => 3
[1,4,3,2] => [3,1]
=> 10010 => 2
[2,1,4,3] => [2,2]
=> 1100 => 2
[2,3,4,1] => [2,1,1]
=> 10110 => 3
[2,4,3,1] => [3,1]
=> 10010 => 2
[3,1,4,2] => [2,2]
=> 1100 => 2
[3,2,4,1] => [3,1]
=> 10010 => 2
[3,4,2,1] => [3,1]
=> 10010 => 2
[4,1,3,2] => [3,1]
=> 10010 => 2
[4,2,3,1] => [3,1]
=> 10010 => 2
[4,3,2,1] => [4]
=> 10000 => 1
[1,2,3,5,4] => [2,1,1,1]
=> 101110 => 4
[1,2,4,5,3] => [2,1,1,1]
=> 101110 => 4
[1,2,5,4,3] => [3,1,1]
=> 100110 => 3
[1,3,2,5,4] => [2,2,1]
=> 11010 => 3
[1,3,4,5,2] => [2,1,1,1]
=> 101110 => 4
[1,3,5,4,2] => [3,1,1]
=> 100110 => 3
[1,4,2,5,3] => [2,2,1]
=> 11010 => 3
[1,4,3,5,2] => [3,1,1]
=> 100110 => 3
[1,4,5,3,2] => [3,1,1]
=> 100110 => 3
[1,5,2,4,3] => [3,1,1]
=> 100110 => 3
[1,5,3,4,2] => [3,1,1]
=> 100110 => 3
[1,5,4,3,2] => [4,1]
=> 100010 => 2
[2,1,3,5,4] => [2,2,1]
=> 11010 => 3
[2,1,4,5,3] => [2,2,1]
=> 11010 => 3
[2,1,5,4,3] => [3,2]
=> 10100 => 2
[2,3,1,5,4] => [2,2,1]
=> 11010 => 3
[2,3,4,5,1] => [2,1,1,1]
=> 101110 => 4
[2,3,5,4,1] => [3,1,1]
=> 100110 => 3
[2,4,1,5,3] => [2,2,1]
=> 11010 => 3
[2,4,3,5,1] => [3,1,1]
=> 100110 => 3
[2,4,5,3,1] => [3,1,1]
=> 100110 => 3
[2,5,1,4,3] => [3,1,1]
=> 100110 => 3
[2,5,3,4,1] => [3,1,1]
=> 100110 => 3
[2,5,4,3,1] => [4,1]
=> 100010 => 2
[3,1,2,5,4] => [2,2,1]
=> 11010 => 3
[3,1,4,5,2] => [2,2,1]
=> 11010 => 3
[3,1,5,4,2] => [3,2]
=> 10100 => 2
[3,2,1,5,4] => [3,2]
=> 10100 => 2
[3,2,4,5,1] => [3,1,1]
=> 100110 => 3
[3,2,5,4,1] => [3,2]
=> 10100 => 2
[3,4,1,5,2] => [2,2,1]
=> 11010 => 3
[3,4,2,5,1] => [3,1,1]
=> 100110 => 3
[3,4,5,2,1] => [3,1,1]
=> 100110 => 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ 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] => [1]
=> [1]
=> 1
[2,1] => [2]
=> [1,1]
=> 1
[1,3,2] => [2,1]
=> [3]
=> 2
[2,3,1] => [2,1]
=> [3]
=> 2
[3,2,1] => [3]
=> [1,1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [2,2]
=> 3
[1,3,4,2] => [2,1,1]
=> [2,2]
=> 3
[1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,4,1] => [2,1,1]
=> [2,2]
=> 3
[2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,4,2] => [2,2]
=> [4]
=> 2
[3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [4,1]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [4,1]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [4,1]
=> 3
[1,5,3,4,2] => [3,1,1]
=> [4,1]
=> 3
[1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[2,1,4,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[2,1,5,4,3] => [3,2]
=> [5]
=> 2
[2,3,1,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> 4
[2,3,5,4,1] => [3,1,1]
=> [4,1]
=> 3
[2,4,1,5,3] => [2,2,1]
=> [2,2,1]
=> 3
[2,4,3,5,1] => [3,1,1]
=> [4,1]
=> 3
[2,4,5,3,1] => [3,1,1]
=> [4,1]
=> 3
[2,5,1,4,3] => [3,1,1]
=> [4,1]
=> 3
[2,5,3,4,1] => [3,1,1]
=> [4,1]
=> 3
[2,5,4,3,1] => [4,1]
=> [2,1,1,1]
=> 2
[3,1,2,5,4] => [2,2,1]
=> [2,2,1]
=> 3
[3,1,4,5,2] => [2,2,1]
=> [2,2,1]
=> 3
[3,1,5,4,2] => [3,2]
=> [5]
=> 2
[3,2,1,5,4] => [3,2]
=> [5]
=> 2
[3,2,4,5,1] => [3,1,1]
=> [4,1]
=> 3
[3,2,5,4,1] => [3,2]
=> [5]
=> 2
[3,4,1,5,2] => [2,2,1]
=> [2,2,1]
=> 3
[3,4,2,5,1] => [3,1,1]
=> [4,1]
=> 3
[3,4,5,2,1] => [3,1,1]
=> [4,1]
=> 3
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 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> [1,0]
=> 1
[2,1] => [[.,.],.]
=> [1,1,0,0]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 2
[3,2,1] => [[[.,.],.],.]
=> [1,1,0,1,0,0]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,4,3,2] => [.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,4,3,1] => [[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 3
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> [1,1,0,1,1,0,0,0,1,0]
=> 3
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[3,4,5,2,1] => [[[.,[.,[.,.]]],.],.]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
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: St000733
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1
[2,1] => [2]
=> [[1,2]]
=> 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2
[2,3,1] => [2,1]
=> [[1,2],[3]]
=> 2
[3,2,1] => [3]
=> [[1,2,3]]
=> 1
[1,2,4,3] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,3,4,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[1,4,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2
[2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 2
[2,3,4,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[2,4,3,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[3,1,4,2] => [2,2]
=> [[1,2],[3,4]]
=> 2
[3,2,4,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[3,4,2,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,1,3,2] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,2,3,1] => [3,1]
=> [[1,2,3],[4]]
=> 2
[4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,2,4,5,3] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,2,5,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,3,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,3,4,5,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[1,3,5,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,2,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[1,4,3,5,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,4,5,3,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,5,2,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,5,3,4,2] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[1,5,4,3,2] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[2,1,3,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[2,1,4,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[2,1,5,4,3] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[2,3,1,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[2,3,4,5,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[2,3,5,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,4,1,5,3] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[2,4,3,5,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,4,5,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,5,1,4,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,5,3,4,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[2,5,4,3,1] => [4,1]
=> [[1,2,3,4],[5]]
=> 2
[3,1,2,5,4] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[3,1,4,5,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[3,1,5,4,2] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[3,2,1,5,4] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[3,2,4,5,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2,5,4,1] => [3,2]
=> [[1,2,3],[4,5]]
=> 2
[3,4,1,5,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[3,4,2,5,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,4,5,2,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000786
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,3,4,5,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[1,3,5,4,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,3,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,4,5,3,2] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,3,4,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,5,4,3,2] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,1,3,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,4,5,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,1,5,4,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[2,3,1,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,3,4,5,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4
[2,3,5,4,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,1,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,3,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,4,5,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,1,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,3,4,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[2,5,4,3,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,1,2,5,4] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,4,5,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,1,5,4,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,1,5,4] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,2,4,5,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,2,5,4,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,5,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,2,5,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[3,4,5,2,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
The following 268 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000025The number of initial rises of a Dyck path. St000087The number of induced subgraphs. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000381The largest part of an integer composition. St000382The first part of an integer composition. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000722The number of different neighbourhoods in a graph. 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. St000808The number of up steps of the associated bargraph. St000822The Hadwiger number of the graph. St000926The clique-coclique number of a graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-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. 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:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001316The domatic number of a graph. St001330The hat guessing 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. St001389The number of partitions of the same length below the given integer partition. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. 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. St001670The connected partition 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. St001733The number of weak left to right maxima of a Dyck path. St001746The coalition number of a graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000089The absolute variation of a composition. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000211The rank of the set partition. St000225Difference between largest and smallest parts in a partition. St000234The number of global ascents of a permutation. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000272The treewidth 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. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. 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. St000439The position of the first down step of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St000691The number of changes of a binary word. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001176The size of a partition minus its first part. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001358The largest degree of a regular subgraph of a graph. St001372The length of a longest cyclic run of ones of a binary word. St001391The disjunction number of a graph. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000806The semiperimeter of the associated bargraph. St000325The width of the tree associated to a permutation. St000444The length of the maximal rise of a Dyck path. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000708The product of the parts of an integer partition. St000925The number of topologically connected components of a set partition. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001128The exponens consonantiae of a partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000248The number of anti-singletons of a set partition. St000316The number of non-left-to-right-maxima of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St000083The number of left oriented leafs of a binary tree except the first one. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St001812The biclique partition number of a graph. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000519The largest length of a factor maximising the subword complexity. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000653The last descent of a permutation. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000356The number of occurrences of the pattern 13-2. St001323The independence gap of a graph. St000006The dinv of a Dyck path. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000619The number of cyclic descents of a permutation. St000527The width of the poset. St000632The jump number of the poset. St000167The number of leaves of an ordered tree. St000015The number of peaks of a Dyck path. St001530The depth of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. 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. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000216The absolute length of a permutation. St000306The bounce count of a Dyck path. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. 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. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000159The number of distinct parts of the integer partition. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000308The height of the tree associated to a permutation. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St000141The maximum drop size of a permutation. St001427The number of descents of a signed permutation. St000214The number of adjacencies of a permutation. St000153The number of adjacent cycles of a permutation. St000332The positive inversions of an alternating sign matrix. St001152The number of pairs with even minimum in a perfect matching. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St000067The inversion number of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. 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. St000168The number of internal nodes of an ordered tree. 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. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000240The number of indices that are not small excedances. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima 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. 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. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. 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. St001180Number of indecomposable injective modules with projective dimension at most 1. 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 $eA$. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000061The number of nodes on the left branch of a binary tree. St001674The number of vertices of the largest induced star graph in the graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St000731The number of double exceedences of a permutation. St000746The number of pairs with odd minimum in a perfect matching. St001435The number of missing boxes in the first row. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001726The number of visible inversions of a permutation. St000039The number of crossings of a permutation. St000299The number of nonisomorphic vertex-induced subtrees. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001960The number of descents of a permutation minus one if its first entry is not one. 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 $eA$. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001720The minimal length of a chain of small intervals in a lattice.
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