Your data matches 144 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Mp00108: Permutations cycle typeInteger partitions
St001280: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 1 - 1
[1,2] => [1,1]
=> 0 = 1 - 1
[2,1] => [2]
=> 1 = 2 - 1
[1,2,3] => [1,1,1]
=> 0 = 1 - 1
[1,3,2] => [2,1]
=> 1 = 2 - 1
[2,1,3] => [2,1]
=> 1 = 2 - 1
[2,3,1] => [3]
=> 1 = 2 - 1
[3,1,2] => [3]
=> 1 = 2 - 1
[3,2,1] => [2,1]
=> 1 = 2 - 1
[1,2,3,4] => [1,1,1,1]
=> 0 = 1 - 1
[1,2,4,3] => [2,1,1]
=> 1 = 2 - 1
[1,3,2,4] => [2,1,1]
=> 1 = 2 - 1
[1,3,4,2] => [3,1]
=> 1 = 2 - 1
[1,4,2,3] => [3,1]
=> 1 = 2 - 1
[1,4,3,2] => [2,1,1]
=> 1 = 2 - 1
[2,1,3,4] => [2,1,1]
=> 1 = 2 - 1
[2,1,4,3] => [2,2]
=> 2 = 3 - 1
[2,3,1,4] => [3,1]
=> 1 = 2 - 1
[2,3,4,1] => [4]
=> 1 = 2 - 1
[2,4,1,3] => [4]
=> 1 = 2 - 1
[2,4,3,1] => [3,1]
=> 1 = 2 - 1
[3,1,2,4] => [3,1]
=> 1 = 2 - 1
[3,1,4,2] => [4]
=> 1 = 2 - 1
[3,2,1,4] => [2,1,1]
=> 1 = 2 - 1
[3,2,4,1] => [3,1]
=> 1 = 2 - 1
[3,4,1,2] => [2,2]
=> 2 = 3 - 1
[3,4,2,1] => [4]
=> 1 = 2 - 1
[4,1,2,3] => [4]
=> 1 = 2 - 1
[4,1,3,2] => [3,1]
=> 1 = 2 - 1
[4,2,1,3] => [3,1]
=> 1 = 2 - 1
[4,2,3,1] => [2,1,1]
=> 1 = 2 - 1
[4,3,1,2] => [4]
=> 1 = 2 - 1
[4,3,2,1] => [2,2]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,1,1,1,1]
=> 0 = 1 - 1
[1,2,3,5,4] => [2,1,1,1]
=> 1 = 2 - 1
[1,2,4,3,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,2,4,5,3] => [3,1,1]
=> 1 = 2 - 1
[1,2,5,3,4] => [3,1,1]
=> 1 = 2 - 1
[1,2,5,4,3] => [2,1,1,1]
=> 1 = 2 - 1
[1,3,2,4,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,3,2,5,4] => [2,2,1]
=> 2 = 3 - 1
[1,3,4,2,5] => [3,1,1]
=> 1 = 2 - 1
[1,3,4,5,2] => [4,1]
=> 1 = 2 - 1
[1,3,5,2,4] => [4,1]
=> 1 = 2 - 1
[1,3,5,4,2] => [3,1,1]
=> 1 = 2 - 1
[1,4,2,3,5] => [3,1,1]
=> 1 = 2 - 1
[1,4,2,5,3] => [4,1]
=> 1 = 2 - 1
[1,4,3,2,5] => [2,1,1,1]
=> 1 = 2 - 1
[1,4,3,5,2] => [3,1,1]
=> 1 = 2 - 1
[1,4,5,2,3] => [2,2,1]
=> 2 = 3 - 1
Description
The number of parts of an integer partition that are at least two.
Mp00108: Permutations cycle typeInteger partitions
Mp00308: Integer partitions Bulgarian solitaireInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1
[1,2] => [1,1]
=> [2]
=> 1
[2,1] => [2]
=> [1,1]
=> 2
[1,2,3] => [1,1,1]
=> [3]
=> 1
[1,3,2] => [2,1]
=> [2,1]
=> 2
[2,1,3] => [2,1]
=> [2,1]
=> 2
[2,3,1] => [3]
=> [2,1]
=> 2
[3,1,2] => [3]
=> [2,1]
=> 2
[3,2,1] => [2,1]
=> [2,1]
=> 2
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 1
[1,2,4,3] => [2,1,1]
=> [3,1]
=> 2
[1,3,2,4] => [2,1,1]
=> [3,1]
=> 2
[1,3,4,2] => [3,1]
=> [2,2]
=> 2
[1,4,2,3] => [3,1]
=> [2,2]
=> 2
[1,4,3,2] => [2,1,1]
=> [3,1]
=> 2
[2,1,3,4] => [2,1,1]
=> [3,1]
=> 2
[2,1,4,3] => [2,2]
=> [2,1,1]
=> 3
[2,3,1,4] => [3,1]
=> [2,2]
=> 2
[2,3,4,1] => [4]
=> [3,1]
=> 2
[2,4,1,3] => [4]
=> [3,1]
=> 2
[2,4,3,1] => [3,1]
=> [2,2]
=> 2
[3,1,2,4] => [3,1]
=> [2,2]
=> 2
[3,1,4,2] => [4]
=> [3,1]
=> 2
[3,2,1,4] => [2,1,1]
=> [3,1]
=> 2
[3,2,4,1] => [3,1]
=> [2,2]
=> 2
[3,4,1,2] => [2,2]
=> [2,1,1]
=> 3
[3,4,2,1] => [4]
=> [3,1]
=> 2
[4,1,2,3] => [4]
=> [3,1]
=> 2
[4,1,3,2] => [3,1]
=> [2,2]
=> 2
[4,2,1,3] => [3,1]
=> [2,2]
=> 2
[4,2,3,1] => [2,1,1]
=> [3,1]
=> 2
[4,3,1,2] => [4]
=> [3,1]
=> 2
[4,3,2,1] => [2,2]
=> [2,1,1]
=> 3
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> 2
[1,2,4,5,3] => [3,1,1]
=> [3,2]
=> 2
[1,2,5,3,4] => [3,1,1]
=> [3,2]
=> 2
[1,2,5,4,3] => [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> 2
[1,3,2,5,4] => [2,2,1]
=> [3,1,1]
=> 3
[1,3,4,2,5] => [3,1,1]
=> [3,2]
=> 2
[1,3,4,5,2] => [4,1]
=> [3,2]
=> 2
[1,3,5,2,4] => [4,1]
=> [3,2]
=> 2
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> 2
[1,4,2,3,5] => [3,1,1]
=> [3,2]
=> 2
[1,4,2,5,3] => [4,1]
=> [3,2]
=> 2
[1,4,3,2,5] => [2,1,1,1]
=> [4,1]
=> 2
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> 2
[1,4,5,2,3] => [2,2,1]
=> [3,1,1]
=> 3
Description
The length of the partition.
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [[1]]
=> 1
[1,2] => [1,2] => [[1,2]]
=> 2
[2,1] => [2,1] => [[1],[2]]
=> 1
[1,2,3] => [1,3,2] => [[1,2],[3]]
=> 2
[1,3,2] => [1,3,2] => [[1,2],[3]]
=> 2
[2,1,3] => [2,1,3] => [[1,3],[2]]
=> 2
[2,3,1] => [2,3,1] => [[1,2],[3]]
=> 2
[3,1,2] => [3,1,2] => [[1,3],[2]]
=> 2
[3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 1
[1,2,3,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,2,4,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,3,2,4] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,3,4,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,4,3,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[2,1,3,4] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,1,4,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[2,3,1,4] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[2,3,4,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[2,4,1,3] => [2,4,1,3] => [[1,2],[3,4]]
=> 3
[2,4,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[3,1,2,4] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,1,4,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[3,4,1,2] => [3,4,1,2] => [[1,2],[3,4]]
=> 3
[3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[4,1,2,3] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[4,1,3,2] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[4,2,1,3] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[4,2,3,1] => [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[4,3,1,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 2
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
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000093: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => ([],1)
=> 1
[1,2] => [1,2] => [2] => ([],2)
=> 2
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
=> 2
[2,3,1] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,1,2] => [1,2] => ([(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,2,4,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,2,4] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,3,4,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[2,4,3,1] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[1,2,3,4,5] => [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
[1,2,3,5,4] => [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
[1,2,4,3,5] => [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
[1,2,4,5,3] => [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
[1,2,5,3,4] => [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
[1,2,5,4,3] => [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
[1,3,2,4,5] => [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
[1,3,2,5,4] => [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
[1,3,4,2,5] => [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
[1,3,4,5,2] => [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
[1,3,5,2,4] => [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
[1,3,5,4,2] => [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
[1,4,2,3,5] => [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
[1,4,2,5,3] => [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
[1,4,3,2,5] => [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
[1,4,3,5,2] => [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
[1,4,5,2,3] => [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
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 $G$.
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00235: Permutations descent views to invisible inversion bottomsPermutations
Mp00240: Permutations weak exceedance partitionSet partitions
St000105: Set partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => {{1}}
=> 1
[1,2] => [1,2] => [1,2] => {{1},{2}}
=> 2
[2,1] => [2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,3,2] => [1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[2,1,3] => [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,3,1] => [2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,1,2] => [3,1,2] => [3,1,2] => {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => [2,3,1] => {{1,2,3}}
=> 1
[1,2,3,4] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,2,4,3] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,2,4] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[2,1,3,4] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 2
[2,3,1,4] => [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[2,3,4,1] => [2,4,3,1] => [3,2,4,1] => {{1,3,4},{2}}
=> 2
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3
[2,4,3,1] => [2,4,3,1] => [3,2,4,1] => {{1,3,4},{2}}
=> 2
[3,1,2,4] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,1,4,2] => [3,1,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2
[3,2,4,1] => [3,2,4,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 2
[3,4,1,2] => [3,4,1,2] => [4,1,3,2] => {{1,4},{2},{3}}
=> 3
[3,4,2,1] => [3,4,2,1] => [2,4,3,1] => {{1,2,4},{3}}
=> 2
[4,1,2,3] => [4,1,3,2] => [4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,1,3,2] => [4,1,3,2] => [4,3,1,2] => {{1,4},{2,3}}
=> 2
[4,2,1,3] => [4,2,1,3] => [2,4,1,3] => {{1,2,4},{3}}
=> 2
[4,2,3,1] => [4,2,3,1] => [3,4,2,1] => {{1,3},{2,4}}
=> 2
[4,3,1,2] => [4,3,1,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 2
[4,3,2,1] => [4,3,2,1] => [2,3,4,1] => {{1,2,3,4}}
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2
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].
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00044: Integer partitions conjugateInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,2] => [1,2] => [1,1]
=> [2]
=> 2
[2,1] => [2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,3,2] => [2,1]
=> [2,1]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [2,1]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [2,1]
=> 2
[2,3,1] => [2,3,1] => [2,1]
=> [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1]
=> [2,1]
=> 2
[3,2,1] => [3,2,1] => [3]
=> [1,1,1]
=> 1
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2,2]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [3,1]
=> 3
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [3,1]
=> 3
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2,2]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2,2]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [2,1,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [3,1]
=> 3
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [2,1,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
Description
The largest part of an integer partition.
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00095: Integer partitions to binary wordBinary words
St000288: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 10 => 1
[1,2] => [1,2] => [1,1]
=> 110 => 2
[2,1] => [2,1] => [2]
=> 100 => 1
[1,2,3] => [1,3,2] => [2,1]
=> 1010 => 2
[1,3,2] => [1,3,2] => [2,1]
=> 1010 => 2
[2,1,3] => [2,1,3] => [2,1]
=> 1010 => 2
[2,3,1] => [2,3,1] => [2,1]
=> 1010 => 2
[3,1,2] => [3,1,2] => [2,1]
=> 1010 => 2
[3,2,1] => [3,2,1] => [3]
=> 1000 => 1
[1,2,3,4] => [1,4,3,2] => [3,1]
=> 10010 => 2
[1,2,4,3] => [1,4,3,2] => [3,1]
=> 10010 => 2
[1,3,2,4] => [1,4,3,2] => [3,1]
=> 10010 => 2
[1,3,4,2] => [1,4,3,2] => [3,1]
=> 10010 => 2
[1,4,2,3] => [1,4,3,2] => [3,1]
=> 10010 => 2
[1,4,3,2] => [1,4,3,2] => [3,1]
=> 10010 => 2
[2,1,3,4] => [2,1,4,3] => [2,2]
=> 1100 => 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> 1100 => 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> 10110 => 3
[2,3,4,1] => [2,4,3,1] => [3,1]
=> 10010 => 2
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> 10110 => 3
[2,4,3,1] => [2,4,3,1] => [3,1]
=> 10010 => 2
[3,1,2,4] => [3,1,4,2] => [2,2]
=> 1100 => 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> 1100 => 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> 10010 => 2
[3,2,4,1] => [3,2,4,1] => [3,1]
=> 10010 => 2
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> 10110 => 3
[3,4,2,1] => [3,4,2,1] => [3,1]
=> 10010 => 2
[4,1,2,3] => [4,1,3,2] => [3,1]
=> 10010 => 2
[4,1,3,2] => [4,1,3,2] => [3,1]
=> 10010 => 2
[4,2,1,3] => [4,2,1,3] => [3,1]
=> 10010 => 2
[4,2,3,1] => [4,2,3,1] => [3,1]
=> 10010 => 2
[4,3,1,2] => [4,3,1,2] => [3,1]
=> 10010 => 2
[4,3,2,1] => [4,3,2,1] => [4]
=> 10000 => 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> 100010 => 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Matching statistic: St000321
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00064: Permutations reversePermutations
Mp00204: Permutations LLPSInteger partitions
St000321: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2,1] => [2]
=> 2
[2,1] => [2,1] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[1,3,2] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => [2,1]
=> 2
[2,3,1] => [2,3,1] => [1,3,2] => [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1,3] => [2,1]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,1,1]
=> 1
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 3
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 3
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [2,1,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [2,1,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,2]
=> 3
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [2,1,1]
=> 2
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,1,1]
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [2,1,1]
=> 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [2,1,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
Description
The number of integer partitions of n that are dominated by an integer partition. A partition $\lambda = (\lambda_1,\ldots,\lambda_n) \vdash n$ dominates a partition $\mu = (\mu_1,\ldots,\mu_n) \vdash n$ if $\sum_{i=1}^k (\lambda_i - \mu_i) \geq 0$ for all $k$.
Matching statistic: St000345
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00064: Permutations reversePermutations
Mp00204: Permutations LLPSInteger partitions
St000345: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1]
=> 1
[1,2] => [1,2] => [2,1] => [2]
=> 2
[2,1] => [2,1] => [1,2] => [1,1]
=> 1
[1,2,3] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[1,3,2] => [1,3,2] => [2,3,1] => [2,1]
=> 2
[2,1,3] => [2,1,3] => [3,1,2] => [2,1]
=> 2
[2,3,1] => [2,3,1] => [1,3,2] => [2,1]
=> 2
[3,1,2] => [3,1,2] => [2,1,3] => [2,1]
=> 2
[3,2,1] => [3,2,1] => [1,2,3] => [1,1,1]
=> 1
[1,2,3,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,2,4,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,3,2,4] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[1,4,3,2] => [1,4,3,2] => [2,3,4,1] => [2,1,1]
=> 2
[2,1,3,4] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 2
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [2,1,1]
=> 2
[2,3,1,4] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 3
[2,3,4,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 2
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [2,2]
=> 3
[2,4,3,1] => [2,4,3,1] => [1,3,4,2] => [2,1,1]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [2,1,1]
=> 2
[3,2,1,4] => [3,2,1,4] => [4,1,2,3] => [2,1,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [1,4,2,3] => [2,1,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [2,2]
=> 3
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [2,1,1]
=> 2
[4,1,2,3] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [2,3,1,4] => [2,1,1]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1,2,4] => [2,1,1]
=> 2
[4,2,3,1] => [4,2,3,1] => [1,3,2,4] => [2,1,1]
=> 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [2,1,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [2,3,4,5,1] => [2,1,1,1]
=> 2
Description
The number of refinements of a partition. A partition $\lambda$ refines a partition $\mu$ if the parts of $\mu$ can be subdivided to obtain the parts of $\lambda$.
Matching statistic: St000378
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00204: Permutations LLPSInteger partitions
Mp00322: Integer partitions Loehr-WarringtonInteger partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> 1
[1,2] => [1,2] => [1,1]
=> [2]
=> 2
[2,1] => [2,1] => [2]
=> [1,1]
=> 1
[1,2,3] => [1,3,2] => [2,1]
=> [3]
=> 2
[1,3,2] => [1,3,2] => [2,1]
=> [3]
=> 2
[2,1,3] => [2,1,3] => [2,1]
=> [3]
=> 2
[2,3,1] => [2,3,1] => [2,1]
=> [3]
=> 2
[3,1,2] => [3,1,2] => [2,1]
=> [3]
=> 2
[3,2,1] => [3,2,1] => [3]
=> [1,1,1]
=> 1
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [2,1,1]
=> 2
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [4]
=> 2
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [4]
=> 2
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [2,2]
=> 3
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [2,2]
=> 3
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [2,1,1]
=> 2
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [4]
=> 2
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [4]
=> 2
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [2,1,1]
=> 2
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [2,1,1]
=> 2
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [2,2]
=> 3
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [2,1,1]
=> 2
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [2,1,1]
=> 2
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [2,1,1]
=> 2
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [2,1,1]
=> 2
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [2,1,1]
=> 2
[4,3,2,1] => [4,3,2,1] => [4]
=> [1,1,1,1]
=> 1
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> 2
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].
The following 134 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000676The number of odd rises of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000935The number of ordered refinements of an integer partition. 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. St001389The number of partitions of the same length below the given integer partition. St000012The area of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000291The number of descents of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000519The largest length of a factor maximising the subword complexity. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000834The number of right outer peaks of a permutation. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001657The number of twos in an integer partition. St000251The number of nonsingleton blocks of a set partition. St000919The number of maximal left branches of a binary tree. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000658The number of rises of length 2 of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000984The number of boxes below precisely one peak. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001333The cardinality of a minimal edge-isolating set of a graph. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000470The number of runs in a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. 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. St000374The number of exclusive right-to-left minima of a permutation. St000670The reversal length of a permutation. St000260The radius of a connected graph. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000035The number of left outer peaks of a permutation. St000259The diameter of a connected graph. St000167The number of leaves of an ordered tree. St000354The number of recoils of a permutation. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000386The number of factors DDU in a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001665The number of pure excedances of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000662The staircase size of the code of a permutation. 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 $eA$. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000264The girth of a graph, which is not a tree. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St001427The number of descents of a signed permutation. St001726The number of visible inversions of a permutation. St001060The distinguishing index of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000702The number of weak deficiencies of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001330The hat guessing number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000542The number of left-to-right-minima of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000021The number of descents of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000015The number of peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. 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. 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. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001298The number of repeated entries in the Lehmer code of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001874Lusztig's a-function for the symmetric group. St000083The number of left oriented leafs of a binary tree except the first one. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001480The number of simple summands of the module J^2/J^3. St000454The largest eigenvalue of a graph if it is integral. St000256The number of parts from which one can substract 2 and still get an integer partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000455The second largest eigenvalue of a graph if it is integral. St001720The minimal length of a chain of small intervals in a lattice. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St000711The number of big exceedences of a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001935The number of ascents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001597The Frobenius rank of a skew partition. St000862The number of parts of the shifted shape of a permutation. St001624The breadth of a lattice.