Your data matches 440 different statistics following compositions of up to 3 maps.
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Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00284: Standard tableaux rowsSet partitions
St001151: Set partitions ⟶ 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}}
=> 1
[2,1] => [[1],[2]]
=> {{1},{2}}
=> 1
[1,2,3] => [[1,2,3]]
=> {{1,2,3}}
=> 1
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[2,1,3] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2
[3,1,2] => [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,2,1] => [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2
[1,2,3,4] => [[1,2,3,4]]
=> {{1,2,3,4}}
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,1,3,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,1,2,4] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,2,1,4] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[4,1,2,3] => [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,1,3] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[5,3,1,2,4] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[5,4,1,2,3] => [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2
[2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 2
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 2
Description
The number of blocks with odd minimum. See [[St000746]] for the analogous statistic on perfect matchings.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001716: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 1
[1,2] => [2] => ([],2)
=> 1
[2,1] => [1,1] => ([(0,1)],2)
=> 1
[1,2,3] => [3] => ([],3)
=> 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,1,3] => [1,2] => ([(1,2)],3)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
[3,1,2] => [1,2] => ([(1,2)],3)
=> 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [4] => ([],4)
=> 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,2,3,4,5] => [5] => ([],5)
=> 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 1
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 1
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 2
[1,2,3,4,5,6] => [6] => ([],6)
=> 1
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,4,2,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,5,2,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[1,6,2,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
=> 1
[2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
[2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 2
Description
The 1-improper chromatic number of a graph. This is the least number of colours in a vertex-colouring, such that each vertex has at most one neighbour with the same colour.
Mp00071: Permutations descent compositionInteger compositions
Mp00094: Integer compositions to binary wordBinary words
St001414: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1 => 0 = 1 - 1
[1,2] => [2] => 10 => 0 = 1 - 1
[2,1] => [1,1] => 11 => 0 = 1 - 1
[1,2,3] => [3] => 100 => 0 = 1 - 1
[1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,2,1] => [1,1,1] => 111 => 1 = 2 - 1
[1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[2,1,3,4] => [1,3] => 1100 => 0 = 1 - 1
[2,3,1,4] => [2,2] => 1010 => 1 = 2 - 1
[2,4,1,3] => [2,2] => 1010 => 1 = 2 - 1
[3,1,2,4] => [1,3] => 1100 => 0 = 1 - 1
[3,2,1,4] => [1,1,2] => 1110 => 1 = 2 - 1
[3,4,1,2] => [2,2] => 1010 => 1 = 2 - 1
[4,1,2,3] => [1,3] => 1100 => 0 = 1 - 1
[4,2,1,3] => [1,1,2] => 1110 => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1110 => 1 = 2 - 1
[1,2,3,4,5] => [5] => 10000 => 0 = 1 - 1
[1,3,2,4,5] => [2,3] => 10100 => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 10100 => 1 = 2 - 1
[1,5,2,3,4] => [2,3] => 10100 => 1 = 2 - 1
[2,1,3,4,5] => [1,4] => 11000 => 0 = 1 - 1
[2,3,1,4,5] => [2,3] => 10100 => 1 = 2 - 1
[2,4,1,3,5] => [2,3] => 10100 => 1 = 2 - 1
[2,5,1,3,4] => [2,3] => 10100 => 1 = 2 - 1
[3,1,2,4,5] => [1,4] => 11000 => 0 = 1 - 1
[3,2,1,4,5] => [1,1,3] => 11100 => 1 = 2 - 1
[3,4,1,2,5] => [2,3] => 10100 => 1 = 2 - 1
[3,5,1,2,4] => [2,3] => 10100 => 1 = 2 - 1
[4,1,2,3,5] => [1,4] => 11000 => 0 = 1 - 1
[4,2,1,3,5] => [1,1,3] => 11100 => 1 = 2 - 1
[4,3,1,2,5] => [1,1,3] => 11100 => 1 = 2 - 1
[4,5,1,2,3] => [2,3] => 10100 => 1 = 2 - 1
[5,1,2,3,4] => [1,4] => 11000 => 0 = 1 - 1
[5,2,1,3,4] => [1,1,3] => 11100 => 1 = 2 - 1
[5,3,1,2,4] => [1,1,3] => 11100 => 1 = 2 - 1
[5,4,1,2,3] => [1,1,3] => 11100 => 1 = 2 - 1
[1,2,3,4,5,6] => [6] => 100000 => 0 = 1 - 1
[1,3,2,4,5,6] => [2,4] => 101000 => 1 = 2 - 1
[1,4,2,3,5,6] => [2,4] => 101000 => 1 = 2 - 1
[1,5,2,3,4,6] => [2,4] => 101000 => 1 = 2 - 1
[1,6,2,3,4,5] => [2,4] => 101000 => 1 = 2 - 1
[2,1,3,4,5,6] => [1,5] => 110000 => 0 = 1 - 1
[2,3,1,4,5,6] => [2,4] => 101000 => 1 = 2 - 1
[2,4,1,3,5,6] => [2,4] => 101000 => 1 = 2 - 1
[2,5,1,3,4,6] => [2,4] => 101000 => 1 = 2 - 1
Description
Half the length of the longest odd length palindromic prefix of a binary word. More precisely, this statistic is the largest number $k$ such that the word has a palindromic prefix of length $2k+1$.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001723: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,2,3,4,5,6] => [6] => ([],6)
=> 0 = 1 - 1
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,2,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,2,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
[2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The differential of a graph. The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$. The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$. The differential of a graph is the maximal differential of a set of vertices.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St001724: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> 0 = 1 - 1
[1,2] => [2] => ([],2)
=> 0 = 1 - 1
[2,1] => [1,1] => ([(0,1)],2)
=> 0 = 1 - 1
[1,2,3] => [3] => ([],3)
=> 0 = 1 - 1
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[2,1,3] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 2 - 1
[3,1,2] => [1,2] => ([(1,2)],3)
=> 0 = 1 - 1
[3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 2 - 1
[1,2,3,4] => [4] => ([],4)
=> 0 = 1 - 1
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,1,3,4] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,1,2,4] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,1,2,3] => [1,3] => ([(2,3)],4)
=> 0 = 1 - 1
[4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 1 = 2 - 1
[1,2,3,4,5] => [5] => ([],5)
=> 0 = 1 - 1
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,1,3,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,1,2,4,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[3,2,1,4,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,1,2,3,5] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[4,2,1,3,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,3,1,2,5] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,1,2,3,4] => [1,4] => ([(3,4)],5)
=> 0 = 1 - 1
[5,2,1,3,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,3,1,2,4] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[5,4,1,2,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> 1 = 2 - 1
[1,2,3,4,5,6] => [6] => ([],6)
=> 0 = 1 - 1
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,2,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,6,2,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,1,3,4,5,6] => [1,5] => ([(4,5)],6)
=> 0 = 1 - 1
[2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The 2-packing differential of a graph. The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$. The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$. A set $S\subseteq V(G)$ is $2$-packing if the closed neighbourhoods of any two vertices in $S$ have empty intersection. The $2$-packing differential of a graph is the maximal differential of any $2$-packing set of vertices.
Matching statistic: St000040
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000040: Permutations ⟶ 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] => 1
[2,1] => [[1],[2]]
=> [2,1] => [1,2] => 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,3,1,2,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,4,1,2,3] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
Description
The number of regions of the inversion arrangement of a permutation. The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$. Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1]. Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$. For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.
Matching statistic: St000047
Mp00064: Permutations reversePermutations
Mp00068: Permutations Simion-Schmidt mapPermutations
Mp00071: Permutations descent compositionInteger compositions
St000047: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => [1] => 1
[1,2] => [2,1] => [2,1] => [1,1] => 1
[2,1] => [1,2] => [1,2] => [2] => 1
[1,2,3] => [3,2,1] => [3,2,1] => [1,1,1] => 1
[1,3,2] => [2,3,1] => [2,3,1] => [2,1] => 2
[2,1,3] => [3,1,2] => [3,1,2] => [1,2] => 1
[2,3,1] => [1,3,2] => [1,3,2] => [2,1] => 2
[3,1,2] => [2,1,3] => [2,1,3] => [1,2] => 1
[3,2,1] => [1,2,3] => [1,3,2] => [2,1] => 2
[1,2,3,4] => [4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1
[1,3,2,4] => [4,2,3,1] => [4,2,3,1] => [1,2,1] => 2
[1,4,2,3] => [3,2,4,1] => [3,2,4,1] => [1,2,1] => 2
[2,1,3,4] => [4,3,1,2] => [4,3,1,2] => [1,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [4,1,3,2] => [1,2,1] => 2
[2,4,1,3] => [3,1,4,2] => [3,1,4,2] => [1,2,1] => 2
[3,1,2,4] => [4,2,1,3] => [4,2,1,3] => [1,1,2] => 1
[3,2,1,4] => [4,1,2,3] => [4,1,3,2] => [1,2,1] => 2
[3,4,1,2] => [2,1,4,3] => [2,1,4,3] => [1,2,1] => 2
[4,1,2,3] => [3,2,1,4] => [3,2,1,4] => [1,1,2] => 1
[4,2,1,3] => [3,1,2,4] => [3,1,4,2] => [1,2,1] => 2
[4,3,1,2] => [2,1,3,4] => [2,1,4,3] => [1,2,1] => 2
[1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1] => 1
[1,3,2,4,5] => [5,4,2,3,1] => [5,4,2,3,1] => [1,1,2,1] => 2
[1,4,2,3,5] => [5,3,2,4,1] => [5,3,2,4,1] => [1,1,2,1] => 2
[1,5,2,3,4] => [4,3,2,5,1] => [4,3,2,5,1] => [1,1,2,1] => 2
[2,1,3,4,5] => [5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,2] => 1
[2,3,1,4,5] => [5,4,1,3,2] => [5,4,1,3,2] => [1,1,2,1] => 2
[2,4,1,3,5] => [5,3,1,4,2] => [5,3,1,4,2] => [1,1,2,1] => 2
[2,5,1,3,4] => [4,3,1,5,2] => [4,3,1,5,2] => [1,1,2,1] => 2
[3,1,2,4,5] => [5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,2] => 1
[3,2,1,4,5] => [5,4,1,2,3] => [5,4,1,3,2] => [1,1,2,1] => 2
[3,4,1,2,5] => [5,2,1,4,3] => [5,2,1,4,3] => [1,1,2,1] => 2
[3,5,1,2,4] => [4,2,1,5,3] => [4,2,1,5,3] => [1,1,2,1] => 2
[4,1,2,3,5] => [5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,2] => 1
[4,2,1,3,5] => [5,3,1,2,4] => [5,3,1,4,2] => [1,1,2,1] => 2
[4,3,1,2,5] => [5,2,1,3,4] => [5,2,1,4,3] => [1,1,2,1] => 2
[4,5,1,2,3] => [3,2,1,5,4] => [3,2,1,5,4] => [1,1,2,1] => 2
[5,1,2,3,4] => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,2] => 1
[5,2,1,3,4] => [4,3,1,2,5] => [4,3,1,5,2] => [1,1,2,1] => 2
[5,3,1,2,4] => [4,2,1,3,5] => [4,2,1,5,3] => [1,1,2,1] => 2
[5,4,1,2,3] => [3,2,1,4,5] => [3,2,1,5,4] => [1,1,2,1] => 2
[1,2,3,4,5,6] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1] => 1
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [6,5,4,2,3,1] => [1,1,1,2,1] => 2
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [6,5,3,2,4,1] => [1,1,1,2,1] => 2
[1,5,2,3,4,6] => [6,4,3,2,5,1] => [6,4,3,2,5,1] => [1,1,1,2,1] => 2
[1,6,2,3,4,5] => [5,4,3,2,6,1] => [5,4,3,2,6,1] => [1,1,1,2,1] => 2
[2,1,3,4,5,6] => [6,5,4,3,1,2] => [6,5,4,3,1,2] => [1,1,1,1,2] => 1
[2,3,1,4,5,6] => [6,5,4,1,3,2] => [6,5,4,1,3,2] => [1,1,1,2,1] => 2
[2,4,1,3,5,6] => [6,5,3,1,4,2] => [6,5,3,1,4,2] => [1,1,1,2,1] => 2
[2,5,1,3,4,6] => [6,4,3,1,5,2] => [6,4,3,1,5,2] => [1,1,1,2,1] => 2
Description
The number of standard immaculate tableaux of a given shape. See Proposition 3.13 of [2] for a hook-length counting formula of these tableaux.
Matching statistic: St000058
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000058: Permutations ⟶ 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] => 1
[2,1] => [[1],[2]]
=> [2,1] => [1,2] => 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,3,1,2,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,4,1,2,3] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
Description
The order of a permutation. $\operatorname{ord}(\pi)$ is given by the minimial $k$ for which $\pi^k$ is the identity permutation.
Matching statistic: St000078
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000078: Permutations ⟶ 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] => 1
[2,1] => [[1],[2]]
=> [2,1] => [1,2] => 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,3,1,2,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,4,1,2,3] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
Description
The number of alternating sign matrices whose left key is the permutation. The left key of an alternating sign matrix was defined by Lascoux in [2] and is obtained by successively removing all the `-1`'s until what remains is a permutation matrix. This notion corresponds to the notion of left key for semistandard tableaux.
Matching statistic: St000110
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
Mp00090: Permutations cycle-as-one-line notationPermutations
St000110: Permutations ⟶ 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] => 1
[2,1] => [[1],[2]]
=> [2,1] => [1,2] => 1
[1,2,3] => [[1,2,3]]
=> [1,2,3] => [1,2,3] => 1
[1,3,2] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[2,1,3] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[2,3,1] => [[1,2],[3]]
=> [3,1,2] => [1,3,2] => 2
[3,1,2] => [[1,3],[2]]
=> [2,1,3] => [1,2,3] => 1
[3,2,1] => [[1],[2],[3]]
=> [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [[1,2,3,4]]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,1,3,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1,2,4] => [1,3,2,4] => 2
[2,4,1,3] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[3,1,2,4] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,4,1,2] => [1,3,2,4] => 2
[4,1,2,3] => [[1,3,4],[2]]
=> [2,1,3,4] => [1,2,3,4] => 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,2,1,4] => [1,3,2,4] => 2
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [1,2,3,4,5] => [1,2,3,4,5] => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,1,2,4,5] => [1,3,2,4,5] => 2
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,1,2,4,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[3,2,1,4,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[4,1,2,3,5] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[4,2,1,3,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,3,1,2,5] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [3,4,1,2,5] => [1,3,2,4,5] => 2
[5,1,2,3,4] => [[1,3,4,5],[2]]
=> [2,1,3,4,5] => [1,2,3,4,5] => 1
[5,2,1,3,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,3,1,2,4] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[5,4,1,2,3] => [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => [1,3,2,4,5] => 2
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,1,3,4,5,6] => [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => [1,2,3,4,5,6] => 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> [3,1,2,4,5,6] => [1,3,2,4,5,6] => 2
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => [1,3,2,4,5,6] => 2
Description
The number of permutations less than or equal to a permutation in left weak order. This is the same as the number of permutations less than or equal to the given permutation in right weak order.
The following 430 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000255The number of reduced Kogan faces with the permutation as type. St000451The length of the longest pattern of the form k 1 2. St000470The number of runs in a permutation. St000644The number of graphs with given frequency partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000862The number of parts of the shifted shape of a permutation. St000883The number of longest increasing subsequences of a permutation. St000920The logarithmic height of a Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001624The breadth of a lattice. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000057The Shynar inversion number of a standard tableau. St000141The maximum drop size of a permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000157The number of descents of a standard tableau. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000214The number of adjacencies of a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000237The number of small exceedances. St000245The number of ascents of a permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000336The leg major index of a standard tableau. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000374The number of exclusive right-to-left minima of a permutation. St000377The dinv defect of an integer partition. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000463The number of admissible inversions of a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000766The number of inversions of an integer composition. St000834The number of right outer peaks of a permutation. St000884The number of isolated descents of a permutation. St000921The number of internal inversions of a binary word. 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. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001092The number of distinct even parts of a partition. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001115The number of even descents of a permutation. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001320The minimal number of occurrences of the path-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001335The cardinality of a minimal cycle-isolating set of a graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001377The major index minus the number of inversions of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001469The holeyness of a permutation. St001489The maximum of the number of descents and the number of inverse descents. 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. St001638The book thickness of a graph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001665The number of pure excedances of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001699The major index of a standard tableau minus the weighted size of its shape. St001726The number of visible inversions of a permutation. St001729The number of visible descents of a permutation. St001736The total number of cycles in a graph. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001781The interlacing number of a set partition. St001797The number of overfull subgraphs of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001928The number of non-overlapping descents in a permutation. St000982The length of the longest constant subword. St000292The number of ascents of a binary word. St000062The length of the longest increasing subsequence of the permutation. St000288The number of ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000381The largest part of an integer composition. St000383The last part of an integer composition. St000485The length of the longest cycle of a permutation. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000808The number of up steps of the associated bargraph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001081The number of minimal length factorizations of a permutation into star transpositions. St001246The maximal difference between two consecutive entries of a permutation. St001313The number of Dyck paths above the lattice path given by a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000290The major index of a binary word. St000291The number of descents of a binary word. St000304The load of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000348The non-inversion sum of a binary word. St000354The number of recoils of a permutation. St000446The disorder of a permutation. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000494The number of inversions of distance at most 3 of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000497The lcb statistic of a set partition. St000538The number of even inversions of a permutation. St000539The number of odd inversions of a permutation. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000628The balance of a binary word. St000661The number of rises of length 3 of a Dyck path. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000795The mad of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000836The number of descents of distance 2 of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000956The maximal displacement of a permutation. St000957The number of Bruhat lower covers of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. 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). St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001298The number of repeated entries in the Lehmer code of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001485The modular major index of a binary word. St001874Lusztig's a-function for the symmetric group. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000452The number of distinct eigenvalues of a graph. St000453The number of distinct Laplacian eigenvalues of a graph. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001093The detour number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St001315The dissociation number of a 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. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001674The number of vertices of the largest induced star graph in the graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000387The matching number of a graph. St000535The rank-width of a graph. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001071The beta invariant of the graph. St001271The competition number of a graph. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000668The least common multiple of the parts of the partition. St001118The acyclic chromatic index of a graph. St001727The number of invisible inversions of a permutation. St000353The number of inner valleys of a permutation. St000647The number of big descents of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000939The number of characters of the symmetric group whose value on the partition is positive. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000264The girth of a graph, which is not a tree. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St000567The sum of the products of all pairs of parts. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000455The second largest eigenvalue of a graph if it is integral. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000886The number of permutations with the same antidiagonal sums. St000486The number of cycles of length at least 3 of a permutation. St001728The number of invisible descents of a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000092The number of outer peaks of a permutation. St000109The number of elements less than or equal to the given element in Bruhat order. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000055The inversion sum of a permutation. St000095The number of triangles of a graph. St000224The sorting index of a permutation. St000238The number of indices that are not small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St000216The absolute length of a permutation. St000806The semiperimeter of the associated bargraph. St000837The number of ascents of distance 2 of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001731The factorization defect of a permutation. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000474Dyson's crank of a partition. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001651The Frankl number of a lattice. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001427The number of descents of a signed permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001684The reduced word complexity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. 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 $eA$. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001060The distinguishing index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St000477The weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St000997The even-odd crank of an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001281The normalized isoperimetric number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001592The maximal number of simple paths between any two different vertices of a graph. St001896The number of right descents of a signed permutations. St001768The number of reduced words of a signed permutation. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.