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Mp00109: Permutations descent wordBinary words
St001355: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => 01 => 1
[2,3,1] => 01 => 1
[1,2,4,3] => 001 => 0
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 0
[1,4,2,3] => 010 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 0
[2,4,1,3] => 010 => 1
[3,4,1,2] => 010 => 1
[1,2,4,3,5] => 0010 => 0
[1,2,5,3,4] => 0010 => 0
[1,3,2,4,5] => 0100 => 1
[1,3,4,2,5] => 0010 => 0
[1,3,5,2,4] => 0010 => 0
[1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => 0010 => 0
[1,5,2,3,4] => 0100 => 1
[2,3,1,4,5] => 0100 => 1
[2,3,4,1,5] => 0010 => 0
[2,3,5,1,4] => 0010 => 0
[2,4,1,3,5] => 0100 => 1
[2,4,5,1,3] => 0010 => 0
[2,5,1,3,4] => 0100 => 1
[3,4,1,2,5] => 0100 => 1
[3,4,5,1,2] => 0010 => 0
[3,5,1,2,4] => 0100 => 1
[4,5,1,2,3] => 0100 => 1
[1,2,4,3,5,6] => 00100 => 0
[1,2,5,3,4,6] => 00100 => 0
[1,2,6,3,4,5] => 00100 => 0
[1,3,2,4,5,6] => 01000 => 1
[1,3,4,2,5,6] => 00100 => 0
[1,3,5,2,4,6] => 00100 => 0
[1,3,6,2,4,5] => 00100 => 0
[1,4,2,3,5,6] => 01000 => 1
[1,4,5,2,3,6] => 00100 => 0
[1,4,6,2,3,5] => 00100 => 0
[1,5,2,3,4,6] => 01000 => 1
[1,5,6,2,3,4] => 00100 => 0
[1,6,2,3,4,5] => 01000 => 1
[2,3,1,4,5,6] => 01000 => 1
[2,3,4,1,5,6] => 00100 => 0
[2,3,5,1,4,6] => 00100 => 0
[2,3,6,1,4,5] => 00100 => 0
[2,4,1,3,5,6] => 01000 => 1
[2,4,5,1,3,6] => 00100 => 0
[2,4,6,1,3,5] => 00100 => 0
[2,5,1,3,4,6] => 01000 => 1
[2,5,6,1,3,4] => 00100 => 0
Description
Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. Graphically, this is the number of returns to the main diagonal of the monotone lattice path of a binary word.
Mp00109: Permutations descent wordBinary words
St000792: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => 01 => 2 = 1 + 1
[2,3,1] => 01 => 2 = 1 + 1
[1,2,4,3] => 001 => 1 = 0 + 1
[1,3,2,4] => 010 => 2 = 1 + 1
[1,3,4,2] => 001 => 1 = 0 + 1
[1,4,2,3] => 010 => 2 = 1 + 1
[2,3,1,4] => 010 => 2 = 1 + 1
[2,3,4,1] => 001 => 1 = 0 + 1
[2,4,1,3] => 010 => 2 = 1 + 1
[3,4,1,2] => 010 => 2 = 1 + 1
[1,2,4,3,5] => 0010 => 1 = 0 + 1
[1,2,5,3,4] => 0010 => 1 = 0 + 1
[1,3,2,4,5] => 0100 => 2 = 1 + 1
[1,3,4,2,5] => 0010 => 1 = 0 + 1
[1,3,5,2,4] => 0010 => 1 = 0 + 1
[1,4,2,3,5] => 0100 => 2 = 1 + 1
[1,4,5,2,3] => 0010 => 1 = 0 + 1
[1,5,2,3,4] => 0100 => 2 = 1 + 1
[2,3,1,4,5] => 0100 => 2 = 1 + 1
[2,3,4,1,5] => 0010 => 1 = 0 + 1
[2,3,5,1,4] => 0010 => 1 = 0 + 1
[2,4,1,3,5] => 0100 => 2 = 1 + 1
[2,4,5,1,3] => 0010 => 1 = 0 + 1
[2,5,1,3,4] => 0100 => 2 = 1 + 1
[3,4,1,2,5] => 0100 => 2 = 1 + 1
[3,4,5,1,2] => 0010 => 1 = 0 + 1
[3,5,1,2,4] => 0100 => 2 = 1 + 1
[4,5,1,2,3] => 0100 => 2 = 1 + 1
[1,2,4,3,5,6] => 00100 => 1 = 0 + 1
[1,2,5,3,4,6] => 00100 => 1 = 0 + 1
[1,2,6,3,4,5] => 00100 => 1 = 0 + 1
[1,3,2,4,5,6] => 01000 => 2 = 1 + 1
[1,3,4,2,5,6] => 00100 => 1 = 0 + 1
[1,3,5,2,4,6] => 00100 => 1 = 0 + 1
[1,3,6,2,4,5] => 00100 => 1 = 0 + 1
[1,4,2,3,5,6] => 01000 => 2 = 1 + 1
[1,4,5,2,3,6] => 00100 => 1 = 0 + 1
[1,4,6,2,3,5] => 00100 => 1 = 0 + 1
[1,5,2,3,4,6] => 01000 => 2 = 1 + 1
[1,5,6,2,3,4] => 00100 => 1 = 0 + 1
[1,6,2,3,4,5] => 01000 => 2 = 1 + 1
[2,3,1,4,5,6] => 01000 => 2 = 1 + 1
[2,3,4,1,5,6] => 00100 => 1 = 0 + 1
[2,3,5,1,4,6] => 00100 => 1 = 0 + 1
[2,3,6,1,4,5] => 00100 => 1 = 0 + 1
[2,4,1,3,5,6] => 01000 => 2 = 1 + 1
[2,4,5,1,3,6] => 00100 => 1 = 0 + 1
[2,4,6,1,3,5] => 00100 => 1 = 0 + 1
[2,5,1,3,4,6] => 01000 => 2 = 1 + 1
[2,5,6,1,3,4] => 00100 => 1 = 0 + 1
Description
The Grundy value for the game of ruler on a binary word. Two players alternately may switch any consecutive sequence of numbers that ends with a 1. The player facing the word which has only 0's looses.
Mp00109: Permutations descent wordBinary words
Mp00135: Binary words rotate front-to-backBinary words
St000297: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => 01 => 10 => 1
[2,3,1] => 01 => 10 => 1
[1,2,4,3] => 001 => 010 => 0
[1,3,2,4] => 010 => 100 => 1
[1,3,4,2] => 001 => 010 => 0
[1,4,2,3] => 010 => 100 => 1
[2,3,1,4] => 010 => 100 => 1
[2,3,4,1] => 001 => 010 => 0
[2,4,1,3] => 010 => 100 => 1
[3,4,1,2] => 010 => 100 => 1
[1,2,4,3,5] => 0010 => 0100 => 0
[1,2,5,3,4] => 0010 => 0100 => 0
[1,3,2,4,5] => 0100 => 1000 => 1
[1,3,4,2,5] => 0010 => 0100 => 0
[1,3,5,2,4] => 0010 => 0100 => 0
[1,4,2,3,5] => 0100 => 1000 => 1
[1,4,5,2,3] => 0010 => 0100 => 0
[1,5,2,3,4] => 0100 => 1000 => 1
[2,3,1,4,5] => 0100 => 1000 => 1
[2,3,4,1,5] => 0010 => 0100 => 0
[2,3,5,1,4] => 0010 => 0100 => 0
[2,4,1,3,5] => 0100 => 1000 => 1
[2,4,5,1,3] => 0010 => 0100 => 0
[2,5,1,3,4] => 0100 => 1000 => 1
[3,4,1,2,5] => 0100 => 1000 => 1
[3,4,5,1,2] => 0010 => 0100 => 0
[3,5,1,2,4] => 0100 => 1000 => 1
[4,5,1,2,3] => 0100 => 1000 => 1
[1,2,4,3,5,6] => 00100 => 01000 => 0
[1,2,5,3,4,6] => 00100 => 01000 => 0
[1,2,6,3,4,5] => 00100 => 01000 => 0
[1,3,2,4,5,6] => 01000 => 10000 => 1
[1,3,4,2,5,6] => 00100 => 01000 => 0
[1,3,5,2,4,6] => 00100 => 01000 => 0
[1,3,6,2,4,5] => 00100 => 01000 => 0
[1,4,2,3,5,6] => 01000 => 10000 => 1
[1,4,5,2,3,6] => 00100 => 01000 => 0
[1,4,6,2,3,5] => 00100 => 01000 => 0
[1,5,2,3,4,6] => 01000 => 10000 => 1
[1,5,6,2,3,4] => 00100 => 01000 => 0
[1,6,2,3,4,5] => 01000 => 10000 => 1
[2,3,1,4,5,6] => 01000 => 10000 => 1
[2,3,4,1,5,6] => 00100 => 01000 => 0
[2,3,5,1,4,6] => 00100 => 01000 => 0
[2,3,6,1,4,5] => 00100 => 01000 => 0
[2,4,1,3,5,6] => 01000 => 10000 => 1
[2,4,5,1,3,6] => 00100 => 01000 => 0
[2,4,6,1,3,5] => 00100 => 01000 => 0
[2,5,1,3,4,6] => 01000 => 10000 => 1
[2,5,6,1,3,4] => 00100 => 01000 => 0
Description
The number of leading ones in a binary word.
Mp00071: Permutations descent compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000313: Graphs ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[2,3,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1
[1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 0
[2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
=> 1
[1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[2,3,1,4,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[2,3,4,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,3,5,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,4,1,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[2,4,5,1,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[2,5,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[3,4,1,2,5] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[3,4,5,1,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 0
[3,5,1,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[4,5,1,2,3] => [2,3] => ([(2,4),(3,4)],5)
=> 1
[1,2,4,3,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,2,5,3,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,2,6,3,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,3,2,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[1,3,4,2,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,3,5,2,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,3,6,2,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,4,2,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[1,4,5,2,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,4,6,2,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,5,2,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[1,5,6,2,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[1,6,2,3,4,5] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[2,3,1,4,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[2,3,4,1,5,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[2,3,5,1,4,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[2,3,6,1,4,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[2,4,1,3,5,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[2,4,5,1,3,6] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[2,4,6,1,3,5] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
[2,5,1,3,4,6] => [2,4] => ([(3,5),(4,5)],6)
=> 1
[2,5,6,1,3,4] => [3,3] => ([(2,5),(3,5),(4,5)],6)
=> 0
Description
The number of degree 2 vertices of a graph. A vertex has degree 2 if and only if it lies on a unique maximal path.
Mp00109: Permutations descent wordBinary words
Mp00158: Binary words alternating inverseBinary words
St001413: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => 01 => 00 => 1
[2,3,1] => 01 => 00 => 1
[1,2,4,3] => 001 => 011 => 0
[1,3,2,4] => 010 => 000 => 1
[1,3,4,2] => 001 => 011 => 0
[1,4,2,3] => 010 => 000 => 1
[2,3,1,4] => 010 => 000 => 1
[2,3,4,1] => 001 => 011 => 0
[2,4,1,3] => 010 => 000 => 1
[3,4,1,2] => 010 => 000 => 1
[1,2,4,3,5] => 0010 => 0111 => 0
[1,2,5,3,4] => 0010 => 0111 => 0
[1,3,2,4,5] => 0100 => 0001 => 1
[1,3,4,2,5] => 0010 => 0111 => 0
[1,3,5,2,4] => 0010 => 0111 => 0
[1,4,2,3,5] => 0100 => 0001 => 1
[1,4,5,2,3] => 0010 => 0111 => 0
[1,5,2,3,4] => 0100 => 0001 => 1
[2,3,1,4,5] => 0100 => 0001 => 1
[2,3,4,1,5] => 0010 => 0111 => 0
[2,3,5,1,4] => 0010 => 0111 => 0
[2,4,1,3,5] => 0100 => 0001 => 1
[2,4,5,1,3] => 0010 => 0111 => 0
[2,5,1,3,4] => 0100 => 0001 => 1
[3,4,1,2,5] => 0100 => 0001 => 1
[3,4,5,1,2] => 0010 => 0111 => 0
[3,5,1,2,4] => 0100 => 0001 => 1
[4,5,1,2,3] => 0100 => 0001 => 1
[1,2,4,3,5,6] => 00100 => 01110 => 0
[1,2,5,3,4,6] => 00100 => 01110 => 0
[1,2,6,3,4,5] => 00100 => 01110 => 0
[1,3,2,4,5,6] => 01000 => 00010 => 1
[1,3,4,2,5,6] => 00100 => 01110 => 0
[1,3,5,2,4,6] => 00100 => 01110 => 0
[1,3,6,2,4,5] => 00100 => 01110 => 0
[1,4,2,3,5,6] => 01000 => 00010 => 1
[1,4,5,2,3,6] => 00100 => 01110 => 0
[1,4,6,2,3,5] => 00100 => 01110 => 0
[1,5,2,3,4,6] => 01000 => 00010 => 1
[1,5,6,2,3,4] => 00100 => 01110 => 0
[1,6,2,3,4,5] => 01000 => 00010 => 1
[2,3,1,4,5,6] => 01000 => 00010 => 1
[2,3,4,1,5,6] => 00100 => 01110 => 0
[2,3,5,1,4,6] => 00100 => 01110 => 0
[2,3,6,1,4,5] => 00100 => 01110 => 0
[2,4,1,3,5,6] => 01000 => 00010 => 1
[2,4,5,1,3,6] => 00100 => 01110 => 0
[2,4,6,1,3,5] => 00100 => 01110 => 0
[2,5,1,3,4,6] => 01000 => 00010 => 1
[2,5,6,1,3,4] => 00100 => 01110 => 0
Description
Half the length of the longest even length palindromic prefix of a binary word.
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
[1,3,2] => [2,1] => 101 => 1
[2,3,1] => [2,1] => 101 => 1
[1,2,4,3] => [3,1] => 1001 => 0
[1,3,2,4] => [2,2] => 1010 => 1
[1,3,4,2] => [3,1] => 1001 => 0
[1,4,2,3] => [2,2] => 1010 => 1
[2,3,1,4] => [2,2] => 1010 => 1
[2,3,4,1] => [3,1] => 1001 => 0
[2,4,1,3] => [2,2] => 1010 => 1
[3,4,1,2] => [2,2] => 1010 => 1
[1,2,4,3,5] => [3,2] => 10010 => 0
[1,2,5,3,4] => [3,2] => 10010 => 0
[1,3,2,4,5] => [2,3] => 10100 => 1
[1,3,4,2,5] => [3,2] => 10010 => 0
[1,3,5,2,4] => [3,2] => 10010 => 0
[1,4,2,3,5] => [2,3] => 10100 => 1
[1,4,5,2,3] => [3,2] => 10010 => 0
[1,5,2,3,4] => [2,3] => 10100 => 1
[2,3,1,4,5] => [2,3] => 10100 => 1
[2,3,4,1,5] => [3,2] => 10010 => 0
[2,3,5,1,4] => [3,2] => 10010 => 0
[2,4,1,3,5] => [2,3] => 10100 => 1
[2,4,5,1,3] => [3,2] => 10010 => 0
[2,5,1,3,4] => [2,3] => 10100 => 1
[3,4,1,2,5] => [2,3] => 10100 => 1
[3,4,5,1,2] => [3,2] => 10010 => 0
[3,5,1,2,4] => [2,3] => 10100 => 1
[4,5,1,2,3] => [2,3] => 10100 => 1
[1,2,4,3,5,6] => [3,3] => 100100 => 0
[1,2,5,3,4,6] => [3,3] => 100100 => 0
[1,2,6,3,4,5] => [3,3] => 100100 => 0
[1,3,2,4,5,6] => [2,4] => 101000 => 1
[1,3,4,2,5,6] => [3,3] => 100100 => 0
[1,3,5,2,4,6] => [3,3] => 100100 => 0
[1,3,6,2,4,5] => [3,3] => 100100 => 0
[1,4,2,3,5,6] => [2,4] => 101000 => 1
[1,4,5,2,3,6] => [3,3] => 100100 => 0
[1,4,6,2,3,5] => [3,3] => 100100 => 0
[1,5,2,3,4,6] => [2,4] => 101000 => 1
[1,5,6,2,3,4] => [3,3] => 100100 => 0
[1,6,2,3,4,5] => [2,4] => 101000 => 1
[2,3,1,4,5,6] => [2,4] => 101000 => 1
[2,3,4,1,5,6] => [3,3] => 100100 => 0
[2,3,5,1,4,6] => [3,3] => 100100 => 0
[2,3,6,1,4,5] => [3,3] => 100100 => 0
[2,4,1,3,5,6] => [2,4] => 101000 => 1
[2,4,5,1,3,6] => [3,3] => 100100 => 0
[2,4,6,1,3,5] => [3,3] => 100100 => 0
[2,5,1,3,4,6] => [2,4] => 101000 => 1
[2,5,6,1,3,4] => [3,3] => 100100 => 0
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 $2k+1$.
Mp00109: Permutations descent wordBinary words
Mp00273: Binary words Gray previousBinary words
St000392: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => 01 => 11 => 2 = 1 + 1
[2,3,1] => 01 => 11 => 2 = 1 + 1
[1,2,4,3] => 001 => 101 => 1 = 0 + 1
[1,3,2,4] => 010 => 110 => 2 = 1 + 1
[1,3,4,2] => 001 => 101 => 1 = 0 + 1
[1,4,2,3] => 010 => 110 => 2 = 1 + 1
[2,3,1,4] => 010 => 110 => 2 = 1 + 1
[2,3,4,1] => 001 => 101 => 1 = 0 + 1
[2,4,1,3] => 010 => 110 => 2 = 1 + 1
[3,4,1,2] => 010 => 110 => 2 = 1 + 1
[1,2,4,3,5] => 0010 => 1010 => 1 = 0 + 1
[1,2,5,3,4] => 0010 => 1010 => 1 = 0 + 1
[1,3,2,4,5] => 0100 => 1100 => 2 = 1 + 1
[1,3,4,2,5] => 0010 => 1010 => 1 = 0 + 1
[1,3,5,2,4] => 0010 => 1010 => 1 = 0 + 1
[1,4,2,3,5] => 0100 => 1100 => 2 = 1 + 1
[1,4,5,2,3] => 0010 => 1010 => 1 = 0 + 1
[1,5,2,3,4] => 0100 => 1100 => 2 = 1 + 1
[2,3,1,4,5] => 0100 => 1100 => 2 = 1 + 1
[2,3,4,1,5] => 0010 => 1010 => 1 = 0 + 1
[2,3,5,1,4] => 0010 => 1010 => 1 = 0 + 1
[2,4,1,3,5] => 0100 => 1100 => 2 = 1 + 1
[2,4,5,1,3] => 0010 => 1010 => 1 = 0 + 1
[2,5,1,3,4] => 0100 => 1100 => 2 = 1 + 1
[3,4,1,2,5] => 0100 => 1100 => 2 = 1 + 1
[3,4,5,1,2] => 0010 => 1010 => 1 = 0 + 1
[3,5,1,2,4] => 0100 => 1100 => 2 = 1 + 1
[4,5,1,2,3] => 0100 => 1100 => 2 = 1 + 1
[1,2,4,3,5,6] => 00100 => 10100 => 1 = 0 + 1
[1,2,5,3,4,6] => 00100 => 10100 => 1 = 0 + 1
[1,2,6,3,4,5] => 00100 => 10100 => 1 = 0 + 1
[1,3,2,4,5,6] => 01000 => 11000 => 2 = 1 + 1
[1,3,4,2,5,6] => 00100 => 10100 => 1 = 0 + 1
[1,3,5,2,4,6] => 00100 => 10100 => 1 = 0 + 1
[1,3,6,2,4,5] => 00100 => 10100 => 1 = 0 + 1
[1,4,2,3,5,6] => 01000 => 11000 => 2 = 1 + 1
[1,4,5,2,3,6] => 00100 => 10100 => 1 = 0 + 1
[1,4,6,2,3,5] => 00100 => 10100 => 1 = 0 + 1
[1,5,2,3,4,6] => 01000 => 11000 => 2 = 1 + 1
[1,5,6,2,3,4] => 00100 => 10100 => 1 = 0 + 1
[1,6,2,3,4,5] => 01000 => 11000 => 2 = 1 + 1
[2,3,1,4,5,6] => 01000 => 11000 => 2 = 1 + 1
[2,3,4,1,5,6] => 00100 => 10100 => 1 = 0 + 1
[2,3,5,1,4,6] => 00100 => 10100 => 1 = 0 + 1
[2,3,6,1,4,5] => 00100 => 10100 => 1 = 0 + 1
[2,4,1,3,5,6] => 01000 => 11000 => 2 = 1 + 1
[2,4,5,1,3,6] => 00100 => 10100 => 1 = 0 + 1
[2,4,6,1,3,5] => 00100 => 10100 => 1 = 0 + 1
[2,5,1,3,4,6] => 01000 => 11000 => 2 = 1 + 1
[2,5,6,1,3,4] => 00100 => 10100 => 1 = 0 + 1
Description
The length of the longest run of ones in a binary word.
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
Mp00153: Standard tableaux inverse promotionStandard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2 = 1 + 1
[2,3,1] => [[1,2],[3]]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,3,4,2] => [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[1,4,2,3] => [[1,2,4],[3]]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,3,1,4] => [[1,2,4],[3]]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[2,3,4,1] => [[1,2,3],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[2,4,1,3] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 1 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1 = 0 + 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [[1,2,4,5],[3]]
=> 1 = 0 + 1
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> [[1,2,5],[3,4]]
=> 1 = 0 + 1
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> 2 = 1 + 1
[1,2,4,3,5,6] => [[1,2,3,5,6],[4]]
=> [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[1,2,5,3,4,6] => [[1,2,3,5,6],[4]]
=> [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[1,2,6,3,4,5] => [[1,2,3,5,6],[4]]
=> [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,3,4,2,5,6] => [[1,2,3,5,6],[4]]
=> [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[1,3,5,2,4,6] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[1,3,6,2,4,5] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[1,4,6,2,3,5] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[1,5,6,2,3,4] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> [[1,3,4,5,6],[2]]
=> 2 = 1 + 1
[2,3,4,1,5,6] => [[1,2,3,5,6],[4]]
=> [[1,2,4,5,6],[3]]
=> 1 = 0 + 1
[2,3,5,1,4,6] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[2,3,6,1,4,5] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[2,4,5,1,3,6] => [[1,2,3,6],[4,5]]
=> [[1,2,5,6],[3,4]]
=> 1 = 0 + 1
[2,4,6,1,3,5] => [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> 1 = 0 + 1
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> [[1,3,4,5],[2,6]]
=> 2 = 1 + 1
[2,5,6,1,3,4] => [[1,2,3],[4,5,6]]
=> [[1,2,5],[3,4,6]]
=> 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
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
[1,3,2] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 1 + 1
[2,3,1] => [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[1,3,4,2] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[1,4,2,3] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[2,3,1,4] => [[1,2,4],[3]]
=> {{1,2,4},{3}}
=> 2 = 1 + 1
[2,3,4,1] => [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
[2,4,1,3] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 0 + 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 0 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 0 + 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> {{1,2,4,5},{3}}
=> 2 = 1 + 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> {{1,2,3,5},{4}}
=> 1 = 0 + 1
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[3,4,1,2,5] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[3,4,5,1,2] => [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
[3,5,1,2,4] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[4,5,1,2,3] => [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2 = 1 + 1
[1,2,4,3,5,6] => [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[1,2,5,3,4,6] => [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[1,2,6,3,4,5] => [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[1,3,2,4,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[1,3,4,2,5,6] => [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[1,3,5,2,4,6] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[1,3,6,2,4,5] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[1,4,2,3,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[1,4,5,2,3,6] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[1,4,6,2,3,5] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[1,5,2,3,4,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[1,5,6,2,3,4] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[1,6,2,3,4,5] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[2,3,1,4,5,6] => [[1,2,4,5,6],[3]]
=> {{1,2,4,5,6},{3}}
=> 2 = 1 + 1
[2,3,4,1,5,6] => [[1,2,3,5,6],[4]]
=> {{1,2,3,5,6},{4}}
=> 1 = 0 + 1
[2,3,5,1,4,6] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[2,3,6,1,4,5] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[2,4,1,3,5,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 2 = 1 + 1
[2,4,5,1,3,6] => [[1,2,3,6],[4,5]]
=> {{1,2,3,6},{4,5}}
=> 1 = 0 + 1
[2,4,6,1,3,5] => [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 1 = 0 + 1
[2,5,1,3,4,6] => [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 2 = 1 + 1
[2,5,6,1,3,4] => [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 1 = 0 + 1
Description
The number of blocks with odd minimum. See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St000090
Mp00064: Permutations reversePermutations
Mp00071: Permutations descent compositionInteger compositions
Mp00172: Integer compositions rotate back to frontInteger compositions
St000090: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,3,2] => [2,3,1] => [2,1] => [1,2] => 1
[2,3,1] => [1,3,2] => [2,1] => [1,2] => 1
[1,2,4,3] => [3,4,2,1] => [2,1,1] => [1,2,1] => 0
[1,3,2,4] => [4,2,3,1] => [1,2,1] => [1,1,2] => 1
[1,3,4,2] => [2,4,3,1] => [2,1,1] => [1,2,1] => 0
[1,4,2,3] => [3,2,4,1] => [1,2,1] => [1,1,2] => 1
[2,3,1,4] => [4,1,3,2] => [1,2,1] => [1,1,2] => 1
[2,3,4,1] => [1,4,3,2] => [2,1,1] => [1,2,1] => 0
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [1,1,2] => 1
[3,4,1,2] => [2,1,4,3] => [1,2,1] => [1,1,2] => 1
[1,2,4,3,5] => [5,3,4,2,1] => [1,2,1,1] => [1,1,2,1] => 0
[1,2,5,3,4] => [4,3,5,2,1] => [1,2,1,1] => [1,1,2,1] => 0
[1,3,2,4,5] => [5,4,2,3,1] => [1,1,2,1] => [1,1,1,2] => 1
[1,3,4,2,5] => [5,2,4,3,1] => [1,2,1,1] => [1,1,2,1] => 0
[1,3,5,2,4] => [4,2,5,3,1] => [1,2,1,1] => [1,1,2,1] => 0
[1,4,2,3,5] => [5,3,2,4,1] => [1,1,2,1] => [1,1,1,2] => 1
[1,4,5,2,3] => [3,2,5,4,1] => [1,2,1,1] => [1,1,2,1] => 0
[1,5,2,3,4] => [4,3,2,5,1] => [1,1,2,1] => [1,1,1,2] => 1
[2,3,1,4,5] => [5,4,1,3,2] => [1,1,2,1] => [1,1,1,2] => 1
[2,3,4,1,5] => [5,1,4,3,2] => [1,2,1,1] => [1,1,2,1] => 0
[2,3,5,1,4] => [4,1,5,3,2] => [1,2,1,1] => [1,1,2,1] => 0
[2,4,1,3,5] => [5,3,1,4,2] => [1,1,2,1] => [1,1,1,2] => 1
[2,4,5,1,3] => [3,1,5,4,2] => [1,2,1,1] => [1,1,2,1] => 0
[2,5,1,3,4] => [4,3,1,5,2] => [1,1,2,1] => [1,1,1,2] => 1
[3,4,1,2,5] => [5,2,1,4,3] => [1,1,2,1] => [1,1,1,2] => 1
[3,4,5,1,2] => [2,1,5,4,3] => [1,2,1,1] => [1,1,2,1] => 0
[3,5,1,2,4] => [4,2,1,5,3] => [1,1,2,1] => [1,1,1,2] => 1
[4,5,1,2,3] => [3,2,1,5,4] => [1,1,2,1] => [1,1,1,2] => 1
[1,2,4,3,5,6] => [6,5,3,4,2,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,2,5,3,4,6] => [6,4,3,5,2,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,2,6,3,4,5] => [5,4,3,6,2,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,3,2,4,5,6] => [6,5,4,2,3,1] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[1,3,4,2,5,6] => [6,5,2,4,3,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,3,5,2,4,6] => [6,4,2,5,3,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,4,2,3,5,6] => [6,5,3,2,4,1] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[1,4,5,2,3,6] => [6,3,2,5,4,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,4,6,2,3,5] => [5,3,2,6,4,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,5,2,3,4,6] => [6,4,3,2,5,1] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[1,5,6,2,3,4] => [4,3,2,6,5,1] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[1,6,2,3,4,5] => [5,4,3,2,6,1] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[2,3,1,4,5,6] => [6,5,4,1,3,2] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[2,3,4,1,5,6] => [6,5,1,4,3,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[2,3,5,1,4,6] => [6,4,1,5,3,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[2,3,6,1,4,5] => [5,4,1,6,3,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[2,4,1,3,5,6] => [6,5,3,1,4,2] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[2,4,5,1,3,6] => [6,3,1,5,4,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[2,4,6,1,3,5] => [5,3,1,6,4,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
[2,5,1,3,4,6] => [6,4,3,1,5,2] => [1,1,1,2,1] => [1,1,1,1,2] => 1
[2,5,6,1,3,4] => [4,3,1,6,5,2] => [1,1,2,1,1] => [1,1,1,2,1] => 0
Description
The variation of a composition.
The following 191 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000137The Grundy value of an integer partition. St000142The number of even parts of a partition. St000148The number of odd parts of a partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000292The number of ascents of a binary word. St000389The number of runs of ones of odd length in a binary word. St000445The number of rises of length 1 of a Dyck path. St000475The number of parts equal to 1 in a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000549The number of odd partial sums of an integer partition. St000552The number of cut vertices of a graph. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000648The number of 2-excedences of a permutation. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001092The number of distinct even parts of a partition. St001115The number of even descents of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001153The number of blocks with even minimum in a set partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001383The BG-rank of an integer partition. St001395The number of strictly unfriendly partitions of a graph. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001479The number of bridges of a graph. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001691The number of kings in a graph. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001797The number of overfull subgraphs of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001826The maximal number of leaves on a vertex of a graph. St001828The Euler characteristic of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St001957The number of Hasse diagrams with a given underlying undirected graph. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000383The last part of an integer composition. St000553The number of blocks of a graph. St000617The number of global maxima of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000759The smallest missing part in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000917The open packing number of a graph. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001672The restrained domination number of a graph. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001733The number of weak left to right maxima of a Dyck path. St001743The discrepancy of a graph. St000145The Dyson rank of a partition. St000381The largest part of an integer composition. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000981The length of the longest zigzag subpath. St001111The weak 2-dynamic chromatic number of a graph. St001267The length of the Lyndon factorization of the binary word. St001415The length of the longest palindromic prefix of a binary word. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000146The Andrews-Garvan crank of a partition. St000474Dyson's crank of a partition. St000997The even-odd crank of an integer partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000237The number of small exceedances. St001114The number of odd descents of a permutation. St000990The first ascent of a permutation. St000461The rix statistic of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000928The sum of the coefficients of the character polynomial of an integer partition. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001877Number of indecomposable injective modules with projective dimension 2. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000681The Grundy value of Chomp on Ferrers diagrams. 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. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. 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. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. 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. St000477The weight of a partition according to Alladi. 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. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. 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. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000873The aix statistic of a permutation. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000241The number of cyclical small excedances. St000534The number of 2-rises of a permutation. St000663The number of right floats of a permutation. St000989The number of final rises of a permutation. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001827The number of two-component spanning forests of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000654The first descent of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001765The number of connected components of the friends and strangers graph. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001367The smallest number which does not occur as degree of a vertex in a graph. St001377The major index minus the number of inversions of a permutation. St001625The Möbius invariant of a lattice. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001754The number of tolerances of a finite lattice. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001644The dimension of a graph. St001557The number of inversions of the second entry of a permutation. St001948The number of augmented double ascents of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001403The number of vertical separators in a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St000422The energy of a graph, if it is integral. St001645The pebbling number of a connected graph.