Processing math: 97%

Your data matches 113 different statistics following compositions of up to 3 maps.
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St000008: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,1] => 1
[2] => 0
[1,1,1] => 3
[1,2] => 1
[2,1] => 2
[3] => 0
[1,1,1,1] => 6
[1,1,2] => 3
[1,2,1] => 4
[1,3] => 1
[2,1,1] => 5
[2,2] => 2
[3,1] => 3
[4] => 0
[1,1,1,1,1] => 10
[1,1,1,2] => 6
[1,1,2,1] => 7
[1,1,3] => 3
[1,2,1,1] => 8
[1,2,2] => 4
[1,3,1] => 5
[1,4] => 1
[2,1,1,1] => 9
[2,1,2] => 5
[2,2,1] => 6
[2,3] => 2
[3,1,1] => 7
[3,2] => 3
[4,1] => 4
[5] => 0
[1,1,1,1,1,1] => 15
[1,1,1,1,2] => 10
[1,1,1,2,1] => 11
[1,1,1,3] => 6
[1,1,2,1,1] => 12
[1,1,2,2] => 7
[1,1,3,1] => 8
[1,1,4] => 3
[1,2,1,1,1] => 13
[1,2,1,2] => 8
[1,2,2,1] => 9
[1,2,3] => 4
[1,3,1,1] => 10
[1,3,2] => 5
[1,4,1] => 6
[1,5] => 1
[2,1,1,1,1] => 14
[2,1,1,2] => 9
[2,1,2,1] => 10
Description
The major index of the composition. The descents of a composition [c1,c2,,ck] are the partial sums c1,c1+c2,,c1++ck1, excluding the sum of all parts. The major index of a composition is the sum of its descents. For details about the major index see [[Permutations/Descents-Major]].
Mp00094: Integer compositions to binary wordBinary words
Mp00104: Binary words reverseBinary words
Mp00105: Binary words complementBinary words
St000391: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1] => 1 => 1 => 0 => 0
[1,1] => 11 => 11 => 00 => 0
[2] => 10 => 01 => 10 => 1
[1,1,1] => 111 => 111 => 000 => 0
[1,2] => 110 => 011 => 100 => 1
[2,1] => 101 => 101 => 010 => 2
[3] => 100 => 001 => 110 => 3
[1,1,1,1] => 1111 => 1111 => 0000 => 0
[1,1,2] => 1110 => 0111 => 1000 => 1
[1,2,1] => 1101 => 1011 => 0100 => 2
[1,3] => 1100 => 0011 => 1100 => 3
[2,1,1] => 1011 => 1101 => 0010 => 3
[2,2] => 1010 => 0101 => 1010 => 4
[3,1] => 1001 => 1001 => 0110 => 5
[4] => 1000 => 0001 => 1110 => 6
[1,1,1,1,1] => 11111 => 11111 => 00000 => 0
[1,1,1,2] => 11110 => 01111 => 10000 => 1
[1,1,2,1] => 11101 => 10111 => 01000 => 2
[1,1,3] => 11100 => 00111 => 11000 => 3
[1,2,1,1] => 11011 => 11011 => 00100 => 3
[1,2,2] => 11010 => 01011 => 10100 => 4
[1,3,1] => 11001 => 10011 => 01100 => 5
[1,4] => 11000 => 00011 => 11100 => 6
[2,1,1,1] => 10111 => 11101 => 00010 => 4
[2,1,2] => 10110 => 01101 => 10010 => 5
[2,2,1] => 10101 => 10101 => 01010 => 6
[2,3] => 10100 => 00101 => 11010 => 7
[3,1,1] => 10011 => 11001 => 00110 => 7
[3,2] => 10010 => 01001 => 10110 => 8
[4,1] => 10001 => 10001 => 01110 => 9
[5] => 10000 => 00001 => 11110 => 10
[1,1,1,1,1,1] => 111111 => 111111 => 000000 => 0
[1,1,1,1,2] => 111110 => 011111 => 100000 => 1
[1,1,1,2,1] => 111101 => 101111 => 010000 => 2
[1,1,1,3] => 111100 => 001111 => 110000 => 3
[1,1,2,1,1] => 111011 => 110111 => 001000 => 3
[1,1,2,2] => 111010 => 010111 => 101000 => 4
[1,1,3,1] => 111001 => 100111 => 011000 => 5
[1,1,4] => 111000 => 000111 => 111000 => 6
[1,2,1,1,1] => 110111 => 111011 => 000100 => 4
[1,2,1,2] => 110110 => 011011 => 100100 => 5
[1,2,2,1] => 110101 => 101011 => 010100 => 6
[1,2,3] => 110100 => 001011 => 110100 => 7
[1,3,1,1] => 110011 => 110011 => 001100 => 7
[1,3,2] => 110010 => 010011 => 101100 => 8
[1,4,1] => 110001 => 100011 => 011100 => 9
[1,5] => 110000 => 000011 => 111100 => 10
[2,1,1,1,1] => 101111 => 111101 => 000010 => 5
[2,1,1,2] => 101110 => 011101 => 100010 => 6
[2,1,2,1] => 101101 => 101101 => 010010 => 7
Description
The sum of the positions of the ones in a binary word.
Mp00231: Integer compositions bounce pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000009: Standard tableaux ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 0
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,2,3],[4],[5],[6]]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,3,4],[2],[5],[6]]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,2,3,4],[5],[6]]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,4,5],[2],[3],[6]]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 5
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? ∊ {8,9,10,10,12,12,13,14,14,15,15,16,16,16,17,17,19,21}
Description
The charge of a standard tableau.
Mp00231: Integer compositions bounce pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000169: Standard tableaux ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 7
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,2,3],[4],[5],[6]]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,3,4],[2],[5],[6]]
=> 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,2,3,4],[5],[6]]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,4,5],[2],[3],[6]]
=> 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 10
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? ∊ {7,9,11,11,12,12,12,13,13,14,14,15,16,16,18,18,19,20}
Description
The cocharge of a standard tableau. The '''cocharge''' of a standard tableau T, denoted cc(T), is defined to be the cocharge of the reading word of the tableau. The cocharge of a permutation w1w2wn can be computed by the following algorithm: 1) Starting from wn, scan the entries right-to-left until finding the entry 1 with a superscript 0. 2) Continue scanning until the 2 is found, and label this with a superscript 1. Then scan until the 3 is found, labeling with a 2, and so on, incrementing the label each time, until the beginning of the word is reached. Then go back to the end and scan again from right to left, and *do not* increment the superscript label for the first number found in the next scan. Then continue scanning and labeling, each time incrementing the superscript only if we have not cycled around the word since the last labeling. 3) The cocharge is defined as the sum of the superscript labels on the letters.
Mp00231: Integer compositions bounce pathDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
Mp00070: Permutations Robinson-Schensted recording tableauStandard tableaux
St000330: Standard tableaux ⟶ ℤResult quality: 93% values known / values provided: 93%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [[1]]
=> 0
[1,1] => [1,0,1,0]
=> [2,1] => [[1],[2]]
=> 1
[2] => [1,1,0,0]
=> [1,2] => [[1,2]]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => [[1],[2],[3]]
=> 3
[1,2] => [1,0,1,1,0,0]
=> [2,3,1] => [[1,2],[3]]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [3,1,2] => [[1,3],[2]]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [[1,2,3]]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[1],[2],[3],[4]]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[1,2],[3],[4]]
=> 5
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[1,3],[2],[4]]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[1,2,3],[4]]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[1,4],[2],[3]]
=> 3
[2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[1,2],[3,4]]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[1,3,4],[2]]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 9
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [[1,3],[2],[4],[5]]
=> 8
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [[1,4],[2],[3],[5]]
=> 7
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [[1,2],[3,4],[5]]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [[1,3,4],[2],[5]]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 4
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [[1,2],[3,5],[4]]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [[1,3],[2,5],[4]]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 3
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [[1],[2],[3],[4],[5],[6]]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [[1,2],[3],[4],[5],[6]]
=> 14
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [[1,3],[2],[4],[5],[6]]
=> 13
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [[1,2,3],[4],[5],[6]]
=> 12
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [[1,4],[2],[3],[5],[6]]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [[1,2],[3,4],[5],[6]]
=> 11
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [[1,3,4],[2],[5],[6]]
=> 10
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [[1,2,3,4],[5],[6]]
=> 9
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [[1,5],[2],[3],[4],[6]]
=> 11
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [[1,2],[3,5],[4],[6]]
=> 10
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [[1,3],[2,5],[4],[6]]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [[1,2,3],[4,5],[6]]
=> 8
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [6,5,2,3,4,1] => [[1,4,5],[2],[3],[6]]
=> 8
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [5,6,2,3,4,1] => [[1,2,5],[3,4],[6]]
=> 7
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [6,2,3,4,5,1] => [[1,3,4,5],[2],[6]]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> 5
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [[1,6],[2],[3],[4],[5]]
=> 10
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => [[1,2],[3,6],[4],[5]]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => [[1,3],[2,6],[4],[5]]
=> 8
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,3,2,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,2,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,3,4,2,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [6,7,8,5,4,2,3,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [7,8,5,6,4,2,3,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,2,3,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [6,7,8,4,5,2,3,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[1,3,1,1,2] => [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [7,8,6,5,2,3,4,1] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [8,5,6,7,4,3,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [8,7,4,5,6,3,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [7,8,4,5,6,3,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,1,4,1] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [8,4,5,6,7,3,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,3,4,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [7,8,6,3,4,5,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [6,7,8,3,4,5,1,2] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [8,6,7,5,4,1,2,3] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [8,7,5,6,4,1,2,3] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [7,8,6,4,5,1,2,3] => ?
=> ? ∊ {9,10,11,12,12,13,13,14,14,15,16,16,17,18,19,20,20,21}
Description
The (standard) major index of a standard tableau. A descent of a standard tableau T is an index i such that i+1 appears in a row strictly below the row of i. The (standard) major index is the the sum of the descents.
Mp00231: Integer compositions bounce pathDyck paths
Mp00229: Dyck paths Delest-ViennotDyck paths
St000946: Dyck paths ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 97%
Values
[1] => [1,0]
=> [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 1
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 4
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 6
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 5
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 5
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 8
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 4
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 6
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 10
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 7
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 3
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 6
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 10
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 5
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 8
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 13
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 9
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 4
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> 6
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 11
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 15
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 10
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 7
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 12
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 8
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 7
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 11
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 6
[1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,4,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,1,5] => [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {1,5,6,7,8,8,10,11,12,12,13,14,14,15,16,17,18,19,19,20,21,23,26}
Description
The sum of the skew hook positions in a Dyck path. A skew hook is an occurrence of a down step followed by two up steps or of an up step followed by a down step. Write Ui for the i-th up step and Dj for the j-th down step in the Dyck path. Then the skew hook set is the set H={j:Ui1UiDj is a skew hook}{i:Di1DiUj is a skew hook}. This statistic is the sum of all elements in H.
Mp00231: Integer compositions bounce pathDyck paths
St001161: Dyck paths ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
Description
The major index north count of a Dyck path. The descent set des(D) of a Dyck path D=D1D2n with Di{N,E} is given by all indices i such that Di=E and Di+1=N. This is, the positions of the valleys of D. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, ides(D)i, see [[St000027]]. The '''major index north count''' is given by ides(D)#{jiDj=N}.
Mp00231: Integer compositions bounce pathDyck paths
St000947: Dyck paths ⟶ ℤResult quality: 87% values known / values provided: 87%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 2
[3] => [1,1,1,0,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> 0
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 10
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 4
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 9
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 4
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 15
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 10
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 11
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 12
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 3
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 13
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 9
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 4
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 10
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 5
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 14
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 9
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 10
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 5
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[3,5] => [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[4,4] => [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[5,3] => [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[6,2] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
[8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,3,4,5,6,7,7,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,16,17,18,18,19,20,21,22,25}
Description
The major index east count of a Dyck path. The descent set des(D) of a Dyck path D=D1D2n with Di{N,E} is given by all indices i such that Di=E and Di+1=N. This is, the positions of the valleys of D. The '''major index''' of a Dyck path is then the sum of the positions of the valleys, ides(D)i, see [[St000027]]. The '''major index east count''' is given by ides(D)#{jiDj=E}.
Mp00180: Integer compositions to ribbonSkew partitions
Mp00182: Skew partitions outer shapeInteger partitions
Mp00202: Integer partitions first row removalInteger partitions
St000867: Integer partitions ⟶ ℤResult quality: 82% values known / values provided: 82%distinct values known / distinct values provided: 93%
Values
[1] => [[1],[]]
=> [1]
=> []
=> 0
[1,1] => [[1,1],[]]
=> [1,1]
=> [1]
=> 1
[2] => [[2],[]]
=> [2]
=> []
=> 0
[1,1,1] => [[1,1,1],[]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2] => [[2,1],[]]
=> [2,1]
=> [1]
=> 1
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 3
[3] => [[3],[]]
=> [3]
=> []
=> 0
[1,1,1,1] => [[1,1,1,1],[]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[1,1,2] => [[2,1,1],[]]
=> [2,1,1]
=> [1,1]
=> 2
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 4
[1,3] => [[3,1],[]]
=> [3,1]
=> [1]
=> 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 5
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 3
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 6
[4] => [[4],[]]
=> [4]
=> []
=> 0
[1,1,1,1,1] => [[1,1,1,1,1],[]]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,2] => [[2,1,1,1],[]]
=> [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 5
[1,1,3] => [[3,1,1],[]]
=> [3,1,1]
=> [1,1]
=> 2
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 6
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 4
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 7
[1,4] => [[4,1],[]]
=> [4,1]
=> [1]
=> 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 7
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 5
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 8
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 3
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 9
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 6
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 10
[5] => [[5],[]]
=> [5]
=> []
=> 0
[1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5
[1,1,1,1,2] => [[2,1,1,1,1],[]]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 6
[1,1,1,3] => [[3,1,1,1],[]]
=> [3,1,1,1]
=> [1,1,1]
=> 3
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 7
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 5
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 8
[1,1,4] => [[4,1,1],[]]
=> [4,1,1]
=> [1,1]
=> 2
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 8
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 6
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 9
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 4
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 10
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 7
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 11
[1,5] => [[5,1],[]]
=> [5,1]
=> [1]
=> 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 9
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 7
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 10
[1,1,1,1,3,1] => [[3,3,1,1,1,1],[2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,1,2,3] => [[4,2,1,1,1],[1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,1,3,1,1] => [[3,3,3,1,1,1],[2,2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,1,4,1] => [[4,4,1,1,1],[3]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,2,4] => [[5,2,1,1],[1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,4,2] => [[5,4,1,1],[3]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,1,5,1] => [[5,5,1,1],[4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,2,1,1,3] => [[4,2,2,2,1],[1,1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,2,5] => [[6,2,1],[1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,3,1,1,1,1] => [[3,3,3,3,3,1],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,3,4] => [[6,3,1],[2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,4,1,1,1] => [[4,4,4,4,1],[3,3,3]]
=> [4,4,4,4,1]
=> [4,4,4,1]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,5,1,1] => [[5,5,5,1],[4,4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,5,2] => [[6,5,1],[4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[1,6,1] => [[6,6,1],[5]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,1,1,1,3] => [[4,2,2,2,2],[1,1,1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,1,1,4] => [[5,2,2,2],[1,1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,1,2,1,1,1] => [[3,3,3,3,2,2],[2,2,2,1,1]]
=> [3,3,3,3,2,2]
=> [3,3,3,2,2]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,1,5] => [[6,2,2],[1,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,2,1,1,1,1] => [[3,3,3,3,3,2],[2,2,2,2,1]]
=> [3,3,3,3,3,2]
=> [3,3,3,3,2]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,3,1,1,1] => [[4,4,4,4,2],[3,3,3,1]]
=> [4,4,4,4,2]
=> [4,4,4,2]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[2,6] => [[7,2],[1]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,1,1,1,2] => [[4,3,3,3,3],[2,2,2,2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,1,1,2,1] => [[4,4,3,3,3],[3,2,2,2]]
=> [4,4,3,3,3]
=> [4,3,3,3]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> [4,4,4,3,3]
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,1,4] => [[6,3,3],[2,2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> [4,4,4,4,3]
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> [5,5,5,3]
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[3,5] => [[7,3],[2]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [4,4,4,4,4]
=> [4,4,4,4]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> [5,5,4,4]
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> [5,5,5,4]
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [5,5,5,5]
=> [5,5,5]
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[5,1,2] => [[6,5,5],[4,4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[5,3] => [[7,5],[4]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
[6,2] => [[7,6],[5]]
=> ?
=> ?
=> ? ∊ {3,4,5,5,6,6,6,7,7,7,8,8,9,9,10,12,12,13,14,15,15,16,16,16,16,17,17,18,18,19,19,19,20,20,20,21,21,21,22,22,23,23,23,24,25,26}
Description
The sum of the hook lengths in the first row of an integer partition. For a cell in the Ferrers diagram of a partition, the hook length is given by the number of boxes to its right plus the number of boxes below plus one. This statistic is the sum of the hook lengths of the first row of a partition. Put differently, for a partition of size n with first parth λ1, this is \binom{\lambda_1}{2} + n.
Mp00231: Integer compositions bounce pathDyck paths
Mp00032: Dyck paths inverse zeta mapDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000548: Integer partitions ⟶ ℤResult quality: 76% values known / values provided: 76%distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> []
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 4
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 3
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 6
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 7
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 5
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 9
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 5
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 4
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 8
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 3
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 7
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 10
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 5
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> 4
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 9
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> 8
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 12
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> 6
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> 11
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [2,2,1]
=> 5
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> 10
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1]
=> 9
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 14
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> 6
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> 5
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,4,3,3,2,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,2,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [4,4,3,2,1,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,2,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [4,3,3,2,1,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [4,3,2,2,1,1]
=> ? ∊ {13,13,14,14,15,15,16,17}
[1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,2,2,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,1,1,4] => [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,1,2,3] => [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [3,3,3,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,2,1,3] => [1,0,1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,2,4] => [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,3,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,3,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,3,3] => [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,1,4,2] => [1,0,1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [4,4,4,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,1,1,3] => [1,0,1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,1,2,2] => [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,1,4] => [1,0,1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,2,1,2] => [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [3,3,2,2,1,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,2,3] => [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,3,2] => [1,0,1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,4,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,1,0,0,0,0]
=> [4,4,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,2,5] => [1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,3,2,2] => [1,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,3,3,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,3,4] => [1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,4,1,2] => [1,0,1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,4,2,1] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [4,4,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,4,3] => [1,0,1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [5,5,4,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,1,1,1,3] => [1,1,0,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,2,2,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,1,1,4] => [1,1,0,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,2,4] => [1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,3,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,1,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [4,3,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,4,3,2,1,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [4,3,2,2,2,2,1]
=> ? ∊ {13,13,13,13,13,13,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,19,19,19,19,19,20,20,20,20,21,21,21,21,22,22,23,24}
Description
The number of different non-empty partial sums of an integer partition.
The following 103 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000012The area of a Dyck path. St000378The diagonal inversion number of an integer partition. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000492The rob statistic of a set partition. St000499The rcb statistic of a set partition. St001094The depth index of a set partition. St000493The los statistic of a set partition. St000498The lcs statistic of a set partition. St000081The number of edges of a graph. St000041The number of nestings of a perfect matching. St000490The intertwining number of a set partition. St001956The comajor index for set-valued two-row standard Young tableaux. St001814The number of partitions interlacing the given partition. St001759The Rajchgot index of a permutation. St000228The size of a partition. St000147The largest part of an integer partition. St000459The hook length of the base cell of a partition. St000010The length of the partition. St000148The number of odd parts of a partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St000018The number of inversions of a permutation. St000246The number of non-inversions of a permutation. St000692Babson and Steingrímsson's statistic of a permutation. 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. St001933The largest multiplicity of a part in an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000784The maximum of the length and the largest part of the integer partition. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St000063The number of linear extensions of a certain poset defined for an integer partition. St000532The total number of rook placements on a Ferrers board. St000160The multiplicity of the smallest part of a partition. St000475The number of parts equal to 1 in a partition. St001127The sum of the squares of the parts of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001671Haglund's hag of a permutation. St000161The sum of the sizes of the right subtrees of a binary tree. St000446The disorder of a permutation. St000794The mak of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000993The multiplicity of the largest part of an integer partition. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001571The Cartan determinant of the integer partition. St000108The number of partitions contained in the given partition. St000472The sum of the ascent bottoms of a permutation. St000067The inversion number of the alternating sign matrix. St000057The Shynar inversion number of a standard tableau. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000921The number of internal inversions of a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000456The monochromatic index of a connected graph. St000004The major index of a permutation. St000133The "bounce" of a permutation. St000154The sum of the descent bottoms of a permutation. St000156The Denert index of a permutation. St000304The load of a permutation. St000305The inverse major index of a permutation. St000796The stat' of a permutation. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000795The mad of a permutation. St000448The number of pairs of vertices of a graph with distance 2. St001341The number of edges in the center of a graph. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001311The cyclomatic number of a graph. St000357The number of occurrences of the pattern 12-3. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001746The coalition number of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001725The harmonious chromatic number of a graph. St001645The pebbling number of a connected graph. St000719The number of alignments in a perfect matching. St001622The number of join-irreducible elements of a lattice. St000450The number of edges minus the number of vertices plus 2 of a graph. St000360The number of occurrences of the pattern 32-1. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001330The hat guessing number of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001621The number of atoms of a lattice. St000454The largest eigenvalue of a graph if it is integral.