- FindStat

Your data matches 141 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000157
(load all 14 compositions to match this statistic)

Mapping

Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1] => [[1]]
 => 0
[1,2] => [[1,2]]
 => 0
[2,1] => [[1],[2]]
 => 1
[1,2,3] => [[1,2,3]]
 => 0
[1,3,2] => [[1,2],[3]]
 => 1
[2,1,3] => [[1,3],[2]]
 => 1
[2,3,1] => [[1,3],[2]]
 => 1
[3,1,2] => [[1,2],[3]]
 => 1
[3,2,1] => [[1],[2],[3]]
 => 2
[1,2,3,4] => [[1,2,3,4]]
 => 0
[1,2,4,3] => [[1,2,3],[4]]
 => 1
[1,3,2,4] => [[1,2,4],[3]]
 => 1
[1,3,4,2] => [[1,2,4],[3]]
 => 1
[1,4,2,3] => [[1,2,3],[4]]
 => 1
[1,4,3,2] => [[1,2],[3],[4]]
 => 2
[2,1,3,4] => [[1,3,4],[2]]
 => 1
[2,1,4,3] => [[1,3],[2,4]]
 => 2
[2,3,1,4] => [[1,3,4],[2]]
 => 1
[2,3,4,1] => [[1,3,4],[2]]
 => 1
[2,4,1,3] => [[1,3],[2,4]]
 => 2
[2,4,3,1] => [[1,3],[2],[4]]
 => 2
[3,1,2,4] => [[1,2,4],[3]]
 => 1
[3,1,4,2] => [[1,2],[3,4]]
 => 1
[3,2,1,4] => [[1,4],[2],[3]]
 => 2
[3,2,4,1] => [[1,4],[2],[3]]
 => 2
[3,4,1,2] => [[1,2],[3,4]]
 => 1
[3,4,2,1] => [[1,4],[2],[3]]
 => 2
[4,1,2,3] => [[1,2,3],[4]]
 => 1
[4,1,3,2] => [[1,2],[3],[4]]
 => 2
[4,2,1,3] => [[1,3],[2],[4]]
 => 2
[4,2,3,1] => [[1,3],[2],[4]]
 => 2
[4,3,1,2] => [[1,2],[3],[4]]
 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
 => 3
[1,2,3,4,5] => [[1,2,3,4,5]]
 => 0
[1,2,3,5,4] => [[1,2,3,4],[5]]
 => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
 => 1
[1,2,4,5,3] => [[1,2,3,5],[4]]
 => 1
[1,2,5,3,4] => [[1,2,3,4],[5]]
 => 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
 => 2
[1,3,2,4,5] => [[1,2,4,5],[3]]
 => 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
 => 2
[1,3,4,2,5] => [[1,2,4,5],[3]]
 => 1
[1,3,4,5,2] => [[1,2,4,5],[3]]
 => 1
[1,3,5,2,4] => [[1,2,4],[3,5]]
 => 2
[1,3,5,4,2] => [[1,2,4],[3],[5]]
 => 2
[1,4,2,3,5] => [[1,2,3,5],[4]]
 => 1
[1,4,2,5,3] => [[1,2,3],[4,5]]
 => 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
 => 2
[1,4,3,5,2] => [[1,2,5],[3],[4]]
 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
 => 1

Description

The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.

Matching statistic: St000010
(load all 4 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1] => [1]
 => 1 = 0 + 1
[1,2] => [1,2] => [2] => [2]
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => [1,1]
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => [3]
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [2,1] => [2,1]
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1] => [1,1,1]
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4] => [4]
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => [3,1]
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => [2,2]
 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [3,1] => [3,1]
 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => [3,1]
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => [3,1]
 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => [3,1]
 => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[3,1,2,4] => [2,3,1,4] => [2,2] => [2,2]
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,2] => [2,2]
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [2,2] => [2,2]
 => 2 = 1 + 1
[3,4,2,1] => [4,3,1,2] => [1,1,2] => [2,1,1]
 => 3 = 2 + 1
[4,1,2,3] => [2,3,4,1] => [3,1] => [3,1]
 => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => [2,1,1]
 => 3 = 2 + 1
[4,3,1,2] => [3,4,2,1] => [2,1,1] => [2,1,1]
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => [1,1,1,1]
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => [5]
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => [4,1]
 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => [3,1,1]
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => [3,2]
 => 2 = 1 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => [2,2,1]
 => 3 = 2 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => [3,2]
 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => [2,2,1]
 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => [3,2]
 => 2 = 1 + 1
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ? => ?
 => ? = 4 + 1
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ? => ?
 => ? = 4 + 1
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ? => ?
 => ? = 3 + 1
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ? => ?
 => ? = 4 + 1
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ? => ?
 => ? = 3 + 1
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ? => ?
 => ? = 4 + 1
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ? => ?
 => ? = 4 + 1
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ? => ?
 => ? = 3 + 1
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ? => ?
 => ? = 3 + 1
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ? => ?
 => ? = 3 + 1
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ? => ?
 => ? = 4 + 1
[7,6,5,3,2,4,8,1] => [8,5,4,6,3,2,1,7] => ? => ?
 => ? = 5 + 1
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ? => ?
 => ? = 4 + 1
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ? => ?
 => ? = 4 + 1
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ? => ?
 => ? = 4 + 1
[8,7,5,6,2,3,1,4] => [7,5,6,8,3,4,2,1] => ? => ?
 => ? = 4 + 1
[8,5,6,7,2,3,1,4] => [7,5,6,8,2,3,4,1] => ? => ?
 => ? = 3 + 1
[7,8,6,4,1,2,3,5] => [5,6,7,4,8,3,1,2] => ? => ?
 => ? = 3 + 1
[7,6,8,2,3,1,4,5] => [6,4,5,7,8,2,1,3] => ? => ?
 => ? = 3 + 1
[8,7,4,5,2,3,1,6] => [7,5,6,3,4,8,2,1] => ? => ?
 => ? = 4 + 1
[8,7,3,4,5,1,2,6] => [6,7,3,4,5,8,2,1] => ? => ?
 => ? = 3 + 1
[7,8,4,2,3,1,5,6] => [6,4,5,3,7,8,1,2] => ? => ?
 => ? = 3 + 1
[8,7,3,4,1,2,5,6] => [5,6,3,4,7,8,2,1] => ? => ?
 => ? = 3 + 1
[8,6,3,2,4,5,1,7] => [7,4,3,5,6,2,8,1] => ? => ?
 => ? = 4 + 1
[8,6,4,5,3,1,2,7] => [6,7,5,3,4,2,8,1] => ? => ?
 => ? = 4 + 1
[8,5,3,4,6,1,2,7] => [6,7,3,4,2,5,8,1] => ? => ?
 => ? = 3 + 1
[8,4,5,6,2,1,3,7] => [6,5,7,2,3,4,8,1] => ? => ?
 => ? = 3 + 1
[8,4,2,3,5,1,6,7] => [6,3,4,2,5,7,8,1] => ? => ?
 => ? = 3 + 1
[8,3,4,5,1,2,6,7] => [5,6,2,3,4,7,8,1] => ? => ?
 => ? = 2 + 1
[7,6,3,2,4,5,1,8] => [7,4,3,5,6,2,1,8] => ? => ?
 => ? = 4 + 1
[7,4,2,3,5,6,1,8] => [7,3,4,2,5,6,1,8] => ? => ?
 => ? = 3 + 1
[5,6,4,3,2,7,1,8] => ? => ? => ?
 => ? = 4 + 1
[6,4,2,3,5,7,1,8] => [7,3,4,2,5,1,6,8] => ? => ?
 => ? = 3 + 1
[5,4,6,7,2,1,3,8] => [6,5,7,2,1,3,4,8] => ? => ?
 => ? = 3 + 1
[7,6,5,2,3,1,4,8] => [6,4,5,7,3,2,1,8] => ? => ?
 => ? = 4 + 1
[6,7,5,2,3,1,4,8] => [6,4,5,7,3,1,2,8] => ? => ?
 => ? = 3 + 1
[7,5,6,2,3,1,4,8] => [6,4,5,7,2,3,1,8] => ? => ?
 => ? = 3 + 1
[6,5,7,2,3,1,4,8] => [6,4,5,7,2,1,3,8] => ? => ?
 => ? = 3 + 1
[7,6,3,4,2,1,5,8] => [6,5,3,4,7,2,1,8] => ? => ?
 => ? = 4 + 1
[7,6,3,2,4,1,5,8] => [6,4,3,5,7,2,1,8] => ? => ?
 => ? = 4 + 1
[7,6,2,3,4,1,5,8] => [6,3,4,5,7,2,1,8] => ? => ?
 => ? = 3 + 1
[6,7,3,2,1,4,5,8] => [5,4,3,6,7,1,2,8] => ? => ?
 => ? = 3 + 1
[7,6,2,3,1,4,5,8] => [5,3,4,6,7,2,1,8] => ? => ?
 => ? = 3 + 1
[7,4,5,3,2,1,6,8] => [6,5,4,2,3,7,1,8] => ? => ?
 => ? = 4 + 1
[7,5,4,2,3,1,6,8] => [6,4,5,3,2,7,1,8] => ? => ?
 => ? = 4 + 1
[7,5,4,3,1,2,6,8] => ? => ? => ?
 => ? = 4 + 1
[5,6,2,3,4,1,7,8] => [6,3,4,5,1,2,7,8] => ? => ?
 => ? = 2 + 1
[4,3,5,6,1,2,7,8] => [5,6,2,1,3,4,7,8] => ? => ?
 => ? = 2 + 1
[4,5,6,2,1,3,7,8] => [5,4,6,1,2,3,7,8] => ? => ?
 => ? = 2 + 1
[5,6,2,1,3,4,7,8] => [4,3,5,6,1,2,7,8] => ? => ?
 => ? = 2 + 1

Description

The length of the partition.

Matching statistic: St000288
(load all 42 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 92%●distinct values known / distinct values provided: 82%

Values

[1] => [1] => [1] => 1 => 1 = 0 + 1
[1,2] => [1,2] => [2] => 10 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => 11 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => 100 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => 101 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => 110 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => 110 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [2,1] => 101 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1] => 111 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4] => 1000 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => 1001 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => 1010 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => 1010 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [3,1] => 1001 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => 1011 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => 1100 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => 1101 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => 1100 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => 1100 => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => 1101 => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => [1,2,1] => 1101 => 3 = 2 + 1
[3,1,2,4] => [2,3,1,4] => [2,2] => 1010 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,2] => 1010 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => 1110 => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [1,1,2] => 1110 => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [2,2] => 1010 => 2 = 1 + 1
[3,4,2,1] => [4,3,1,2] => [1,1,2] => 1110 => 3 = 2 + 1
[4,1,2,3] => [2,3,4,1] => [3,1] => 1001 => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => [2,1,1] => 1011 => 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [1,2,1] => 1101 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => 1101 => 3 = 2 + 1
[4,3,1,2] => [3,4,2,1] => [2,1,1] => 1011 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => 1111 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => 10000 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => 10001 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => 10010 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => 10010 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => 10001 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => 10011 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => 10100 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => 10101 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => 10100 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => 10100 => 2 = 1 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => 10101 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => 10101 => 3 = 2 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => 10010 => 2 = 1 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => 10010 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => 10110 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => 10110 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => 10010 => 2 = 1 + 1
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ? => ? => ? = 4 + 1
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ? => ? => ? = 4 + 1
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ? => ? => ? = 3 + 1
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ? => ? => ? = 4 + 1
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ? => ? => ? = 3 + 1
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ? => ? => ? = 4 + 1
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ? => ? => ? = 4 + 1
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ? => ? => ? = 3 + 1
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ? => ? => ? = 3 + 1
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ? => ? => ? = 3 + 1
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ? => ? => ? = 4 + 1
[7,6,5,3,2,4,8,1] => [8,5,4,6,3,2,1,7] => ? => ? => ? = 5 + 1
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ? => ? => ? = 4 + 1
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ? => ? => ? = 4 + 1
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ? => ? => ? = 4 + 1
[8,7,5,6,2,3,1,4] => [7,5,6,8,3,4,2,1] => ? => ? => ? = 4 + 1
[8,5,6,7,2,3,1,4] => [7,5,6,8,2,3,4,1] => ? => ? => ? = 3 + 1
[7,8,6,4,1,2,3,5] => [5,6,7,4,8,3,1,2] => ? => ? => ? = 3 + 1
[7,6,8,2,3,1,4,5] => [6,4,5,7,8,2,1,3] => ? => ? => ? = 3 + 1
[8,7,4,5,2,3,1,6] => [7,5,6,3,4,8,2,1] => ? => ? => ? = 4 + 1
[8,7,3,4,5,1,2,6] => [6,7,3,4,5,8,2,1] => ? => ? => ? = 3 + 1
[7,8,4,2,3,1,5,6] => [6,4,5,3,7,8,1,2] => ? => ? => ? = 3 + 1
[8,7,3,4,1,2,5,6] => [5,6,3,4,7,8,2,1] => ? => ? => ? = 3 + 1
[8,6,3,2,4,5,1,7] => [7,4,3,5,6,2,8,1] => ? => ? => ? = 4 + 1
[8,6,4,5,3,1,2,7] => [6,7,5,3,4,2,8,1] => ? => ? => ? = 4 + 1
[8,5,3,4,6,1,2,7] => [6,7,3,4,2,5,8,1] => ? => ? => ? = 3 + 1
[8,4,5,6,2,1,3,7] => [6,5,7,2,3,4,8,1] => ? => ? => ? = 3 + 1
[8,4,2,3,5,1,6,7] => [6,3,4,2,5,7,8,1] => ? => ? => ? = 3 + 1
[8,3,4,5,1,2,6,7] => [5,6,2,3,4,7,8,1] => ? => ? => ? = 2 + 1
[7,6,3,2,4,5,1,8] => [7,4,3,5,6,2,1,8] => ? => ? => ? = 4 + 1
[7,4,2,3,5,6,1,8] => [7,3,4,2,5,6,1,8] => ? => ? => ? = 3 + 1
[5,6,4,3,2,7,1,8] => ? => ? => ? => ? = 4 + 1
[6,4,2,3,5,7,1,8] => [7,3,4,2,5,1,6,8] => ? => ? => ? = 3 + 1
[5,4,6,7,2,1,3,8] => [6,5,7,2,1,3,4,8] => ? => ? => ? = 3 + 1
[7,6,5,2,3,1,4,8] => [6,4,5,7,3,2,1,8] => ? => ? => ? = 4 + 1
[6,7,5,2,3,1,4,8] => [6,4,5,7,3,1,2,8] => ? => ? => ? = 3 + 1
[7,5,6,2,3,1,4,8] => [6,4,5,7,2,3,1,8] => ? => ? => ? = 3 + 1
[6,5,7,2,3,1,4,8] => [6,4,5,7,2,1,3,8] => ? => ? => ? = 3 + 1
[7,6,3,4,2,1,5,8] => [6,5,3,4,7,2,1,8] => ? => ? => ? = 4 + 1
[7,6,3,2,4,1,5,8] => [6,4,3,5,7,2,1,8] => ? => ? => ? = 4 + 1
[7,6,2,3,4,1,5,8] => [6,3,4,5,7,2,1,8] => ? => ? => ? = 3 + 1
[6,7,3,2,1,4,5,8] => [5,4,3,6,7,1,2,8] => ? => ? => ? = 3 + 1
[7,6,2,3,1,4,5,8] => [5,3,4,6,7,2,1,8] => ? => ? => ? = 3 + 1
[7,4,5,3,2,1,6,8] => [6,5,4,2,3,7,1,8] => ? => ? => ? = 4 + 1
[7,5,4,2,3,1,6,8] => [6,4,5,3,2,7,1,8] => ? => ? => ? = 4 + 1
[7,5,4,3,1,2,6,8] => ? => ? => ? => ? = 4 + 1
[5,6,2,3,4,1,7,8] => [6,3,4,5,1,2,7,8] => ? => ? => ? = 2 + 1
[4,3,5,6,1,2,7,8] => [5,6,2,1,3,4,7,8] => ? => ? => ? = 2 + 1
[4,5,6,2,1,3,7,8] => [5,4,6,1,2,3,7,8] => ? => ? => ? = 2 + 1
[5,6,2,1,3,4,7,8] => [4,3,5,6,1,2,7,8] => ? => ? => ? = 2 + 1

Description

The number of ones in a binary word. This is also known as the Hamming weight of the word.

Matching statistic: St000097

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 91% ●values known / values provided: 92%●distinct values known / distinct values provided: 91%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,2,1] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,1,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ? => ?
 => ? = 4 + 1
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ? => ?
 => ? = 4 + 1
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ? => ?
 => ? = 3 + 1
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ? => ?
 => ? = 4 + 1
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ? => ?
 => ? = 3 + 1
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ? => ?
 => ? = 4 + 1
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ? => ?
 => ? = 4 + 1
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ? => ?
 => ? = 3 + 1
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ? => ?
 => ? = 3 + 1
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ? => ?
 => ? = 3 + 1
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ? => ?
 => ? = 4 + 1
[7,6,5,3,2,4,8,1] => [8,5,4,6,3,2,1,7] => ? => ?
 => ? = 5 + 1
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ? => ?
 => ? = 4 + 1
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ? => ?
 => ? = 4 + 1
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ? => ?
 => ? = 4 + 1
[8,7,5,6,2,3,1,4] => [7,5,6,8,3,4,2,1] => ? => ?
 => ? = 4 + 1
[8,5,6,7,2,3,1,4] => [7,5,6,8,2,3,4,1] => ? => ?
 => ? = 3 + 1
[7,8,6,4,1,2,3,5] => [5,6,7,4,8,3,1,2] => ? => ?
 => ? = 3 + 1
[7,6,8,2,3,1,4,5] => [6,4,5,7,8,2,1,3] => ? => ?
 => ? = 3 + 1
[8,7,4,5,2,3,1,6] => [7,5,6,3,4,8,2,1] => ? => ?
 => ? = 4 + 1
[8,7,3,4,5,1,2,6] => [6,7,3,4,5,8,2,1] => ? => ?
 => ? = 3 + 1
[7,8,4,2,3,1,5,6] => [6,4,5,3,7,8,1,2] => ? => ?
 => ? = 3 + 1
[8,7,3,4,1,2,5,6] => [5,6,3,4,7,8,2,1] => ? => ?
 => ? = 3 + 1
[8,6,3,2,4,5,1,7] => [7,4,3,5,6,2,8,1] => ? => ?
 => ? = 4 + 1
[8,6,4,5,3,1,2,7] => [6,7,5,3,4,2,8,1] => ? => ?
 => ? = 4 + 1
[8,5,3,4,6,1,2,7] => [6,7,3,4,2,5,8,1] => ? => ?
 => ? = 3 + 1
[8,4,5,6,2,1,3,7] => [6,5,7,2,3,4,8,1] => ? => ?
 => ? = 3 + 1
[8,4,2,3,5,1,6,7] => [6,3,4,2,5,7,8,1] => ? => ?
 => ? = 3 + 1
[8,3,4,5,1,2,6,7] => [5,6,2,3,4,7,8,1] => ? => ?
 => ? = 2 + 1
[7,6,3,2,4,5,1,8] => [7,4,3,5,6,2,1,8] => ? => ?
 => ? = 4 + 1
[7,4,2,3,5,6,1,8] => [7,3,4,2,5,6,1,8] => ? => ?
 => ? = 3 + 1
[5,6,4,3,2,7,1,8] => ? => ? => ?
 => ? = 4 + 1
[6,4,2,3,5,7,1,8] => [7,3,4,2,5,1,6,8] => ? => ?
 => ? = 3 + 1
[5,4,6,7,2,1,3,8] => [6,5,7,2,1,3,4,8] => ? => ?
 => ? = 3 + 1
[7,6,5,2,3,1,4,8] => [6,4,5,7,3,2,1,8] => ? => ?
 => ? = 4 + 1
[6,7,5,2,3,1,4,8] => [6,4,5,7,3,1,2,8] => ? => ?
 => ? = 3 + 1
[7,5,6,2,3,1,4,8] => [6,4,5,7,2,3,1,8] => ? => ?
 => ? = 3 + 1
[6,5,7,2,3,1,4,8] => [6,4,5,7,2,1,3,8] => ? => ?
 => ? = 3 + 1
[7,6,3,4,2,1,5,8] => [6,5,3,4,7,2,1,8] => ? => ?
 => ? = 4 + 1
[7,6,3,2,4,1,5,8] => [6,4,3,5,7,2,1,8] => ? => ?
 => ? = 4 + 1
[7,6,2,3,4,1,5,8] => [6,3,4,5,7,2,1,8] => ? => ?
 => ? = 3 + 1
[6,7,3,2,1,4,5,8] => [5,4,3,6,7,1,2,8] => ? => ?
 => ? = 3 + 1
[7,6,2,3,1,4,5,8] => [5,3,4,6,7,2,1,8] => ? => ?
 => ? = 3 + 1
[7,4,5,3,2,1,6,8] => [6,5,4,2,3,7,1,8] => ? => ?
 => ? = 4 + 1
[7,5,4,2,3,1,6,8] => [6,4,5,3,2,7,1,8] => ? => ?
 => ? = 4 + 1
[7,5,4,3,1,2,6,8] => ? => ? => ?
 => ? = 4 + 1
[5,6,2,3,4,1,7,8] => [6,3,4,5,1,2,7,8] => ? => ?
 => ? = 2 + 1
[4,3,5,6,1,2,7,8] => [5,6,2,1,3,4,7,8] => ? => ?
 => ? = 2 + 1
[4,5,6,2,1,3,7,8] => [5,4,6,1,2,3,7,8] => ? => ?
 => ? = 2 + 1
[5,6,2,1,3,4,7,8] => [4,3,5,6,1,2,7,8] => ? => ?
 => ? = 2 + 1

Description

The order of the largest clique of the graph. A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.

Matching statistic: St000377

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => [2]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [2,1]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [3]
 => 1
[2,1,3] => [2,1,3] => [2,1]
 => [3]
 => 1
[2,3,1] => [3,1,2] => [2,1]
 => [3]
 => 1
[3,1,2] => [2,3,1] => [2,1]
 => [3]
 => 1
[3,2,1] => [3,2,1] => [3]
 => [1,1,1]
 => 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [3,1]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [2,2]
 => 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [2,2]
 => 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
 => [2,2]
 => 1
[1,4,2,3] => [1,3,4,2] => [2,1,1]
 => [2,2]
 => 1
[1,4,3,2] => [1,4,3,2] => [3,1]
 => [2,1,1]
 => 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [2,2]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [4]
 => 2
[2,3,1,4] => [3,1,2,4] => [2,1,1]
 => [2,2]
 => 1
[2,3,4,1] => [4,1,2,3] => [2,1,1]
 => [2,2]
 => 1
[2,4,1,3] => [3,1,4,2] => [2,2]
 => [4]
 => 2
[2,4,3,1] => [4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[3,1,2,4] => [2,3,1,4] => [2,1,1]
 => [2,2]
 => 1
[3,1,4,2] => [2,4,1,3] => [2,1,1]
 => [2,2]
 => 1
[3,2,1,4] => [3,2,1,4] => [3,1]
 => [2,1,1]
 => 2
[3,2,4,1] => [4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[3,4,1,2] => [3,4,1,2] => [2,1,1]
 => [2,2]
 => 1
[3,4,2,1] => [4,3,1,2] => [3,1]
 => [2,1,1]
 => 2
[4,1,2,3] => [2,3,4,1] => [2,1,1]
 => [2,2]
 => 1
[4,1,3,2] => [2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [4,2,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,1,2] => [3,4,2,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,2,1] => [4,3,2,1] => [4]
 => [1,1,1,1]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [3,2]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
 => [4,1]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [2,2,1]
 => 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1]
 => [2,2,1]
 => 2
[1,3,5,4,2] => [1,5,2,4,3] => [3,1,1]
 => [4,1]
 => 2
[1,4,2,3,5] => [1,3,4,2,5] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,2,5,3] => [1,3,5,2,4] => [2,1,1,1]
 => [3,1,1]
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
 => [4,1]
 => 2
[1,4,3,5,2] => [1,5,3,2,4] => [3,1,1]
 => [4,1]
 => 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
 => [3,1,1]
 => 1
[7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => ?
 => ?
 => ? = 5
[7,8,6,3,4,5,2,1] => [8,7,4,5,6,3,1,2] => ?
 => ?
 => ? = 4
[7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => ?
 => ?
 => ? = 5
[8,5,3,4,6,7,2,1] => [8,7,3,4,2,5,6,1] => ?
 => ?
 => ? = 4
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ?
 => ?
 => ? = 4
[7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => ?
 => ?
 => ? = 5
[6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ?
 => ?
 => ? = 5
[7,4,5,6,3,8,2,1] => [8,7,5,2,3,4,1,6] => ?
 => ?
 => ? = 4
[7,6,4,3,5,8,2,1] => [8,7,4,3,5,2,1,6] => ?
 => ?
 => ? = 5
[8,7,5,6,3,2,4,1] => [8,6,5,7,3,4,2,1] => ?
 => ?
 => ? = 5
[6,7,5,8,3,2,4,1] => [8,6,5,7,3,1,2,4] => ?
 => ?
 => ? = 4
[7,6,8,5,2,3,4,1] => [8,5,6,7,4,2,1,3] => ?
 => ?
 => ? = 4
[6,7,5,8,2,3,4,1] => [8,5,6,7,3,1,2,4] => ?
 => ?
 => ? = 3
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ?
 => ?
 => ? = 4
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ?
 => ?
 => ? = 3
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ?
 => ?
 => ? = 4
[8,7,5,3,2,4,6,1] => [8,5,4,6,3,7,2,1] => ?
 => ?
 => ? = 5
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ?
 => ?
 => ? = 3
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ?
 => ?
 => ? = 4
[8,5,6,4,2,3,7,1] => [8,5,6,4,2,3,7,1] => ?
 => ?
 => ? = 4
[8,5,4,2,3,6,7,1] => [8,4,5,3,2,6,7,1] => ?
 => ?
 => ? = 4
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ?
 => ?
 => ? = 4
[7,5,3,4,6,2,8,1] => [8,6,3,4,2,5,1,7] => ?
 => ?
 => ? = 4
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ?
 => ?
 => ? = 3
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ?
 => ?
 => ? = 3
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ?
 => ?
 => ? = 3
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ?
 => ?
 => ? = 4
[7,6,5,3,2,4,8,1] => [8,5,4,6,3,2,1,7] => ?
 => ?
 => ? = 5
[6,7,5,3,2,4,8,1] => [8,5,4,6,3,1,2,7] => ?
 => ?
 => ? = 4
[6,5,7,3,2,4,8,1] => [8,5,4,6,2,1,3,7] => ?
 => ?
 => ? = 4
[7,5,4,2,3,6,8,1] => [8,4,5,3,2,6,1,7] => ?
 => ?
 => ? = 4
[7,5,2,3,4,6,8,1] => [8,3,4,5,2,6,1,7] => ?
 => ?
 => ? = 3
[7,4,3,2,5,6,8,1] => [8,4,3,2,5,6,1,7] => ?
 => ?
 => ? = 4
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ?
 => ?
 => ? = 4
[4,5,3,2,6,7,8,1] => [8,4,3,1,2,5,6,7] => ?
 => ?
 => ? = 3
[5,3,2,4,6,7,8,1] => [8,3,2,4,1,5,6,7] => ?
 => ?
 => ? = 3
[8,6,7,4,5,3,1,2] => [7,8,6,4,5,2,3,1] => ?
 => ?
 => ? = 4
[6,7,8,4,5,3,1,2] => [7,8,6,4,5,1,2,3] => ?
 => ?
 => ? = 3
[8,5,6,4,7,3,1,2] => [7,8,6,4,2,3,5,1] => ?
 => ?
 => ? = 4
[7,5,6,4,8,3,1,2] => [7,8,6,4,2,3,1,5] => ?
 => ?
 => ? = 4
[5,4,6,7,8,3,1,2] => [7,8,6,2,1,3,4,5] => ?
 => ?
 => ? = 3
[8,6,5,7,3,4,1,2] => [7,8,5,6,3,2,4,1] => ?
 => ?
 => ? = 4
[8,7,6,3,4,5,1,2] => [7,8,4,5,6,3,2,1] => ?
 => ?
 => ? = 4
[8,7,4,3,5,6,1,2] => [7,8,4,3,5,6,2,1] => ?
 => ?
 => ? = 4
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ?
 => ?
 => ? = 4
[7,6,4,3,5,8,1,2] => [7,8,4,3,5,2,1,6] => ?
 => ?
 => ? = 4
[6,7,5,4,8,2,1,3] => [7,6,8,4,3,1,2,5] => ?
 => ?
 => ? = 4
[6,5,7,4,8,2,1,3] => [7,6,8,4,2,1,3,5] => ?
 => ?
 => ? = 4
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ?
 => ?
 => ? = 4
[5,4,6,7,8,2,1,3] => [7,6,8,2,1,3,4,5] => ?
 => ?
 => ? = 3

Description

The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.

Matching statistic: St001176
(load all 2 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => [1]
 => [1]
 => 0
[1,2] => [1,2] => [1,1]
 => [2]
 => 0
[2,1] => [2,1] => [2]
 => [1,1]
 => 1
[1,2,3] => [1,2,3] => [1,1,1]
 => [3]
 => 0
[1,3,2] => [1,3,2] => [2,1]
 => [2,1]
 => 1
[2,1,3] => [2,1,3] => [2,1]
 => [2,1]
 => 1
[2,3,1] => [3,1,2] => [2,1]
 => [2,1]
 => 1
[3,1,2] => [2,3,1] => [2,1]
 => [2,1]
 => 1
[3,2,1] => [3,2,1] => [3]
 => [1,1,1]
 => 2
[1,2,3,4] => [1,2,3,4] => [1,1,1,1]
 => [4]
 => 0
[1,2,4,3] => [1,2,4,3] => [2,1,1]
 => [3,1]
 => 1
[1,3,2,4] => [1,3,2,4] => [2,1,1]
 => [3,1]
 => 1
[1,3,4,2] => [1,4,2,3] => [2,1,1]
 => [3,1]
 => 1
[1,4,2,3] => [1,3,4,2] => [2,1,1]
 => [3,1]
 => 1
[1,4,3,2] => [1,4,3,2] => [3,1]
 => [2,1,1]
 => 2
[2,1,3,4] => [2,1,3,4] => [2,1,1]
 => [3,1]
 => 1
[2,1,4,3] => [2,1,4,3] => [2,2]
 => [2,2]
 => 2
[2,3,1,4] => [3,1,2,4] => [2,1,1]
 => [3,1]
 => 1
[2,3,4,1] => [4,1,2,3] => [2,1,1]
 => [3,1]
 => 1
[2,4,1,3] => [3,1,4,2] => [2,2]
 => [2,2]
 => 2
[2,4,3,1] => [4,1,3,2] => [3,1]
 => [2,1,1]
 => 2
[3,1,2,4] => [2,3,1,4] => [2,1,1]
 => [3,1]
 => 1
[3,1,4,2] => [2,4,1,3] => [2,1,1]
 => [3,1]
 => 1
[3,2,1,4] => [3,2,1,4] => [3,1]
 => [2,1,1]
 => 2
[3,2,4,1] => [4,2,1,3] => [3,1]
 => [2,1,1]
 => 2
[3,4,1,2] => [3,4,1,2] => [2,1,1]
 => [3,1]
 => 1
[3,4,2,1] => [4,3,1,2] => [3,1]
 => [2,1,1]
 => 2
[4,1,2,3] => [2,3,4,1] => [2,1,1]
 => [3,1]
 => 1
[4,1,3,2] => [2,4,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,1,3] => [3,2,4,1] => [3,1]
 => [2,1,1]
 => 2
[4,2,3,1] => [4,2,3,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,1,2] => [3,4,2,1] => [3,1]
 => [2,1,1]
 => 2
[4,3,2,1] => [4,3,2,1] => [4]
 => [1,1,1,1]
 => 3
[1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
 => [5]
 => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,4,5,3] => [1,2,5,3,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,1,1,1]
 => [4,1]
 => 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
 => [3,1,1]
 => 2
[1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
 => [3,2]
 => 2
[1,3,4,2,5] => [1,4,2,3,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1]
 => [3,2]
 => 2
[1,3,5,4,2] => [1,5,2,4,3] => [3,1,1]
 => [3,1,1]
 => 2
[1,4,2,3,5] => [1,3,4,2,5] => [2,1,1,1]
 => [4,1]
 => 1
[1,4,2,5,3] => [1,3,5,2,4] => [2,1,1,1]
 => [4,1]
 => 1
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
 => [3,1,1]
 => 2
[1,4,3,5,2] => [1,5,3,2,4] => [3,1,1]
 => [3,1,1]
 => 2
[1,4,5,2,3] => [1,4,5,2,3] => [2,1,1,1]
 => [4,1]
 => 1
[7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => ?
 => ?
 => ? = 5
[7,8,6,3,4,5,2,1] => [8,7,4,5,6,3,1,2] => ?
 => ?
 => ? = 4
[7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => ?
 => ?
 => ? = 5
[8,5,3,4,6,7,2,1] => [8,7,3,4,2,5,6,1] => ?
 => ?
 => ? = 4
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ?
 => ?
 => ? = 4
[7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => ?
 => ?
 => ? = 5
[6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ?
 => ?
 => ? = 5
[7,4,5,6,3,8,2,1] => [8,7,5,2,3,4,1,6] => ?
 => ?
 => ? = 4
[7,6,4,3,5,8,2,1] => [8,7,4,3,5,2,1,6] => ?
 => ?
 => ? = 5
[8,7,5,6,3,2,4,1] => [8,6,5,7,3,4,2,1] => ?
 => ?
 => ? = 5
[6,7,5,8,3,2,4,1] => [8,6,5,7,3,1,2,4] => ?
 => ?
 => ? = 4
[7,6,8,5,2,3,4,1] => [8,5,6,7,4,2,1,3] => ?
 => ?
 => ? = 4
[6,7,5,8,2,3,4,1] => [8,5,6,7,3,1,2,4] => ?
 => ?
 => ? = 3
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ?
 => ?
 => ? = 4
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ?
 => ?
 => ? = 3
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ?
 => ?
 => ? = 4
[8,7,5,3,2,4,6,1] => [8,5,4,6,3,7,2,1] => ?
 => ?
 => ? = 5
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ?
 => ?
 => ? = 3
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ?
 => ?
 => ? = 4
[8,5,6,4,2,3,7,1] => [8,5,6,4,2,3,7,1] => ?
 => ?
 => ? = 4
[8,5,4,2,3,6,7,1] => [8,4,5,3,2,6,7,1] => ?
 => ?
 => ? = 4
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ?
 => ?
 => ? = 4
[7,5,3,4,6,2,8,1] => [8,6,3,4,2,5,1,7] => ?
 => ?
 => ? = 4
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ?
 => ?
 => ? = 3
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ?
 => ?
 => ? = 3
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ?
 => ?
 => ? = 3
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ?
 => ?
 => ? = 4
[7,6,5,3,2,4,8,1] => [8,5,4,6,3,2,1,7] => ?
 => ?
 => ? = 5
[6,7,5,3,2,4,8,1] => [8,5,4,6,3,1,2,7] => ?
 => ?
 => ? = 4
[6,5,7,3,2,4,8,1] => [8,5,4,6,2,1,3,7] => ?
 => ?
 => ? = 4
[7,5,4,2,3,6,8,1] => [8,4,5,3,2,6,1,7] => ?
 => ?
 => ? = 4
[7,5,2,3,4,6,8,1] => [8,3,4,5,2,6,1,7] => ?
 => ?
 => ? = 3
[7,4,3,2,5,6,8,1] => [8,4,3,2,5,6,1,7] => ?
 => ?
 => ? = 4
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ?
 => ?
 => ? = 4
[4,5,3,2,6,7,8,1] => [8,4,3,1,2,5,6,7] => ?
 => ?
 => ? = 3
[5,3,2,4,6,7,8,1] => [8,3,2,4,1,5,6,7] => ?
 => ?
 => ? = 3
[8,6,7,4,5,3,1,2] => [7,8,6,4,5,2,3,1] => ?
 => ?
 => ? = 4
[6,7,8,4,5,3,1,2] => [7,8,6,4,5,1,2,3] => ?
 => ?
 => ? = 3
[8,5,6,4,7,3,1,2] => [7,8,6,4,2,3,5,1] => ?
 => ?
 => ? = 4
[7,5,6,4,8,3,1,2] => [7,8,6,4,2,3,1,5] => ?
 => ?
 => ? = 4
[5,4,6,7,8,3,1,2] => [7,8,6,2,1,3,4,5] => ?
 => ?
 => ? = 3
[8,6,5,7,3,4,1,2] => [7,8,5,6,3,2,4,1] => ?
 => ?
 => ? = 4
[8,7,6,3,4,5,1,2] => [7,8,4,5,6,3,2,1] => ?
 => ?
 => ? = 4
[8,7,4,3,5,6,1,2] => [7,8,4,3,5,6,2,1] => ?
 => ?
 => ? = 4
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ?
 => ?
 => ? = 4
[7,6,4,3,5,8,1,2] => [7,8,4,3,5,2,1,6] => ?
 => ?
 => ? = 4
[6,7,5,4,8,2,1,3] => [7,6,8,4,3,1,2,5] => ?
 => ?
 => ? = 4
[6,5,7,4,8,2,1,3] => [7,6,8,4,2,1,3,5] => ?
 => ?
 => ? = 4
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ?
 => ?
 => ? = 4
[5,4,6,7,8,2,1,3] => [7,6,8,2,1,3,4,5] => ?
 => ?
 => ? = 3

Description

The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.

Matching statistic: St000507
(load all 5 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 91%

Values

[1] => [.,.]
 => [1] => [[1]]
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => [2,1] => [[1],[2]]
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,2] => [[1,2]]
 => 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => [[1],[2],[3]]
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => [[1,2],[3]]
 => 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => [[1,2],[3]]
 => 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => [[1,2],[3]]
 => 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => [[1,3],[2]]
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => [[1,2,3]]
 => 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [[1,3],[2],[4]]
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [[1,2],[3],[4]]
 => 2 = 1 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 3 = 2 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [[1,2,3],[4]]
 => 3 = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [[1,3],[2,4]]
 => 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [[1,2,3],[4]]
 => 3 = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [[1,4],[2],[3]]
 => 2 = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [[1,2,4],[3]]
 => 3 = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [[1,2,4],[3]]
 => 3 = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [[1,3,4],[2]]
 => 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => [[1,2,3,4]]
 => 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
 => 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [[1,2,3],[4],[5]]
 => 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [[1,3],[2,4],[5]]
 => 2 = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[7,8,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[8,5,3,4,6,7,2,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[7,3,4,5,6,8,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[8,5,6,7,3,2,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
 => [1,4,3,2,6,5,7,8] => ?
 => ? = 4 + 1
[7,8,5,6,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[8,6,7,3,4,2,5,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[8,6,7,3,2,4,5,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[7,8,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,6,8,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,8,3,4,2,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[7,8,3,2,4,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[8,5,6,3,4,2,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,3,4,5,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,4,2,3,7,1] => ?
 => ? => ?
 => ? = 4 + 1
[8,5,3,4,2,6,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[7,3,4,5,6,2,8,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[7,6,4,5,2,3,8,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[7,5,6,2,3,4,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[7,6,4,2,3,5,8,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[7,6,2,3,4,5,8,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,5,2,3,4,6,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[7,4,3,2,5,6,8,1] => ?
 => ? => ?
 => ? = 4 + 1
[7,3,2,4,5,6,8,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[8,7,6,4,5,3,1,2] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 5 + 1
[8,6,7,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
 => [2,1,3,5,4,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,4,7,3,1,2] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
 => [2,1,3,4,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,4,5,7,3,1,2] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
 => [2,1,3,5,4,7,6,8] => ?
 => ? = 4 + 1
[8,5,4,6,7,3,1,2] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
 => [2,1,3,4,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,5,7,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
 => [2,1,4,3,5,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,7,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
 => [2,1,4,3,7,6,5,8] => ?
 => ? = 3 + 1
[8,7,4,5,3,6,1,2] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
 => [2,1,3,6,5,4,7,8] => ?
 => ? = 4 + 1
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
 => [2,1,3,6,5,4,8,7] => ?
 => ? = 3 + 1
[8,7,4,3,5,6,1,2] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
 => [2,1,3,6,5,4,7,8] => ?
 => ? = 4 + 1
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
 => [2,1,3,6,5,4,8,7] => ?
 => ? = 3 + 1
[8,7,3,4,5,6,1,2] => [[[[.,[.,.]],[.,[.,[.,.]]]],.],.]
 => [2,1,6,5,4,3,7,8] => ?
 => ? = 3 + 1
[8,5,4,3,6,7,1,2] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
 => [2,1,3,4,7,6,5,8] => ?
 => ? = 4 + 1
[8,5,3,4,6,7,1,2] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
 => [2,1,4,3,7,6,5,8] => ?
 => ? = 3 + 1
[8,4,3,5,6,7,1,2] => [[[[.,[.,.]],.],[.,[.,[.,.]]]],.]
 => [2,1,3,7,6,5,4,8] => ?
 => ? = 3 + 1
[7,6,3,4,5,8,1,2] => [[[[.,[.,.]],[.,[.,.]]],.],[.,.]]
 => [2,1,5,4,3,6,8,7] => ?
 => ? = 3 + 1
[7,6,8,4,5,2,1,3] => ?
 => ? => ?
 => ? = 4 + 1
[7,6,4,5,8,2,1,3] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[7,8,5,6,4,1,2,3] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
 => [3,2,1,4,6,5,8,7] => ?
 => ? = 3 + 1
[8,6,5,7,4,1,2,3] => [[[[[.,[.,[.,.]]],.],.],[.,.]],.]
 => [3,2,1,4,5,7,6,8] => ?
 => ? = 4 + 1
[7,5,6,8,4,1,2,3] => [[[[.,[.,[.,.]]],.],[.,.]],[.,.]]
 => [3,2,1,4,6,5,8,7] => ?
 => ? = 3 + 1
[6,7,8,4,5,1,2,3] => [[[.,[.,[.,.]]],[.,.]],[.,[.,.]]]
 => [3,2,1,5,4,8,7,6] => ?
 => ? = 2 + 1
[8,7,5,4,6,1,2,3] => [[[[[.,[.,[.,.]]],.],[.,.]],.],.]
 => [3,2,1,4,6,5,7,8] => ?
 => ? = 4 + 1

Description

The number of ascents of a standard tableau. Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).

Matching statistic: St000147

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 91%

Values

[1] => [.,.]
 => [1] => [1]
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => [2,1] => [1,1]
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,2] => [2]
 => 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [3,2,1] => [1,1,1]
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [2,3,1] => [2,1]
 => 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
 => [1,3,2] => [2,1]
 => 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
 => [2,1,3] => [2,1]
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,2,3] => [3]
 => 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,1,1,1]
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,1,1]
 => 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,1,1]
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [2,4,3,1] => [2,1,1]
 => 2 = 1 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,1,1]
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,1]
 => 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,1]
 => 3 = 2 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => 2 = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,1]
 => 2 = 1 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,1]
 => 3 = 2 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,3,4,2] => [3,1]
 => 3 = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [3,1]
 => 3 = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [3,1]
 => 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,2]
 => 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,2,4,3] => [3,1]
 => 3 = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,1]
 => 2 = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,1]
 => 3 = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1]
 => 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,3,2,4] => [3,1]
 => 3 = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [2,1,3,4] => [3,1]
 => 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,2,3,4] => [4]
 => 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,1,1,1,1]
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,1,1]
 => 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [2,1,1,1]
 => 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,1,1]
 => 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [2,2,1]
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [2,2,1]
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,1,1]
 => 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,1,1]
 => 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [2,2,1]
 => 2 = 1 + 1
[8,5,6,7,3,4,2,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[6,7,8,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[7,8,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[8,5,3,4,6,7,2,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[7,3,4,5,6,8,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[6,3,4,5,7,8,2,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[7,6,8,4,5,2,3,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[8,5,6,7,3,2,4,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
 => [1,4,3,2,6,5,7,8] => ?
 => ? = 4 + 1
[7,8,5,6,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[8,6,7,3,4,2,5,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[6,7,8,3,4,2,5,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[8,6,7,3,2,4,5,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[6,7,8,3,2,4,5,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[7,8,6,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,6,8,2,3,4,5,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,8,3,4,2,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[7,8,3,2,4,5,6,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[8,5,6,3,4,2,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,3,4,5,2,7,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,2,5,4,3,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,4,2,3,7,1] => ?
 => ? => ?
 => ? = 4 + 1
[8,5,3,4,2,6,7,1] => [[[[[.,.],.],[.,.]],[.,[.,.]]],.]
 => [1,2,4,3,7,6,5,8] => ?
 => ? = 4 + 1
[7,3,4,5,6,2,8,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[6,3,4,5,7,2,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[7,6,4,5,2,3,8,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[7,5,6,2,3,4,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[7,6,4,2,3,5,8,1] => [[[[[.,.],[.,.]],[.,.]],.],[.,.]]
 => [1,3,2,5,4,6,8,7] => ?
 => ? = 4 + 1
[6,7,3,2,4,5,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[7,6,2,3,4,5,8,1] => [[[[.,.],[.,[.,[.,.]]]],.],[.,.]]
 => [1,5,4,3,2,6,8,7] => ?
 => ? = 3 + 1
[7,5,2,3,4,6,8,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,4,3,2,6,5,8,7] => ?
 => ? = 3 + 1
[7,4,3,2,5,6,8,1] => ?
 => ? => ?
 => ? = 4 + 1
[7,3,2,4,5,6,8,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,2,6,5,4,3,8,7] => ?
 => ? = 3 + 1
[6,3,4,5,2,7,8,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,2,5,4,3,8,7,6] => ?
 => ? = 3 + 1
[8,7,6,4,5,3,1,2] => [[[[[[.,[.,.]],.],[.,.]],.],.],.]
 => [2,1,3,5,4,6,7,8] => ?
 => ? = 5 + 1
[7,8,6,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
 => [2,1,3,5,4,6,8,7] => ?
 => ? = 4 + 1
[8,6,7,4,5,3,1,2] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
 => [2,1,3,5,4,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,4,7,3,1,2] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
 => [2,1,3,4,7,6,5,8] => ?
 => ? = 4 + 1
[8,6,4,5,7,3,1,2] => [[[[[.,[.,.]],.],[.,.]],[.,.]],.]
 => [2,1,3,5,4,7,6,8] => ?
 => ? = 4 + 1
[8,5,4,6,7,3,1,2] => [[[[[.,[.,.]],.],.],[.,[.,.]]],.]
 => [2,1,3,4,7,6,5,8] => ?
 => ? = 4 + 1
[7,6,4,5,8,3,1,2] => [[[[[.,[.,.]],.],[.,.]],.],[.,.]]
 => [2,1,3,5,4,6,8,7] => ?
 => ? = 4 + 1
[7,8,6,5,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => [2,1,4,3,5,6,8,7] => ?
 => ? = 4 + 1
[7,6,8,5,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => [2,1,4,3,5,6,8,7] => ?
 => ? = 4 + 1
[8,6,5,7,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],[.,.]],.]
 => [2,1,4,3,5,7,6,8] => ?
 => ? = 4 + 1
[8,5,6,7,3,4,1,2] => [[[[.,[.,.]],[.,.]],[.,[.,.]]],.]
 => [2,1,4,3,7,6,5,8] => ?
 => ? = 3 + 1
[7,6,5,8,3,4,1,2] => [[[[[.,[.,.]],[.,.]],.],.],[.,.]]
 => [2,1,4,3,5,6,8,7] => ?
 => ? = 4 + 1
[8,7,4,5,3,6,1,2] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
 => [2,1,3,6,5,4,7,8] => ?
 => ? = 4 + 1
[7,8,4,5,3,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
 => [2,1,3,6,5,4,8,7] => ?
 => ? = 3 + 1
[8,7,4,3,5,6,1,2] => [[[[[.,[.,.]],.],[.,[.,.]]],.],.]
 => [2,1,3,6,5,4,7,8] => ?
 => ? = 4 + 1
[7,8,4,3,5,6,1,2] => [[[[.,[.,.]],.],[.,[.,.]]],[.,.]]
 => [2,1,3,6,5,4,8,7] => ?
 => ? = 3 + 1

Description

The largest part of an integer partition.

Matching statistic: St000676
(load all 7 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 73%

Values

[1] => [.,.]
 => [1,0]
 => [1,0]
 => 1 = 0 + 1
[1,2] => [.,[.,.]]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 0 + 1
[2,1] => [[.,.],.]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2 = 1 + 1
[1,2,3] => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2 = 1 + 1
[2,1,3] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[2,3,1] => [[.,.],[.,.]]
 => [1,1,1,0,0,0]
 => [1,0,1,1,0,0]
 => 2 = 1 + 1
[3,1,2] => [[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 3 = 2 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,3,4,2] => [.,[[.,.],[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,4,2,3] => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,3,1,4] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,3,4,1] => [[.,.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[2,4,1,3] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[2,4,3,1] => [[.,.],[[.,.],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[3,1,2,4] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,1,4,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,2,4,1] => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
 => [1,1,1,0,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[3,4,2,1] => [[[.,.],.],[.,.]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[4,1,2,3] => [[.,[.,[.,.]]],.]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 1 + 1
[4,1,3,2] => [[.,[[.,.],.]],.]
 => [1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[4,2,1,3] => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
 => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[4,3,1,2] => [[[.,[.,.]],.],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 3 = 2 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 2 = 1 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,3,5,2,4] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,3,5,4,2] => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,3,5,2] => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2 = 1 + 1
[6,7,8,5,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[8,5,6,7,4,3,2,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,7,5,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,5,7,8,4,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,6,7,8,4,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[8,7,4,5,6,3,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[8,5,6,4,7,3,2,1] => [[[[[[.,.],.],.],.],[.,[.,.]]],.]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[8,4,5,6,7,3,2,1] => [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[6,7,5,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,5,7,4,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,6,7,4,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[7,4,5,6,8,3,2,1] => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ? = 4 + 1
[6,5,4,7,8,3,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,6,4,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[5,4,6,7,8,3,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[4,5,6,7,8,3,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[8,7,6,3,4,5,2,1] => [[[[[[.,.],.],[.,[.,.]]],.],.],.]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[7,8,6,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[8,6,7,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],[.,.]],.]
 => [1,1,0,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[7,6,8,3,4,5,2,1] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[6,7,8,3,4,5,2,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 3 + 1
[8,7,4,3,5,6,2,1] => [[[[[[.,.],.],.],[.,[.,.]]],.],.]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[8,7,3,4,5,6,2,1] => [[[[[.,.],.],[.,[.,[.,.]]]],.],.]
 => [1,1,0,1,0,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[7,8,3,4,5,6,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => ? = 3 + 1
[8,4,3,5,6,7,2,1] => [[[[[.,.],.],.],[.,[.,[.,.]]]],.]
 => [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[8,3,4,5,6,7,2,1] => [[[[.,.],.],[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[6,7,5,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,5,7,4,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,6,7,4,3,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[7,4,5,6,3,8,2,1] => [[[[[.,.],.],.],[.,[.,.]]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0]
 => ? = 4 + 1
[6,5,4,7,3,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[5,4,6,7,3,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[4,5,6,7,3,8,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[7,6,3,4,5,8,2,1] => [[[[[.,.],.],[.,[.,.]]],.],[.,.]]
 => [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
 => ? = 4 + 1
[7,3,4,5,6,8,2,1] => [[[[.,.],.],[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
 => ? = 3 + 1
[6,5,4,3,7,8,2,1] => [[[[[[.,.],.],.],.],.],[.,[.,.]]]
 => [1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
 => ? = 5 + 1
[6,3,4,5,7,8,2,1] => [[[[.,.],.],[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0]
 => ? = 3 + 1
[5,4,3,6,7,8,2,1] => [[[[[.,.],.],.],.],[.,[.,[.,.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,1,0,0]
 => ? = 4 + 1
[4,3,5,6,7,8,2,1] => [[[[.,.],.],.],[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,1,0,1,0,0]
 => ? = 3 + 1
[3,4,5,6,7,8,2,1] => [[[.,.],.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
 => ? = 2 + 1
[8,7,6,5,2,3,4,1] => [[[[[[.,.],[.,[.,.]]],.],.],.],.]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
 => ? = 5 + 1
[8,6,7,5,2,3,4,1] => [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[7,6,8,5,2,3,4,1] => [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[6,7,8,5,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 3 + 1
[8,7,5,6,2,3,4,1] => [[[[[.,.],[.,[.,.]]],[.,.]],.],.]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[7,8,5,6,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,.]],[.,.]]
 => [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[8,6,5,7,2,3,4,1] => [[[[[.,.],[.,[.,.]]],.],[.,.]],.]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[8,5,6,7,2,3,4,1] => [[[[.,.],[.,[.,.]]],[.,[.,.]]],.]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
 => ? = 3 + 1
[7,6,5,8,2,3,4,1] => [[[[[.,.],[.,[.,.]]],.],.],[.,.]]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[6,7,5,8,2,3,4,1] => [[[[.,.],[.,[.,.]]],.],[.,[.,.]]]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
 => [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
 => ? = 3 + 1

Description

The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].

Matching statistic: St001581

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 62% ●values known / values provided: 62%●distinct values known / distinct values provided: 64%

Values

[1] => [1] => [1] => ([],1)
 => 1 = 0 + 1
[1,2] => [1,2] => [2] => ([],2)
 => 1 = 0 + 1
[2,1] => [2,1] => [1,1] => ([(0,1)],2)
 => 2 = 1 + 1
[1,2,3] => [1,2,3] => [3] => ([],3)
 => 1 = 0 + 1
[1,3,2] => [1,3,2] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[2,1,3] => [2,1,3] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[2,3,1] => [3,1,2] => [1,2] => ([(1,2)],3)
 => 2 = 1 + 1
[3,1,2] => [2,3,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[3,2,1] => [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,2,3,4] => [1,2,3,4] => [4] => ([],4)
 => 1 = 0 + 1
[1,2,4,3] => [1,2,4,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,2,4] => [1,3,2,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,3,4,2] => [1,4,2,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,2,3] => [1,3,4,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,4,3,2] => [1,4,3,2] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,1,3,4] => [2,1,3,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,1,4,3] => [2,1,4,3] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,3,1,4] => [3,1,2,4] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,3,4,1] => [4,1,2,3] => [1,3] => ([(2,3)],4)
 => 2 = 1 + 1
[2,4,1,3] => [3,1,4,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[2,4,3,1] => [4,1,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,1,2,4] => [2,3,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,1,4,2] => [2,4,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,2,1,4] => [3,2,1,4] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,2,4,1] => [4,2,1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[3,4,1,2] => [3,4,1,2] => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 1 + 1
[3,4,2,1] => [4,3,1,2] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,1,2,3] => [2,3,4,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[4,1,3,2] => [2,4,3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,1,3] => [3,2,4,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,2,3,1] => [4,2,3,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,1,2] => [3,4,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3 = 2 + 1
[4,3,2,1] => [4,3,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,2,3,4,5] => [1,2,3,4,5] => [5] => ([],5)
 => 1 = 0 + 1
[1,2,3,5,4] => [1,2,3,5,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,3,5] => [1,2,4,3,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,4,5,3] => [1,2,5,3,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,3,4] => [1,2,4,5,3] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,2,4,5] => [1,3,2,4,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,2,5,4] => [1,3,2,5,4] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,4,2,5] => [1,4,2,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,4,5,2] => [1,5,2,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,3,5,2,4] => [1,4,2,5,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,3,5,4,2] => [1,5,2,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,2,3,5] => [1,3,4,2,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,2,5,3] => [1,3,5,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,4,3,2,5] => [1,4,3,2,5] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,3,5,2] => [1,5,3,2,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3 = 2 + 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => [1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 7 + 1
[7,8,6,5,4,3,2,1] => [8,7,6,5,4,3,1,2] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[8,6,7,5,4,3,2,1] => [8,7,6,5,4,2,3,1] => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,6,8,5,4,3,2,1] => [8,7,6,5,4,2,1,3] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,8,5,4,3,2,1] => [8,7,6,5,4,1,2,3] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,5,6,4,3,2,1] => [8,7,6,5,3,4,2,1] => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,5,6,4,3,2,1] => [8,7,6,5,3,4,1,2] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,5,7,4,3,2,1] => [8,7,6,5,3,2,4,1] => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[8,5,6,7,4,3,2,1] => [8,7,6,5,2,3,4,1] => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,8,4,3,2,1] => [8,7,6,5,3,2,1,4] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,8,4,3,2,1] => [8,7,6,5,3,1,2,4] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,6,8,4,3,2,1] => [8,7,6,5,2,3,1,4] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,8,4,3,2,1] => [8,7,6,5,2,1,3,4] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,6,4,5,3,2,1] => [8,7,6,4,5,3,2,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,6,4,5,3,2,1] => [8,7,6,4,5,3,1,2] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,7,4,5,3,2,1] => [8,7,6,4,5,2,3,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,5,4,6,3,2,1] => [8,7,6,4,3,5,2,1] => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[8,7,4,5,6,3,2,1] => [8,7,6,3,4,5,2,1] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,5,6,4,7,3,2,1] => [8,7,6,4,2,3,5,1] => [1,1,1,1,3,1] => ([(0,7),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,4,5,7,3,2,1] => [8,7,6,3,4,2,5,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,4,8,3,2,1] => [8,7,6,4,3,2,1,5] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,4,8,3,2,1] => [8,7,6,4,3,1,2,5] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,6,4,8,3,2,1] => [8,7,6,4,2,3,1,5] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,4,8,3,2,1] => [8,7,6,4,2,1,3,5] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,4,5,8,3,2,1] => [8,7,6,3,4,2,1,5] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,4,6,8,3,2,1] => [8,7,6,3,2,4,1,5] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,4,7,8,3,2,1] => [8,7,6,3,2,1,4,5] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,6,5,3,4,2,1] => [8,7,5,6,4,3,2,1] => [1,1,2,1,1,1,1] => ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,6,5,3,4,2,1] => [8,7,5,6,4,3,1,2] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,8,5,3,4,2,1] => [8,7,5,6,4,2,1,3] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,5,6,3,4,2,1] => [8,7,5,6,3,4,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,8,3,4,2,1] => [8,7,5,6,3,2,1,4] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,6,4,3,5,2,1] => [8,7,5,4,6,3,2,1] => [1,1,1,2,1,1,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,6,4,3,5,2,1] => [8,7,5,4,6,3,1,2] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => [1,1,1,2,1,2] => ([(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,6,3,4,5,2,1] => [8,7,4,5,6,3,2,1] => [1,1,3,1,1,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,5,4,3,6,2,1] => [8,7,5,4,3,6,2,1] => [1,1,1,1,2,1,1] => ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,5,3,4,6,2,1] => [8,7,4,5,3,6,2,1] => [1,1,2,2,1,1] => ([(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,7,4,3,5,6,2,1] => [8,7,4,3,5,6,2,1] => [1,1,1,3,1,1] => ([(0,6),(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,6,5,4,3,7,2,1] => [8,7,5,4,3,2,6,1] => [1,1,1,1,1,2,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[8,6,4,3,5,7,2,1] => [8,7,4,3,5,2,6,1] => [1,1,1,2,2,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ? => ?
 => ? = 4 + 1
[7,6,5,4,3,8,2,1] => [8,7,5,4,3,2,1,6] => [1,1,1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6 + 1
[6,7,5,4,3,8,2,1] => [8,7,5,4,3,1,2,6] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,5,4,6,3,8,2,1] => [8,7,5,3,2,4,1,6] => [1,1,1,1,2,2] => ([(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[6,5,4,7,3,8,2,1] => [8,7,5,3,2,1,4,6] => [1,1,1,1,1,3] => ([(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1
[7,6,5,3,4,8,2,1] => [8,7,4,5,3,2,1,6] => [1,1,2,1,1,2] => ([(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 5 + 1

Description

The achromatic number of a graph. This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.

The following 131 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000098The chromatic number of a graph. St000011The number of touch points (or returns) of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000306The bounce count of a Dyck path. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000925The number of topologically connected components of a set partition. St000024The number of double up and double down steps of a Dyck path. St000053The number of valleys of the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000211The rank of the set partition. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000105The number of blocks in the set partition. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001494The Alon-Tarsi number of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001963The tree-depth of a graph. St000245The number of ascents of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000528The height of a poset. St000912The number of maximal antichains in a poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000167The number of leaves of an ordered tree. St000159The number of distinct parts of the integer partition. St000354The number of recoils of a permutation. St000470The number of runs in a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St000702The number of weak deficiencies of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St000007The number of saliances of the permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000662The staircase size of the code of a permutation. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000052The number of valleys of a Dyck path not on the x-axis. St000332The positive inversions of an alternating sign matrix. St000809The reduced reflection length of the permutation. St000527The width of the poset. St000632The jump number of the poset. St000292The number of ascents of a binary word. St000164The number of short pairs. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000308The height of the tree associated to a permutation. St001427The number of descents of a signed permutation. St000021The number of descents of a permutation. St000542The number of left-to-right-minima of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000314The number of left-to-right-maxima of a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000080The rank of the poset. St000120The number of left tunnels of a Dyck path. St000238The number of indices that are not small weak excedances. St000316The number of non-left-to-right-maxima of a permutation. St000386The number of factors DDU in a Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000822The Hadwiger number of the graph. St000991The number of right-to-left minima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St001152The number of pairs with even minimum in a perfect matching. St001668The number of points of the poset minus the width of the poset. St001896The number of right descents of a signed permutations. St001864The number of excedances of a signed permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001946The number of descents in a parking function. St001712The number of natural descents of a standard Young tableau. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001777The number of weak descents in an integer composition. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000942The number of critical left to right maxima of the parking functions. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000454The largest eigenvalue of a graph if it is integral.