- FindStat

Your data matches 110 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000157
(load all 16 compositions to match this statistic)

Mapping

Mp00059: Permutations —Robinson-Schensted insertion tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

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: St000288
(load all 16 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 100%

Values

[1] => [1] =>  =>  => ? = 0
[1,2] => [1,2] => 0 => 1 => 1
[2,1] => [2,1] => 1 => 0 => 0
[1,2,3] => [1,2,3] => 00 => 11 => 2
[1,3,2] => [1,3,2] => 01 => 10 => 1
[2,1,3] => [2,1,3] => 10 => 01 => 1
[2,3,1] => [3,1,2] => 01 => 10 => 1
[3,1,2] => [2,3,1] => 01 => 10 => 1
[3,2,1] => [3,2,1] => 11 => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 111 => 3
[1,2,4,3] => [1,2,4,3] => 001 => 110 => 2
[1,3,2,4] => [1,3,2,4] => 010 => 101 => 2
[1,3,4,2] => [1,4,2,3] => 001 => 110 => 2
[1,4,2,3] => [1,3,4,2] => 001 => 110 => 2
[1,4,3,2] => [1,4,3,2] => 011 => 100 => 1
[2,1,3,4] => [2,1,3,4] => 100 => 011 => 2
[2,1,4,3] => [2,1,4,3] => 101 => 010 => 1
[2,3,1,4] => [3,1,2,4] => 010 => 101 => 2
[2,3,4,1] => [4,1,2,3] => 001 => 110 => 2
[2,4,1,3] => [3,1,4,2] => 011 => 100 => 1
[2,4,3,1] => [4,1,3,2] => 011 => 100 => 1
[3,1,2,4] => [2,3,1,4] => 010 => 101 => 2
[3,1,4,2] => [2,4,1,3] => 001 => 110 => 2
[3,2,1,4] => [3,2,1,4] => 110 => 001 => 1
[3,2,4,1] => [4,2,1,3] => 101 => 010 => 1
[3,4,1,2] => [3,4,1,2] => 001 => 110 => 2
[3,4,2,1] => [4,3,1,2] => 011 => 100 => 1
[4,1,2,3] => [2,3,4,1] => 001 => 110 => 2
[4,1,3,2] => [2,4,3,1] => 011 => 100 => 1
[4,2,1,3] => [3,2,4,1] => 011 => 100 => 1
[4,2,3,1] => [4,2,3,1] => 011 => 100 => 1
[4,3,1,2] => [3,4,2,1] => 101 => 010 => 1
[4,3,2,1] => [4,3,2,1] => 111 => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 1111 => 4
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1110 => 3
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1101 => 3
[1,2,4,5,3] => [1,2,5,3,4] => 0001 => 1110 => 3
[1,2,5,3,4] => [1,2,4,5,3] => 0001 => 1110 => 3
[1,2,5,4,3] => [1,2,5,4,3] => 0011 => 1100 => 2
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1011 => 3
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1010 => 2
[1,3,4,2,5] => [1,4,2,3,5] => 0010 => 1101 => 3
[1,3,4,5,2] => [1,5,2,3,4] => 0001 => 1110 => 3
[1,3,5,2,4] => [1,4,2,5,3] => 0011 => 1100 => 2
[1,3,5,4,2] => [1,5,2,4,3] => 0011 => 1100 => 2
[1,4,2,3,5] => [1,3,4,2,5] => 0010 => 1101 => 3
[1,4,2,5,3] => [1,3,5,2,4] => 0001 => 1110 => 3
[1,4,3,2,5] => [1,4,3,2,5] => 0110 => 1001 => 2
[1,4,3,5,2] => [1,5,3,2,4] => 0101 => 1010 => 2
[1,4,5,2,3] => [1,4,5,2,3] => 0001 => 1110 => 3
[1,4,5,3,2] => [1,5,4,2,3] => 0011 => 1100 => 2
[7,6,8,4,3,5,2,1] => [8,7,5,4,6,2,1,3] => ? => ? => ? = 2
[7,8,6,3,4,5,2,1] => [8,7,4,5,6,3,1,2] => ? => ? => ? = 3
[7,8,5,4,3,6,2,1] => [8,7,5,4,3,6,1,2] => ? => ? => ? = 2
[8,5,3,4,6,7,2,1] => [8,7,3,4,2,5,6,1] => ? => ? => ? = 3
[8,4,3,5,6,7,2,1] => [8,7,3,2,4,5,6,1] => ? => ? => ? = 3
[7,5,6,4,3,8,2,1] => [8,7,5,4,2,3,1,6] => ? => ? => ? = 2
[6,5,7,4,3,8,2,1] => [8,7,5,4,2,1,3,6] => ? => ? => ? = 2
[7,4,5,6,3,8,2,1] => [8,7,5,2,3,4,1,6] => ? => ? => ? = 3
[8,7,5,6,3,2,4,1] => [8,6,5,7,3,4,2,1] => ? => ? => ? = 2
[7,6,8,5,2,3,4,1] => [8,5,6,7,4,2,1,3] => ? => ? => ? = 3
[6,7,5,8,2,3,4,1] => [8,5,6,7,3,1,2,4] => ? => ? => ? = 4
[8,6,7,3,2,4,5,1] => [8,5,4,6,7,2,3,1] => ? => ? => ? = 3
[7,8,6,2,3,4,5,1] => [8,4,5,6,7,3,1,2] => ? => ? => ? = 4
[7,8,5,3,4,2,6,1] => [8,6,4,5,3,7,1,2] => ? => ? => ? = 3
[8,7,5,3,2,4,6,1] => [8,5,4,6,3,7,2,1] => ? => ? => ? = 2
[7,8,4,2,3,5,6,1] => [8,4,5,3,6,7,1,2] => ? => ? => ? = 4
[8,4,3,5,6,2,7,1] => [8,6,3,2,4,5,7,1] => ? => ? => ? = 3
[7,6,3,4,5,2,8,1] => [8,6,3,4,5,2,1,7] => ? => ? => ? = 3
[7,5,3,4,6,2,8,1] => [8,6,3,4,2,5,1,7] => ? => ? => ? = 3
[7,3,4,5,6,2,8,1] => [8,6,2,3,4,5,1,7] => ? => ? => ? = 4
[4,5,6,3,7,2,8,1] => [8,6,4,1,2,3,5,7] => ? => ? => ? = 4
[4,5,3,6,7,2,8,1] => [8,6,3,1,2,4,5,7] => ? => ? => ? = 4
[4,3,5,6,7,2,8,1] => [8,6,2,1,3,4,5,7] => ? => ? => ? = 4
[6,7,5,4,2,3,8,1] => [8,5,6,4,3,1,2,7] => ? => ? => ? = 3
[6,7,5,3,2,4,8,1] => [8,5,4,6,3,1,2,7] => ? => ? => ? = 3
[6,5,7,3,2,4,8,1] => [8,5,4,6,2,1,3,7] => ? => ? => ? = 3
[7,5,4,2,3,6,8,1] => [8,4,5,3,2,6,1,7] => ? => ? => ? = 3
[7,5,2,3,4,6,8,1] => [8,3,4,5,2,6,1,7] => ? => ? => ? = 4
[7,4,3,2,5,6,8,1] => [8,4,3,2,5,6,1,7] => ? => ? => ? = 3
[6,5,3,4,2,7,8,1] => [8,5,3,4,2,1,6,7] => ? => ? => ? = 3
[4,5,3,2,6,7,8,1] => [8,4,3,1,2,5,6,7] => ? => ? => ? = 4
[5,3,2,4,6,7,8,1] => [8,3,2,4,1,5,6,7] => ? => ? => ? = 4
[8,6,7,4,5,3,1,2] => [7,8,6,4,5,2,3,1] => ? => ? => ? = 3
[6,7,8,4,5,3,1,2] => [7,8,6,4,5,1,2,3] => ? => ? => ? = 4
[8,5,6,4,7,3,1,2] => [7,8,6,4,2,3,5,1] => ? => ? => ? = 3
[7,5,6,4,8,3,1,2] => [7,8,6,4,2,3,1,5] => ? => ? => ? = 3
[8,6,5,7,3,4,1,2] => [7,8,5,6,3,2,4,1] => ? => ? => ? = 3
[8,7,6,3,4,5,1,2] => [7,8,4,5,6,3,2,1] => ? => ? => ? = 3
[8,7,4,3,5,6,1,2] => [7,8,4,3,5,6,2,1] => ? => ? => ? = 3
[7,5,6,4,3,8,1,2] => [7,8,5,4,2,3,1,6] => ? => ? => ? = 3
[7,6,4,3,5,8,1,2] => [7,8,4,3,5,2,1,6] => ? => ? => ? = 3
[6,7,5,4,8,2,1,3] => [7,6,8,4,3,1,2,5] => ? => ? => ? = 3
[6,5,7,4,8,2,1,3] => [7,6,8,4,2,1,3,5] => ? => ? => ? = 3
[7,6,4,5,8,2,1,3] => [7,6,8,3,4,2,1,5] => ? => ? => ? = 3
[5,4,6,7,8,2,1,3] => [7,6,8,2,1,3,4,5] => ? => ? => ? = 4
[6,7,5,8,4,1,2,3] => [6,7,8,5,3,1,2,4] => ? => ? => ? = 4
[8,7,5,6,2,3,1,4] => [7,5,6,8,3,4,2,1] => ? => ? => ? = 3
[8,6,5,7,2,3,1,4] => [7,5,6,8,3,2,4,1] => ? => ? => ? = 3
[8,5,6,7,2,3,1,4] => [7,5,6,8,2,3,4,1] => ? => ? => ? = 4

Description

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

Matching statistic: St000394

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 82%●distinct values known / distinct values provided: 91%

Values

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

Description

The sum of the heights of the peaks of a Dyck path minus the number of peaks.

Matching statistic: St000010
(load all 5 compositions to match this statistic)

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

Matching statistic: St000733

Mapping

Mp00066: Permutations —inverse⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 91%

Values

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

Description

The row containing the largest entry of a standard tableau.

Matching statistic: St001918

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 91%

Values

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

Description

The degree of the cyclic sieving polynomial corresponding to an integer partition. Let

$\lambda$ be an integer partition of

$n$ and let

$N$ be the least common multiple of the parts of

$\lambda$ . Fix an arbitrary permutation

$\pi$ of cycle type

$\lambda$ . Then

$\pi$ induces a cyclic action of order

$N$ on

$\{1,\dots,n\}$ . The corresponding character can be identified with the cyclic sieving polynomial

$C_\lambda(q)$ of this action, modulo

$q^N-1$ . Explicitly, it is

$\sum_{p\in\lambda} [p]_{q^{N/p}},$ where

$[p]_q = 1+\dots+q^{p-1}$ is the

$q$ -integer. This statistic records the degree of

$C_\lambda(q)$ . Equivalently, it equals

$\left(1 - \frac{1}{\lambda_1}\right) N,$ where

$\lambda_1$ is the largest part of

$\lambda$ . The statistic is undefined for the empty partition.

Matching statistic: St000147

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 91%

Values

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

Description

The largest part of an integer partition.

Matching statistic: St001389

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001389: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 91%

Values

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

Description

The number of partitions of the same length below the given integer partition. For a partition

$\lambda_1 \geq \dots \lambda_k > 0$ , this number is

$\det\left( \binom{\lambda_{k+1-i}}{j-i+1} \right)_{1 \le i,j \le k}.$

Matching statistic: St000668

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 91%

Values

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

Description

The least common multiple of the parts of the partition.

Matching statistic: St000708

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 76% ●values known / values provided: 76%●distinct values known / distinct values provided: 91%

Values

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

Description

The product of the parts of an integer partition.

The following 100 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001176The size of a partition minus its first part. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000093The cardinality of a maximal independent set of vertices of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000676The number of odd rises of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000507The number of ascents of a standard tableau. St001581The achromatic number of a graph. 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. 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. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000105The number of blocks in the set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000362The size of a minimal vertex cover of a graph. St000167The number of leaves of an ordered tree. St000527The width of the poset. St000071The number of maximal chains in a poset. St001304The number of maximally independent sets of vertices of a graph. St000245The number of ascents of a permutation. St000068The number of minimal elements in a poset. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. 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. St000470The number of runs in a permutation. St000354The number of recoils of 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. St000159The number of distinct parts of the integer partition. St000809The reduced reflection length 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. St000702The number of weak deficiencies of a permutation. St000141The maximum drop size of a permutation. St000308The height of the tree associated to a permutation. St001427The number of descents of a signed permutation. St000052The number of valleys of a Dyck path not on the x-axis. St001461The number of topologically connected components of the chord diagram of a permutation. St000332The positive inversions of an alternating sign matrix. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St000021The number of descents of a permutation. St000542The number of left-to-right-minima of a permutation. St000168The number of internal nodes of an ordered tree. St000316The number of non-left-to-right-maxima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to 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. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000238The number of indices that are not small weak excedances. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000062The length of the longest increasing subsequence of the permutation. St000201The number of leaf nodes in a binary tree. St000239The number of small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000991The number of right-to-left minima 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. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St001480The number of simple summands of the module J^2/J^3. St000061The number of nodes on the left branch of a binary tree. St001812The biclique partition number of a graph. St000746The number of pairs with odd minimum in a perfect matching. St001896The number of right descents of a signed permutations. St001863The number of weak excedances of a signed permutation. St001935The number of ascents in a parking function. St001712The number of natural descents of a standard Young tableau. St001946The number of descents in a parking function. St001668The number of points of the poset minus the width of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ .