- FindStat

Your data matches 57 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000201
(load all 7 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of leaf nodes in a binary tree. Equivalently, the number of cherries [1] in the complete binary tree. The number of binary trees of size

$n$ , at least

$1$ , with exactly one leaf node for is

$2^{n-1}$ , see [2]. The number of binary tree of size

$n$ , at least

$3$ , with exactly two leaf nodes is

$n(n+1)2^{n-2}$ , see [3].

Matching statistic: St000291
(load all 11 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a binary word.

Matching statistic: St000390
(load all 6 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 1 => 1
[1,2] => [2] => 10 => 01 => 1
[2,1] => [1,1] => 11 => 11 => 1
[1,2,3] => [3] => 100 => 001 => 1
[1,3,2] => [2,1] => 101 => 101 => 2
[2,1,3] => [1,2] => 110 => 011 => 1
[2,3,1] => [2,1] => 101 => 101 => 2
[3,1,2] => [1,2] => 110 => 011 => 1
[3,2,1] => [1,1,1] => 111 => 111 => 1
[1,2,3,4] => [4] => 1000 => 0001 => 1
[1,2,4,3] => [3,1] => 1001 => 1001 => 2
[1,3,2,4] => [2,2] => 1010 => 0101 => 2
[1,3,4,2] => [3,1] => 1001 => 1001 => 2
[1,4,2,3] => [2,2] => 1010 => 0101 => 2
[1,4,3,2] => [2,1,1] => 1011 => 1101 => 2
[2,1,3,4] => [1,3] => 1100 => 0011 => 1
[2,1,4,3] => [1,2,1] => 1101 => 1011 => 2
[2,3,1,4] => [2,2] => 1010 => 0101 => 2
[2,3,4,1] => [3,1] => 1001 => 1001 => 2
[2,4,1,3] => [2,2] => 1010 => 0101 => 2
[2,4,3,1] => [2,1,1] => 1011 => 1101 => 2
[3,1,2,4] => [1,3] => 1100 => 0011 => 1
[3,1,4,2] => [1,2,1] => 1101 => 1011 => 2
[3,2,1,4] => [1,1,2] => 1110 => 0111 => 1
[3,2,4,1] => [1,2,1] => 1101 => 1011 => 2
[3,4,1,2] => [2,2] => 1010 => 0101 => 2
[3,4,2,1] => [2,1,1] => 1011 => 1101 => 2
[4,1,2,3] => [1,3] => 1100 => 0011 => 1
[4,1,3,2] => [1,2,1] => 1101 => 1011 => 2
[4,2,1,3] => [1,1,2] => 1110 => 0111 => 1
[4,2,3,1] => [1,2,1] => 1101 => 1011 => 2
[4,3,1,2] => [1,1,2] => 1110 => 0111 => 1
[4,3,2,1] => [1,1,1,1] => 1111 => 1111 => 1
[1,2,3,4,5] => [5] => 10000 => 00001 => 1
[1,2,3,5,4] => [4,1] => 10001 => 10001 => 2
[1,2,4,3,5] => [3,2] => 10010 => 01001 => 2
[1,2,4,5,3] => [4,1] => 10001 => 10001 => 2
[1,2,5,3,4] => [3,2] => 10010 => 01001 => 2
[1,2,5,4,3] => [3,1,1] => 10011 => 11001 => 2
[1,3,2,4,5] => [2,3] => 10100 => 00101 => 2
[1,3,2,5,4] => [2,2,1] => 10101 => 10101 => 3
[1,3,4,2,5] => [3,2] => 10010 => 01001 => 2
[1,3,4,5,2] => [4,1] => 10001 => 10001 => 2
[1,3,5,2,4] => [3,2] => 10010 => 01001 => 2
[1,3,5,4,2] => [3,1,1] => 10011 => 11001 => 2
[1,4,2,3,5] => [2,3] => 10100 => 00101 => 2
[1,4,2,5,3] => [2,2,1] => 10101 => 10101 => 3
[1,4,3,2,5] => [2,1,2] => 10110 => 01101 => 2
[1,4,3,5,2] => [2,2,1] => 10101 => 10101 => 3
[1,4,5,2,3] => [3,2] => 10010 => 01001 => 2
[7,6,8,3,2,4,5,1] => ? => ? => ? => ? = 3
[5,6,7,3,2,4,8,1] => ? => ? => ? => ? = 3
[7,8,5,4,6,3,1,2] => ? => ? => ? => ? = 3
[8,4,5,3,6,7,1,2] => ? => ? => ? => ? = 3
[1,5,7,6,4,8,3,2] => ? => ? => ? => ? = 3
[1,4,3,7,6,5,8,2] => ? => ? => ? => ? = 4
[1,6,5,4,7,3,2,8] => ? => ? => ? => ? = 3
[3,4,2,1,7,6,5,8] => ? => ? => ? => ? = 3
[1,3,2,4,6,7,5,8] => ? => ? => ? => ? = 3
[2,4,5,3,6,1,7,8] => ? => ? => ? => ? = 3
[1,3,7,5,4,6,2,8] => ? => ? => ? => ? = 3
[1,3,7,5,6,4,8,2] => ? => ? => ? => ? = 4
[3,2,7,5,4,6,8,1] => ? => ? => ? => ? = 3
[2,5,1,7,3,4,6,8] => ? => ? => ? => ? = 3
[2,5,7,1,3,4,6,8] => ? => ? => ? => ? = 2
[2,1,3,6,8,4,5,7] => ? => ? => ? => ? = 2
[3,4,6,1,2,8,5,7] => ? => ? => ? => ? = 3
[3,4,1,7,2,8,5,6] => ? => ? => ? => ? = 4
[3,1,4,7,8,2,5,6] => ? => ? => ? => ? = 2
[3,1,4,5,2,8,6,7] => ? => ? => ? => ? = 3
[3,1,6,7,2,4,8,5] => ? => ? => ? => ? = 3
[4,1,2,6,7,8,3,5] => ? => ? => ? => ? = 2
[2,1,4,3,6,5,8,7,10,9] => [1,2,2,2,2,1] => 1101010101 => 1010101011 => ? = 5
[2,3,4,5,6,7,8,9,10,1] => [9,1] => 1000000001 => 1000000001 => ? = 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,2,2,2,2,1] => 110101010101 => 101010101011 => ? = 6
[1,2,3,4,5,6,7,8,10,9] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,3,4,5,6,7,8,9,11,10] => [10,1] => 10000000001 => 10000000001 => ? = 2
[1,2,3,4,5,6,7,9,10,8] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,3,4,5,6,8,9,10,7] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,3,4,5,7,8,9,10,6] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,3,4,6,7,8,9,10,5] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,3,5,6,7,8,9,10,4] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,2,4,5,6,7,8,9,10,3] => [9,1] => 1000000001 => 1000000001 => ? = 2
[1,3,4,5,6,7,8,9,10,2] => [9,1] => 1000000001 => 1000000001 => ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [1,10] => 11000000000 => 00000000011 => ? = 1
[4,5,2,3,7,1,6,8] => ? => ? => ? => ? = 3
[3,2,4,5,6,7,8,9,10,1] => [1,8,1] => 1100000001 => 1000000011 => ? = 2
[2,3,4,5,6,7,8,9,10,11,1] => [10,1] => 10000000001 => 10000000001 => ? = 2
[3,1,4,5,6,7,8,9,10,2] => [1,8,1] => 1100000001 => 1000000011 => ? = 2
[2,3,1,7,5,4,8,6] => ? => ? => ? => ? = 4
[1,2,7,8,5,3,6,4] => ? => ? => ? => ? = 3
[6,4,8,2,1,3,7,5] => ? => ? => ? => ? = 3
[2,4,7,5,1,6,8,3] => ? => ? => ? => ? = 3
[2,4,7,5,1,6,3,8] => ? => ? => ? => ? = 3
[8,2,4,1,7,5,3,6] => ? => ? => ? => ? = 3
[1,3,4,5,6,7,8,9,10,11,2] => [10,1] => 10000000001 => 10000000001 => ? = 2
[3,1,4,2,6,5,8,7,10,9] => [1,2,2,2,2,1] => 1101010101 => 1010101011 => ? = 5
[3,1,5,2,6,4,8,7,10,9] => [1,2,2,2,2,1] => 1101010101 => 1010101011 => ? = 5
[3,1,5,2,7,4,8,6,10,9] => [1,2,2,2,2,1] => 1101010101 => 1010101011 => ? = 5
[3,1,5,2,7,4,9,6,10,8] => [1,2,2,2,2,1] => 1101010101 => 1010101011 => ? = 5

Description

The number of runs of ones in a binary word.

Matching statistic: St000292
(load all 13 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%

Values

[1] => [1] => 1 => 0 = 1 - 1
[1,2] => [2] => 10 => 0 = 1 - 1
[2,1] => [1,1] => 11 => 0 = 1 - 1
[1,2,3] => [3] => 100 => 0 = 1 - 1
[1,3,2] => [2,1] => 101 => 1 = 2 - 1
[2,1,3] => [1,2] => 110 => 0 = 1 - 1
[2,3,1] => [2,1] => 101 => 1 = 2 - 1
[3,1,2] => [1,2] => 110 => 0 = 1 - 1
[3,2,1] => [1,1,1] => 111 => 0 = 1 - 1
[1,2,3,4] => [4] => 1000 => 0 = 1 - 1
[1,2,4,3] => [3,1] => 1001 => 1 = 2 - 1
[1,3,2,4] => [2,2] => 1010 => 1 = 2 - 1
[1,3,4,2] => [3,1] => 1001 => 1 = 2 - 1
[1,4,2,3] => [2,2] => 1010 => 1 = 2 - 1
[1,4,3,2] => [2,1,1] => 1011 => 1 = 2 - 1
[2,1,3,4] => [1,3] => 1100 => 0 = 1 - 1
[2,1,4,3] => [1,2,1] => 1101 => 1 = 2 - 1
[2,3,1,4] => [2,2] => 1010 => 1 = 2 - 1
[2,3,4,1] => [3,1] => 1001 => 1 = 2 - 1
[2,4,1,3] => [2,2] => 1010 => 1 = 2 - 1
[2,4,3,1] => [2,1,1] => 1011 => 1 = 2 - 1
[3,1,2,4] => [1,3] => 1100 => 0 = 1 - 1
[3,1,4,2] => [1,2,1] => 1101 => 1 = 2 - 1
[3,2,1,4] => [1,1,2] => 1110 => 0 = 1 - 1
[3,2,4,1] => [1,2,1] => 1101 => 1 = 2 - 1
[3,4,1,2] => [2,2] => 1010 => 1 = 2 - 1
[3,4,2,1] => [2,1,1] => 1011 => 1 = 2 - 1
[4,1,2,3] => [1,3] => 1100 => 0 = 1 - 1
[4,1,3,2] => [1,2,1] => 1101 => 1 = 2 - 1
[4,2,1,3] => [1,1,2] => 1110 => 0 = 1 - 1
[4,2,3,1] => [1,2,1] => 1101 => 1 = 2 - 1
[4,3,1,2] => [1,1,2] => 1110 => 0 = 1 - 1
[4,3,2,1] => [1,1,1,1] => 1111 => 0 = 1 - 1
[1,2,3,4,5] => [5] => 10000 => 0 = 1 - 1
[1,2,3,5,4] => [4,1] => 10001 => 1 = 2 - 1
[1,2,4,3,5] => [3,2] => 10010 => 1 = 2 - 1
[1,2,4,5,3] => [4,1] => 10001 => 1 = 2 - 1
[1,2,5,3,4] => [3,2] => 10010 => 1 = 2 - 1
[1,2,5,4,3] => [3,1,1] => 10011 => 1 = 2 - 1
[1,3,2,4,5] => [2,3] => 10100 => 1 = 2 - 1
[1,3,2,5,4] => [2,2,1] => 10101 => 2 = 3 - 1
[1,3,4,2,5] => [3,2] => 10010 => 1 = 2 - 1
[1,3,4,5,2] => [4,1] => 10001 => 1 = 2 - 1
[1,3,5,2,4] => [3,2] => 10010 => 1 = 2 - 1
[1,3,5,4,2] => [3,1,1] => 10011 => 1 = 2 - 1
[1,4,2,3,5] => [2,3] => 10100 => 1 = 2 - 1
[1,4,2,5,3] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,3,2,5] => [2,1,2] => 10110 => 1 = 2 - 1
[1,4,3,5,2] => [2,2,1] => 10101 => 2 = 3 - 1
[1,4,5,2,3] => [3,2] => 10010 => 1 = 2 - 1
[7,6,8,3,2,4,5,1] => ? => ? => ? = 3 - 1
[5,6,7,3,2,4,8,1] => ? => ? => ? = 3 - 1
[7,8,5,4,6,3,1,2] => ? => ? => ? = 3 - 1
[8,4,5,3,6,7,1,2] => ? => ? => ? = 3 - 1
[1,5,7,6,4,8,3,2] => ? => ? => ? = 3 - 1
[1,4,3,7,6,5,8,2] => ? => ? => ? = 4 - 1
[1,6,5,4,7,3,2,8] => ? => ? => ? = 3 - 1
[3,4,2,1,7,6,5,8] => ? => ? => ? = 3 - 1
[1,3,2,4,6,7,5,8] => ? => ? => ? = 3 - 1
[2,4,5,3,6,1,7,8] => ? => ? => ? = 3 - 1
[1,3,7,5,4,6,2,8] => ? => ? => ? = 3 - 1
[1,3,7,5,6,4,8,2] => ? => ? => ? = 4 - 1
[3,2,7,5,4,6,8,1] => ? => ? => ? = 3 - 1
[2,5,1,7,3,4,6,8] => ? => ? => ? = 3 - 1
[2,5,7,1,3,4,6,8] => ? => ? => ? = 2 - 1
[2,1,3,6,8,4,5,7] => ? => ? => ? = 2 - 1
[3,4,6,1,2,8,5,7] => ? => ? => ? = 3 - 1
[3,4,1,7,2,8,5,6] => ? => ? => ? = 4 - 1
[3,1,4,7,8,2,5,6] => ? => ? => ? = 2 - 1
[3,1,4,5,2,8,6,7] => ? => ? => ? = 3 - 1
[3,1,6,7,2,4,8,5] => ? => ? => ? = 3 - 1
[4,1,2,6,7,8,3,5] => ? => ? => ? = 2 - 1
[2,1,4,3,6,5,8,7,10,9] => [1,2,2,2,2,1] => 1101010101 => ? = 5 - 1
[2,3,4,5,6,7,8,9,10,1] => [9,1] => 1000000001 => ? = 2 - 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,2,2,2,2,1] => 110101010101 => ? = 6 - 1
[1,2,3,4,5,6,7,8,10,9] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,3,4,5,6,7,8,9,11,10] => [10,1] => 10000000001 => ? = 2 - 1
[1,2,3,4,5,6,7,9,10,8] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,3,4,5,6,8,9,10,7] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,3,4,5,7,8,9,10,6] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,3,4,6,7,8,9,10,5] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,3,5,6,7,8,9,10,4] => [9,1] => 1000000001 => ? = 2 - 1
[1,2,4,5,6,7,8,9,10,3] => [9,1] => 1000000001 => ? = 2 - 1
[1,3,4,5,6,7,8,9,10,2] => [9,1] => 1000000001 => ? = 2 - 1
[9,10,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[10,1,2,3,4,5,6,7,8,9] => [1,9] => 1100000000 => ? = 1 - 1
[11,1,2,3,4,5,6,7,8,9,10] => [1,10] => 11000000000 => ? = 1 - 1
[4,5,2,3,7,1,6,8] => ? => ? => ? = 3 - 1
[1,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[1,11,10,9,8,7,6,5,4,3,2] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 2 - 1
[3,2,4,5,6,7,8,9,10,1] => [1,8,1] => 1100000001 => ? = 2 - 1
[2,3,4,5,6,7,8,9,10,11,1] => [10,1] => 10000000001 => ? = 2 - 1
[3,1,4,5,6,7,8,9,10,2] => [1,8,1] => 1100000001 => ? = 2 - 1
[10,11,9,8,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1,1] => 10111111111 => ? = 2 - 1
[8,10,9,7,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[7,10,9,8,6,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[6,10,9,8,7,5,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[5,10,9,8,7,6,4,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[4,10,9,8,7,6,5,3,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1
[3,10,9,8,7,6,5,4,2,1] => [2,1,1,1,1,1,1,1,1] => 1011111111 => ? = 2 - 1

Description

The number of ascents of a binary word.

Matching statistic: St000318

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000318: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of addable cells of the Ferrers diagram of an integer partition.

Matching statistic: St000159

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000159: Integer partitions ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of distinct parts of the integer partition. This statistic is also the number of removeable cells of the partition, and the number of valleys of the Dyck path tracing the shape of the partition.

Matching statistic: St001037
(load all 7 compositions to match this statistic)

Mapping

Mp00069: Permutations —complement⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001037: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 91%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.

Matching statistic: St000386
(load all 10 compositions to match this statistic)

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000386: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%

Values

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

Description

The number of factors DDU in a Dyck path.

Matching statistic: St001124

Mapping

Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 87%●distinct values known / distinct values provided: 50%

Values

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

Description

The multiplicity of the standard representation in the Kronecker square corresponding to a partition. The Kronecker coefficient is the multiplicity

$g_{\mu,\nu}^\lambda$ of the Specht module

$S^\lambda$ in

$S^\mu\otimes S^\nu$ :

$S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda$ This statistic records the Kronecker coefficient

$g_{\lambda,\lambda}^{(n-1)1}$ , for

$\lambda\vdash n > 1$ . For

$n\leq1$ the statistic is undefined. It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.

Matching statistic: St000010

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the partition.

The following 47 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000389The number of runs of ones of odd length in a binary word. St000288The number of ones in a binary word. St000097The order of the largest clique of the graph. St001581The achromatic number of a graph. St000806The semiperimeter of the associated bargraph. St001712The number of natural descents of a standard Young tableau. St000098The chromatic number of a graph. St000071The number of maximal chains in a poset. St000068The number of minimal elements in a poset. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000527The width of the poset. St000568The hook number of a binary tree. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000172The Grundy number of a graph. St001029The size of the core of a graph. St001116The game chromatic number 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. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000387The matching number of a graph. St000536The pathwidth of a graph. St000632The jump number of the poset. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001277The degeneracy of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St000356The number of occurrences of the pattern 13-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000353The number of inner valleys of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000092The number of outer peaks of a permutation. St000822The Hadwiger number of the graph. St000523The number of 2-protected nodes of a rooted tree. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000256The number of parts from which one can substract 2 and still get an integer partition. St001487The number of inner corners of a skew partition. St000354The number of recoils of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001960The number of descents of a permutation minus one if its first entry is not one.