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Matching statistic: St000201
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
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%
St000201: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [.,.]
=> 1
[1,2] => [.,[.,.]]
=> 1
[2,1] => [[.,.],.]
=> 1
[1,2,3] => [.,[.,[.,.]]]
=> 1
[1,3,2] => [.,[[.,.],.]]
=> 1
[2,1,3] => [[.,.],[.,.]]
=> 2
[2,3,1] => [[.,[.,.]],.]
=> 1
[3,1,2] => [[.,.],[.,.]]
=> 2
[3,2,1] => [[[.,.],.],.]
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> 2
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> 2
[1,4,3,2] => [.,[[[.,.],.],.]]
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> 2
[2,1,4,3] => [[.,.],[[.,.],.]]
=> 2
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> 2
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> 2
[2,4,3,1] => [[.,[[.,.],.]],.]
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> 2
[3,1,4,2] => [[.,.],[[.,.],.]]
=> 2
[3,2,1,4] => [[[.,.],.],[.,.]]
=> 2
[3,2,4,1] => [[[.,.],[.,.]],.]
=> 2
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> 2
[3,4,2,1] => [[[.,[.,.]],.],.]
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> 2
[4,1,3,2] => [[.,.],[[.,.],.]]
=> 2
[4,2,1,3] => [[[.,.],.],[.,.]]
=> 2
[4,2,3,1] => [[[.,.],[.,.]],.]
=> 2
[4,3,1,2] => [[[.,.],.],[.,.]]
=> 2
[4,3,2,1] => [[[[.,.],.],.],.]
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> 2
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> 2
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> 2
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> 2
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> 2
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> 2
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 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 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
St000291: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => => ? = 1 - 1
[1,2] => 0 => 0 = 1 - 1
[2,1] => 1 => 0 = 1 - 1
[1,2,3] => 00 => 0 = 1 - 1
[1,3,2] => 01 => 0 = 1 - 1
[2,1,3] => 10 => 1 = 2 - 1
[2,3,1] => 01 => 0 = 1 - 1
[3,1,2] => 10 => 1 = 2 - 1
[3,2,1] => 11 => 0 = 1 - 1
[1,2,3,4] => 000 => 0 = 1 - 1
[1,2,4,3] => 001 => 0 = 1 - 1
[1,3,2,4] => 010 => 1 = 2 - 1
[1,3,4,2] => 001 => 0 = 1 - 1
[1,4,2,3] => 010 => 1 = 2 - 1
[1,4,3,2] => 011 => 0 = 1 - 1
[2,1,3,4] => 100 => 1 = 2 - 1
[2,1,4,3] => 101 => 1 = 2 - 1
[2,3,1,4] => 010 => 1 = 2 - 1
[2,3,4,1] => 001 => 0 = 1 - 1
[2,4,1,3] => 010 => 1 = 2 - 1
[2,4,3,1] => 011 => 0 = 1 - 1
[3,1,2,4] => 100 => 1 = 2 - 1
[3,1,4,2] => 101 => 1 = 2 - 1
[3,2,1,4] => 110 => 1 = 2 - 1
[3,2,4,1] => 101 => 1 = 2 - 1
[3,4,1,2] => 010 => 1 = 2 - 1
[3,4,2,1] => 011 => 0 = 1 - 1
[4,1,2,3] => 100 => 1 = 2 - 1
[4,1,3,2] => 101 => 1 = 2 - 1
[4,2,1,3] => 110 => 1 = 2 - 1
[4,2,3,1] => 101 => 1 = 2 - 1
[4,3,1,2] => 110 => 1 = 2 - 1
[4,3,2,1] => 111 => 0 = 1 - 1
[1,2,3,4,5] => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 0001 => 0 = 1 - 1
[1,2,4,3,5] => 0010 => 1 = 2 - 1
[1,2,4,5,3] => 0001 => 0 = 1 - 1
[1,2,5,3,4] => 0010 => 1 = 2 - 1
[1,2,5,4,3] => 0011 => 0 = 1 - 1
[1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,3,4,2,5] => 0010 => 1 = 2 - 1
[1,3,4,5,2] => 0001 => 0 = 1 - 1
[1,3,5,2,4] => 0010 => 1 = 2 - 1
[1,3,5,4,2] => 0011 => 0 = 1 - 1
[1,4,2,3,5] => 0100 => 1 = 2 - 1
[1,4,2,5,3] => 0101 => 1 = 2 - 1
[1,4,3,2,5] => 0110 => 1 = 2 - 1
[1,4,3,5,2] => 0101 => 1 = 2 - 1
[1,4,5,2,3] => 0010 => 1 = 2 - 1
[1,4,5,3,2] => 0011 => 0 = 1 - 1
[8,5,4,6,3,7,2,1] => ? => ? = 3 - 1
[7,6,4,5,3,8,2,1] => ? => ? = 3 - 1
[6,4,5,7,3,8,1,2] => ? => ? = 4 - 1
[6,5,7,3,4,1,2,8] => ? => ? = 4 - 1
[6,7,3,4,2,1,5,8] => ? => ? = 3 - 1
[6,7,4,2,3,1,5,8] => ? => ? = 3 - 1
[6,3,4,5,2,1,7,8] => ? => ? = 3 - 1
[4,3,5,2,8,7,6,1] => ? => ? = 3 - 1
[2,1,4,3,6,7,5,8] => ? => ? = 4 - 1
[1,4,3,5,2,6,7,8] => ? => ? = 3 - 1
[4,2,3,1,6,5,7,8] => ? => ? = 4 - 1
[4,5,1,2,7,3,8,6] => ? => ? = 3 - 1
[5,4,7,8,1,2,3,6] => ? => ? = 3 - 1
[8,6,2,5,7,1,3,4] => ? => ? = 3 - 1
[6,3,8,2,5,1,4,7] => ? => ? = 4 - 1
[8,4,6,2,5,1,3,7] => ? => ? = 4 - 1
[8,4,7,3,5,1,2,6] => ? => ? = 4 - 1
[7,8,1,6,3,2,5,4] => ? => ? = 3 - 1
[4,6,7,1,3,2,5,8] => ? => ? = 3 - 1
[4,5,1,3,7,2,6,8] => ? => ? = 3 - 1
[3,5,1,4,2,7,6,8] => ? => ? = 4 - 1
[6,3,4,1,2,7,5,8] => ? => ? = 4 - 1
[6,4,5,1,7,2,3,8] => ? => ? = 4 - 1
[5,7,1,8,4,3,2,6] => ? => ? = 3 - 1
[7,6,8,2,4,5,1,3] => ? => ? = 4 - 1
[8,6,7,2,4,1,5,3] => ? => ? = 4 - 1
[8,7,2,1,4,6,3,5] => ? => ? = 3 - 1
[8,5,2,7,1,6,3,4] => ? => ? = 4 - 1
[8,5,2,7,1,3,6,4] => ? => ? = 3 - 1
[8,5,7,4,1,6,2,3] => ? => ? = 4 - 1
[8,5,7,4,1,2,6,3] => ? => ? = 3 - 1
[8,3,2,7,1,5,4,6] => ? => ? = 4 - 1
[8,5,2,7,4,6,1,3] => ? => ? = 4 - 1
[7,4,2,3,1,8,6,5] => ? => ? = 3 - 1
[5,6,1,4,3,2,8,7] => ? => ? = 3 - 1
[5,7,1,8,3,6,2,4] => ? => ? = 4 - 1
[8,7,5,1,3,4,2,6] => ? => ? = 3 - 1
[5,8,1,6,3,7,4,2] => ? => ? = 3 - 1
[4,3,2,5,7,1,8,6] => ? => ? = 3 - 1
[5,1,3,6,4,8,7,2] => ? => ? = 3 - 1
[2,3,1,7,5,4,8,6] => ? => ? = 3 - 1
[2,3,8,6,1,4,7,5] => ? => ? = 2 - 1
[2,1,6,3,5,8,4,7] => ? => ? = 4 - 1
[2,6,1,8,3,5,4,7] => ? => ? = 4 - 1
[2,7,3,1,5,6,4,8] => ? => ? = 3 - 1
[8,3,5,7,1,6,4,2] => ? => ? = 3 - 1
[4,8,1,7,2,3,6,5] => ? => ? = 3 - 1
[3,7,1,8,2,6,4,5] => ? => ? = 4 - 1
[7,8,3,6,1,4,5,2] => ? => ? = 3 - 1
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00109: Permutations —descent word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00104: Binary words —reverse⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => => => ? = 1 - 1
[1,2] => 0 => 0 => 0 = 1 - 1
[2,1] => 1 => 1 => 0 = 1 - 1
[1,2,3] => 00 => 00 => 0 = 1 - 1
[1,3,2] => 01 => 10 => 0 = 1 - 1
[2,1,3] => 10 => 01 => 1 = 2 - 1
[2,3,1] => 01 => 10 => 0 = 1 - 1
[3,1,2] => 10 => 01 => 1 = 2 - 1
[3,2,1] => 11 => 11 => 0 = 1 - 1
[1,2,3,4] => 000 => 000 => 0 = 1 - 1
[1,2,4,3] => 001 => 100 => 0 = 1 - 1
[1,3,2,4] => 010 => 010 => 1 = 2 - 1
[1,3,4,2] => 001 => 100 => 0 = 1 - 1
[1,4,2,3] => 010 => 010 => 1 = 2 - 1
[1,4,3,2] => 011 => 110 => 0 = 1 - 1
[2,1,3,4] => 100 => 001 => 1 = 2 - 1
[2,1,4,3] => 101 => 101 => 1 = 2 - 1
[2,3,1,4] => 010 => 010 => 1 = 2 - 1
[2,3,4,1] => 001 => 100 => 0 = 1 - 1
[2,4,1,3] => 010 => 010 => 1 = 2 - 1
[2,4,3,1] => 011 => 110 => 0 = 1 - 1
[3,1,2,4] => 100 => 001 => 1 = 2 - 1
[3,1,4,2] => 101 => 101 => 1 = 2 - 1
[3,2,1,4] => 110 => 011 => 1 = 2 - 1
[3,2,4,1] => 101 => 101 => 1 = 2 - 1
[3,4,1,2] => 010 => 010 => 1 = 2 - 1
[3,4,2,1] => 011 => 110 => 0 = 1 - 1
[4,1,2,3] => 100 => 001 => 1 = 2 - 1
[4,1,3,2] => 101 => 101 => 1 = 2 - 1
[4,2,1,3] => 110 => 011 => 1 = 2 - 1
[4,2,3,1] => 101 => 101 => 1 = 2 - 1
[4,3,1,2] => 110 => 011 => 1 = 2 - 1
[4,3,2,1] => 111 => 111 => 0 = 1 - 1
[1,2,3,4,5] => 0000 => 0000 => 0 = 1 - 1
[1,2,3,5,4] => 0001 => 1000 => 0 = 1 - 1
[1,2,4,3,5] => 0010 => 0100 => 1 = 2 - 1
[1,2,4,5,3] => 0001 => 1000 => 0 = 1 - 1
[1,2,5,3,4] => 0010 => 0100 => 1 = 2 - 1
[1,2,5,4,3] => 0011 => 1100 => 0 = 1 - 1
[1,3,2,4,5] => 0100 => 0010 => 1 = 2 - 1
[1,3,2,5,4] => 0101 => 1010 => 1 = 2 - 1
[1,3,4,2,5] => 0010 => 0100 => 1 = 2 - 1
[1,3,4,5,2] => 0001 => 1000 => 0 = 1 - 1
[1,3,5,2,4] => 0010 => 0100 => 1 = 2 - 1
[1,3,5,4,2] => 0011 => 1100 => 0 = 1 - 1
[1,4,2,3,5] => 0100 => 0010 => 1 = 2 - 1
[1,4,2,5,3] => 0101 => 1010 => 1 = 2 - 1
[1,4,3,2,5] => 0110 => 0110 => 1 = 2 - 1
[1,4,3,5,2] => 0101 => 1010 => 1 = 2 - 1
[1,4,5,2,3] => 0010 => 0100 => 1 = 2 - 1
[1,4,5,3,2] => 0011 => 1100 => 0 = 1 - 1
[8,5,4,6,3,7,2,1] => ? => ? => ? = 3 - 1
[7,6,4,5,3,8,2,1] => ? => ? => ? = 3 - 1
[6,4,5,7,3,8,1,2] => ? => ? => ? = 4 - 1
[6,5,7,3,4,1,2,8] => ? => ? => ? = 4 - 1
[6,7,3,4,2,1,5,8] => ? => ? => ? = 3 - 1
[6,7,4,2,3,1,5,8] => ? => ? => ? = 3 - 1
[6,3,4,5,2,1,7,8] => ? => ? => ? = 3 - 1
[4,3,5,2,8,7,6,1] => ? => ? => ? = 3 - 1
[2,1,4,3,6,7,5,8] => ? => ? => ? = 4 - 1
[1,4,3,5,2,6,7,8] => ? => ? => ? = 3 - 1
[4,2,3,1,6,5,7,8] => ? => ? => ? = 4 - 1
[4,5,1,2,7,3,8,6] => ? => ? => ? = 3 - 1
[5,4,7,8,1,2,3,6] => ? => ? => ? = 3 - 1
[8,6,2,5,7,1,3,4] => ? => ? => ? = 3 - 1
[6,3,8,2,5,1,4,7] => ? => ? => ? = 4 - 1
[8,4,6,2,5,1,3,7] => ? => ? => ? = 4 - 1
[8,4,7,3,5,1,2,6] => ? => ? => ? = 4 - 1
[7,8,1,6,3,2,5,4] => ? => ? => ? = 3 - 1
[4,6,7,1,3,2,5,8] => ? => ? => ? = 3 - 1
[4,5,1,3,7,2,6,8] => ? => ? => ? = 3 - 1
[3,5,1,4,2,7,6,8] => ? => ? => ? = 4 - 1
[6,3,4,1,2,7,5,8] => ? => ? => ? = 4 - 1
[6,4,5,1,7,2,3,8] => ? => ? => ? = 4 - 1
[5,7,1,8,4,3,2,6] => ? => ? => ? = 3 - 1
[7,6,8,2,4,5,1,3] => ? => ? => ? = 4 - 1
[8,6,7,2,4,1,5,3] => ? => ? => ? = 4 - 1
[8,7,2,1,4,6,3,5] => ? => ? => ? = 3 - 1
[8,5,2,7,1,6,3,4] => ? => ? => ? = 4 - 1
[8,5,2,7,1,3,6,4] => ? => ? => ? = 3 - 1
[8,5,7,4,1,6,2,3] => ? => ? => ? = 4 - 1
[8,5,7,4,1,2,6,3] => ? => ? => ? = 3 - 1
[8,3,2,7,1,5,4,6] => ? => ? => ? = 4 - 1
[8,5,2,7,4,6,1,3] => ? => ? => ? = 4 - 1
[7,4,2,3,1,8,6,5] => ? => ? => ? = 3 - 1
[5,6,1,4,3,2,8,7] => ? => ? => ? = 3 - 1
[5,7,1,8,3,6,2,4] => ? => ? => ? = 4 - 1
[8,7,5,1,3,4,2,6] => ? => ? => ? = 3 - 1
[5,8,1,6,3,7,4,2] => ? => ? => ? = 3 - 1
[4,3,2,5,7,1,8,6] => ? => ? => ? = 3 - 1
[5,1,3,6,4,8,7,2] => ? => ? => ? = 3 - 1
[2,3,1,7,5,4,8,6] => ? => ? => ? = 3 - 1
[2,3,8,6,1,4,7,5] => ? => ? => ? = 2 - 1
[2,1,6,3,5,8,4,7] => ? => ? => ? = 4 - 1
[2,6,1,8,3,5,4,7] => ? => ? => ? = 4 - 1
[2,7,3,1,5,6,4,8] => ? => ? => ? = 3 - 1
[8,3,5,7,1,6,4,2] => ? => ? => ? = 3 - 1
[4,8,1,7,2,3,6,5] => ? => ? => ? = 3 - 1
[3,7,1,8,2,6,4,5] => ? => ? => ? = 4 - 1
[7,8,3,6,1,4,5,2] => ? => ? => ? = 3 - 1
Description
The number of ascents of a binary word.
Matching statistic: St000390
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00071: Permutations —descent composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1] => 1 => 1
[1,2] => [2] => [2] => 10 => 1
[2,1] => [1,1] => [1,1] => 11 => 1
[1,2,3] => [3] => [3] => 100 => 1
[1,3,2] => [2,1] => [1,2] => 110 => 1
[2,1,3] => [1,2] => [2,1] => 101 => 2
[2,3,1] => [2,1] => [1,2] => 110 => 1
[3,1,2] => [1,2] => [2,1] => 101 => 2
[3,2,1] => [1,1,1] => [1,1,1] => 111 => 1
[1,2,3,4] => [4] => [4] => 1000 => 1
[1,2,4,3] => [3,1] => [1,3] => 1100 => 1
[1,3,2,4] => [2,2] => [2,2] => 1010 => 2
[1,3,4,2] => [3,1] => [1,3] => 1100 => 1
[1,4,2,3] => [2,2] => [2,2] => 1010 => 2
[1,4,3,2] => [2,1,1] => [1,1,2] => 1110 => 1
[2,1,3,4] => [1,3] => [3,1] => 1001 => 2
[2,1,4,3] => [1,2,1] => [2,1,1] => 1011 => 2
[2,3,1,4] => [2,2] => [2,2] => 1010 => 2
[2,3,4,1] => [3,1] => [1,3] => 1100 => 1
[2,4,1,3] => [2,2] => [2,2] => 1010 => 2
[2,4,3,1] => [2,1,1] => [1,1,2] => 1110 => 1
[3,1,2,4] => [1,3] => [3,1] => 1001 => 2
[3,1,4,2] => [1,2,1] => [2,1,1] => 1011 => 2
[3,2,1,4] => [1,1,2] => [1,2,1] => 1101 => 2
[3,2,4,1] => [1,2,1] => [2,1,1] => 1011 => 2
[3,4,1,2] => [2,2] => [2,2] => 1010 => 2
[3,4,2,1] => [2,1,1] => [1,1,2] => 1110 => 1
[4,1,2,3] => [1,3] => [3,1] => 1001 => 2
[4,1,3,2] => [1,2,1] => [2,1,1] => 1011 => 2
[4,2,1,3] => [1,1,2] => [1,2,1] => 1101 => 2
[4,2,3,1] => [1,2,1] => [2,1,1] => 1011 => 2
[4,3,1,2] => [1,1,2] => [1,2,1] => 1101 => 2
[4,3,2,1] => [1,1,1,1] => [1,1,1,1] => 1111 => 1
[1,2,3,4,5] => [5] => [5] => 10000 => 1
[1,2,3,5,4] => [4,1] => [1,4] => 11000 => 1
[1,2,4,3,5] => [3,2] => [2,3] => 10100 => 2
[1,2,4,5,3] => [4,1] => [1,4] => 11000 => 1
[1,2,5,3,4] => [3,2] => [2,3] => 10100 => 2
[1,2,5,4,3] => [3,1,1] => [1,1,3] => 11100 => 1
[1,3,2,4,5] => [2,3] => [3,2] => 10010 => 2
[1,3,2,5,4] => [2,2,1] => [2,1,2] => 10110 => 2
[1,3,4,2,5] => [3,2] => [2,3] => 10100 => 2
[1,3,4,5,2] => [4,1] => [1,4] => 11000 => 1
[1,3,5,2,4] => [3,2] => [2,3] => 10100 => 2
[1,3,5,4,2] => [3,1,1] => [1,1,3] => 11100 => 1
[1,4,2,3,5] => [2,3] => [3,2] => 10010 => 2
[1,4,2,5,3] => [2,2,1] => [2,1,2] => 10110 => 2
[1,4,3,2,5] => [2,1,2] => [1,2,2] => 11010 => 2
[1,4,3,5,2] => [2,2,1] => [2,1,2] => 10110 => 2
[1,4,5,2,3] => [3,2] => [2,3] => 10100 => 2
[6,5,7,3,4,1,2,8] => ? => ? => ? => ? = 4
[6,7,3,4,2,1,5,8] => ? => ? => ? => ? = 3
[6,7,4,2,3,1,5,8] => ? => ? => ? => ? = 3
[4,3,5,6,1,2,7,8] => ? => ? => ? => ? = 3
[4,3,5,2,8,7,6,1] => ? => ? => ? => ? = 3
[2,1,4,3,6,7,5,8] => ? => ? => ? => ? = 4
[1,4,3,5,2,6,7,8] => ? => ? => ? => ? = 3
[2,1,4,3,6,5,8,7,10,9] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[9,10,8,7,6,5,4,3,1,2] => [2,1,1,1,1,1,1,2] => [1,1,1,1,1,1,2,2] => 1111111010 => ? = 2
[2,3,4,5,6,7,8,9,10,1] => [9,1] => [1,9] => 1100000000 => ? = 1
[2,1,4,3,6,5,8,7,10,9,12,11] => [1,2,2,2,2,2,1] => [2,2,2,2,2,1,1] => 101010101011 => ? = 6
[2,1,3,4,5,6,7,8,9,10] => [1,9] => [9,1] => 1000000001 => ? = 2
[9,1,2,3,4,5,6,7,10,8] => [1,8,1] => [8,1,1] => 1000000011 => ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [1,10] => [10,1] => 10000000001 => ? = 2
[10,1,2,3,4,5,6,7,8,9] => [1,9] => [9,1] => 1000000001 => ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [1,10] => [10,1] => 10000000001 => ? = 2
[4,6,7,1,3,2,5,8] => ? => ? => ? => ? = 3
[4,5,1,3,7,2,6,8] => ? => ? => ? => ? = 3
[6,3,4,1,2,7,5,8] => ? => ? => ? => ? = 4
[2,3,4,5,6,7,8,9,10,11,1] => [10,1] => [1,10] => 11000000000 => ? = 1
[8,9,7,6,5,4,3,2,1,10] => [2,1,1,1,1,1,1,2] => [1,1,1,1,1,1,2,2] => 1111111010 => ? = 2
[9,1,2,3,4,5,6,7,8,10] => [1,9] => [9,1] => 1000000001 => ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [1,10] => [10,1] => 10000000001 => ? = 2
[8,10,7,6,5,4,3,2,1,9] => [2,1,1,1,1,1,1,2] => [1,1,1,1,1,1,2,2] => 1111111010 => ? = 2
[8,7,2,1,4,6,3,5] => ? => ? => ? => ? = 3
[8,5,7,4,1,2,6,3] => ? => ? => ? => ? = 3
[7,4,2,3,1,8,6,5] => ? => ? => ? => ? = 3
[5,7,1,8,3,6,2,4] => ? => ? => ? => ? = 4
[4,3,2,5,7,1,8,6] => ? => ? => ? => ? = 3
[2,3,1,7,5,4,8,6] => ? => ? => ? => ? = 3
[2,3,8,6,1,4,7,5] => ? => ? => ? => ? = 2
[2,1,6,3,5,8,4,7] => ? => ? => ? => ? = 4
[2,6,1,8,3,5,4,7] => ? => ? => ? => ? = 4
[2,7,3,1,5,6,4,8] => ? => ? => ? => ? = 3
[3,1,4,2,6,5,8,7,10,9] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[3,1,5,2,6,4,8,7,10,9] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[3,1,5,2,7,4,8,6,10,9] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[3,1,5,2,7,4,9,6,10,8] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[6,1,7,2,8,3,9,4,10,5] => [1,2,2,2,2,1] => [2,2,2,2,1,1] => 1010101011 => ? = 5
[7,1,8,2,9,3,10,4,11,5,12,6] => [1,2,2,2,2,2,1] => [2,2,2,2,2,1,1] => 101010101011 => ? = 6
[8,3,5,7,1,6,4,2] => ? => ? => ? => ? = 3
[8,7,4,6,1,2,5,3] => ? => ? => ? => ? = 3
[3,6,8,1,2,7,4,5] => ? => ? => ? => ? = 3
[7,8,6,4,1,2,5,3] => ? => ? => ? => ? = 2
[5,7,4,8,2,1,3,6] => ? => ? => ? => ? = 3
[7,4,6,2,1,5,3,8] => ? => ? => ? => ? = 4
[7,4,8,5,2,3,1,6] => ? => ? => ? => ? = 4
[5,4,7,6,2,8,3,1] => ? => ? => ? => ? = 3
[6,8,5,2,1,7,3,4] => ? => ? => ? => ? = 3
[5,4,2,1,6,7,3,8] => ? => ? => ? => ? = 3
Description
The number of runs of ones in a binary word.
Matching statistic: St000010
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 95% ●values known / values provided: 95%●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]]
=> [3] => [3]
=> 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => [2,1]
=> 2
[2,3,1] => [[1,2],[3]]
=> [3] => [3]
=> 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => [2,1]
=> 2
[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]]
=> [4] => [4]
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => [3,1]
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => [4]
=> 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => [3,1]
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => [4]
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => [2,2]
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => [2,2]
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => [3,1]
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => [4]
=> 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => [3,1]
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => [4]
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => [2,2]
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => [2,2]
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => [3,1]
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => [2,2]
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => [3,1]
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => [4]
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => [2,2]
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => [2,2]
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => [3,1]
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => [2,2]
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => [3,1]
=> 2
[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]]
=> [5] => [5]
=> 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => [4,1]
=> 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => [5]
=> 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => [4,1]
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => [5]
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => [3,2]
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => [3,2]
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => [4,1]
=> 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => [5]
=> 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => [4,1]
=> 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => [5]
=> 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => [3,2]
=> 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => [3,2]
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => [4,1]
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => [3,2]
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => [4,1]
=> 2
[8,5,4,6,3,7,2,1] => ?
=> ? => ?
=> ? = 3
[7,6,4,5,3,8,2,1] => ?
=> ? => ?
=> ? = 3
[6,4,5,7,3,8,1,2] => ?
=> ? => ?
=> ? = 4
[7,5,6,4,8,1,2,3] => ?
=> ? => ?
=> ? = 4
[7,6,4,5,8,1,2,3] => ?
=> ? => ?
=> ? = 3
[6,5,7,3,4,1,2,8] => ?
=> ? => ?
=> ? = 4
[6,7,4,2,3,1,5,8] => ?
=> ? => ?
=> ? = 3
[6,3,4,5,2,1,7,8] => ?
=> ? => ?
=> ? = 3
[4,3,5,2,8,7,6,1] => ?
=> ? => ?
=> ? = 3
[2,1,4,3,6,7,5,8] => ?
=> ? => ?
=> ? = 4
[1,4,3,5,2,6,7,8] => ?
=> ? => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => ?
=> ? => ?
=> ? = 4
[4,5,1,2,7,3,8,6] => ?
=> ? => ?
=> ? = 3
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 2
[5,3,6,7,1,2,4,8] => ?
=> ? => ?
=> ? = 3
[8,6,2,5,7,1,3,4] => ?
=> ? => ?
=> ? = 3
[8,6,3,5,7,1,2,4] => ?
=> ? => ?
=> ? = 3
[6,3,8,2,5,1,4,7] => ?
=> ? => ?
=> ? = 4
[8,4,6,2,5,1,3,7] => ?
=> ? => ?
=> ? = 4
[7,1,2,3,4,5,9,6,8] => [[1,3,4,5,6,7,9],[2,8]]
=> ? => ?
=> ? = 3
[9,1,2,3,4,5,8,6,7] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? => ?
=> ? = 3
[8,3,9,6,1,7,2,4,5] => [[1,3,6,9],[2,4,8],[5,7]]
=> ? => ?
=> ? = 4
[2,3,8,1,5,4,7,6] => ?
=> ? => ?
=> ? = 3
[2,4,8,5,1,3,7,6] => ?
=> ? => ?
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[4,6,7,1,3,2,5,8] => ?
=> ? => ?
=> ? = 3
[4,5,1,3,7,2,6,8] => ?
=> ? => ?
=> ? = 3
[3,5,1,4,2,7,6,8] => ?
=> ? => ?
=> ? = 4
[6,3,4,1,2,7,5,8] => ?
=> ? => ?
=> ? = 4
[6,4,5,1,7,2,3,8] => ?
=> ? => ?
=> ? = 4
[10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ?
=> ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ?
=> ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ?
=> ? = 1
[7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? => ?
=> ? = 3
[5,7,1,8,4,3,2,6] => ?
=> ? => ?
=> ? = 3
[10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[7,3,4,2,1,6,5,8] => ?
=> ? => ?
=> ? = 4
[6,3,5,2,7,1,8,4] => ?
=> ? => ?
=> ? = 4
[7,9,6,5,4,3,2,1,8] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? => ?
=> ? = 2
[8,10,7,6,5,4,3,2,1,9] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 2
[7,6,8,2,4,5,1,3] => ?
=> ? => ?
=> ? = 4
[8,5,2,3,1,7,6,4] => ?
=> ? => ?
=> ? = 3
[8,6,2,3,7,1,5,4] => ?
=> ? => ?
=> ? = 3
[8,6,7,2,4,1,5,3] => ?
=> ? => ?
=> ? = 4
[8,7,2,1,4,6,3,5] => ?
=> ? => ?
=> ? = 3
[8,5,2,7,1,6,3,4] => ?
=> ? => ?
=> ? = 4
[8,5,2,7,1,3,6,4] => ?
=> ? => ?
=> ? = 3
[8,5,7,4,1,6,2,3] => ?
=> ? => ?
=> ? = 4
[8,5,7,4,1,2,6,3] => ?
=> ? => ?
=> ? = 3
Description
The length of the partition.
Matching statistic: St000389
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000389: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1 => 1
[1,2] => [[1,2]]
=> [2] => 10 => 1
[2,1] => [[1],[2]]
=> [2] => 10 => 1
[1,2,3] => [[1,2,3]]
=> [3] => 100 => 1
[1,3,2] => [[1,2],[3]]
=> [3] => 100 => 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => 101 => 2
[2,3,1] => [[1,2],[3]]
=> [3] => 100 => 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => 101 => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 100 => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1000 => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => 1010 => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => 1001 => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => 1010 => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => 1001 => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 10000 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => 10000 => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => 10010 => 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => 10010 => 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => 10001 => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => 10000 => 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => 10010 => 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => 10010 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => 10001 => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => 10010 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => 10001 => 2
[8,5,4,6,3,7,2,1] => ?
=> ? => ? => ? = 3
[7,6,4,5,3,8,2,1] => ?
=> ? => ? => ? = 3
[6,4,5,7,3,8,1,2] => ?
=> ? => ? => ? = 4
[7,5,6,4,8,1,2,3] => ?
=> ? => ? => ? = 4
[7,6,4,5,8,1,2,3] => ?
=> ? => ? => ? = 3
[6,5,7,3,4,1,2,8] => ?
=> ? => ? => ? = 4
[6,7,4,2,3,1,5,8] => ?
=> ? => ? => ? = 3
[6,3,4,5,2,1,7,8] => ?
=> ? => ? => ? = 3
[4,3,5,2,8,7,6,1] => ?
=> ? => ? => ? = 3
[2,1,4,3,6,7,5,8] => ?
=> ? => ? => ? = 4
[1,4,3,5,2,6,7,8] => ?
=> ? => ? => ? = 3
[4,2,3,1,6,5,7,8] => ?
=> ? => ? => ? = 4
[4,5,1,2,7,3,8,6] => ?
=> ? => ? => ? = 3
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ? => ? = 2
[5,3,6,7,1,2,4,8] => ?
=> ? => ? => ? = 3
[8,6,2,5,7,1,3,4] => ?
=> ? => ? => ? = 3
[8,6,3,5,7,1,2,4] => ?
=> ? => ? => ? = 3
[6,3,8,2,5,1,4,7] => ?
=> ? => ? => ? = 4
[8,4,6,2,5,1,3,7] => ?
=> ? => ? => ? = 4
[7,1,2,3,4,5,9,6,8] => [[1,3,4,5,6,7,9],[2,8]]
=> ? => ? => ? = 3
[9,1,2,3,4,5,8,6,7] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? => ? => ? = 3
[8,3,9,6,1,7,2,4,5] => [[1,3,6,9],[2,4,8],[5,7]]
=> ? => ? => ? = 4
[2,3,8,1,5,4,7,6] => ?
=> ? => ? => ? = 3
[2,4,8,5,1,3,7,6] => ?
=> ? => ? => ? = 2
[2,1,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[9,1,2,3,4,5,6,7,10,8] => [[1,3,4,5,6,7,8,9],[2,10]]
=> [2,8] => 1010000000 => ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[10,1,2,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[4,6,7,1,3,2,5,8] => ?
=> ? => ? => ? = 3
[4,5,1,3,7,2,6,8] => ?
=> ? => ? => ? = 3
[3,5,1,4,2,7,6,8] => ?
=> ? => ? => ? = 4
[6,3,4,1,2,7,5,8] => ?
=> ? => ? => ? = 4
[6,4,5,1,7,2,3,8] => ?
=> ? => ? => ? = 4
[10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? => ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[10,9,8,7,6,5,4,3,1,2] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[8,9,7,6,5,4,3,2,1,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? => ? => ? = 3
[5,7,1,8,4,3,2,6] => ?
=> ? => ? => ? = 3
[9,1,2,3,4,5,6,7,8,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[7,3,4,2,1,6,5,8] => ?
=> ? => ? => ? = 4
[6,3,5,2,7,1,8,4] => ?
=> ? => ? => ? = 4
[7,9,6,5,4,3,2,1,8] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? => ? => ? = 2
[8,10,7,6,5,4,3,2,1,9] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ? => ? = 2
[7,6,8,2,4,5,1,3] => ?
=> ? => ? => ? = 4
[8,5,2,3,1,7,6,4] => ?
=> ? => ? => ? = 3
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000288
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[1] => [[1]]
=> [1] => 1 => 1
[1,2] => [[1,2]]
=> [2] => 10 => 1
[2,1] => [[1],[2]]
=> [2] => 10 => 1
[1,2,3] => [[1,2,3]]
=> [3] => 100 => 1
[1,3,2] => [[1,2],[3]]
=> [3] => 100 => 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => 101 => 2
[2,3,1] => [[1,2],[3]]
=> [3] => 100 => 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => 101 => 2
[3,2,1] => [[1],[2],[3]]
=> [3] => 100 => 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => 1000 => 1
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => 1010 => 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => 1001 => 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => 1000 => 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => 1001 => 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => 1010 => 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => 1001 => 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => 1000 => 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => 1010 => 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => 1010 => 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => 1001 => 2
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => 1000 => 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => 10000 => 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => 10000 => 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => 10010 => 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => 10010 => 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => 10001 => 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => 10000 => 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => 10001 => 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => 10000 => 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => 10010 => 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => 10010 => 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => 10001 => 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => 10010 => 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => 10001 => 2
[8,5,4,6,3,7,2,1] => ?
=> ? => ? => ? = 3
[7,6,4,5,3,8,2,1] => ?
=> ? => ? => ? = 3
[6,4,5,7,3,8,1,2] => ?
=> ? => ? => ? = 4
[7,5,6,4,8,1,2,3] => ?
=> ? => ? => ? = 4
[7,6,4,5,8,1,2,3] => ?
=> ? => ? => ? = 3
[6,5,7,3,4,1,2,8] => ?
=> ? => ? => ? = 4
[6,7,4,2,3,1,5,8] => ?
=> ? => ? => ? = 3
[6,3,4,5,2,1,7,8] => ?
=> ? => ? => ? = 3
[4,3,5,2,8,7,6,1] => ?
=> ? => ? => ? = 3
[2,1,4,3,6,7,5,8] => ?
=> ? => ? => ? = 4
[1,4,3,5,2,6,7,8] => ?
=> ? => ? => ? = 3
[4,2,3,1,6,5,7,8] => ?
=> ? => ? => ? = 4
[4,5,1,2,7,3,8,6] => ?
=> ? => ? => ? = 3
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ? => ? = 2
[5,3,6,7,1,2,4,8] => ?
=> ? => ? => ? = 3
[8,6,2,5,7,1,3,4] => ?
=> ? => ? => ? = 3
[8,6,3,5,7,1,2,4] => ?
=> ? => ? => ? = 3
[6,3,8,2,5,1,4,7] => ?
=> ? => ? => ? = 4
[8,4,6,2,5,1,3,7] => ?
=> ? => ? => ? = 4
[7,1,2,3,4,5,9,6,8] => [[1,3,4,5,6,7,9],[2,8]]
=> ? => ? => ? = 3
[9,1,2,3,4,5,8,6,7] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? => ? => ? = 3
[8,3,9,6,1,7,2,4,5] => [[1,3,6,9],[2,4,8],[5,7]]
=> ? => ? => ? = 4
[2,3,8,1,5,4,7,6] => ?
=> ? => ? => ? = 3
[2,4,8,5,1,3,7,6] => ?
=> ? => ? => ? = 2
[2,1,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[9,1,2,3,4,5,6,7,10,8] => [[1,3,4,5,6,7,8,9],[2,10]]
=> [2,8] => 1010000000 => ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,1] => 100000001 => ? = 2
[9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[10,1,2,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[4,6,7,1,3,2,5,8] => ?
=> ? => ? => ? = 3
[4,5,1,3,7,2,6,8] => ?
=> ? => ? => ? = 3
[3,5,1,4,2,7,6,8] => ?
=> ? => ? => ? = 4
[6,3,4,1,2,7,5,8] => ?
=> ? => ? => ? = 4
[6,4,5,1,7,2,3,8] => ?
=> ? => ? => ? = 4
[10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ? => ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? => ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? => ? = 1
[9,8,7,6,5,4,3,1,2] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,1] => 100000001 => ? = 2
[10,9,8,7,6,5,4,3,1,2] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[7,8,6,5,4,3,2,1,9] => [[1,2,9],[3],[4],[5],[6],[7],[8]]
=> [8,1] => 100000001 => ? = 2
[8,9,7,6,5,4,3,2,1,10] => [[1,2,10],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => 1000000001 => ? = 2
[7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? => ? => ? = 3
[5,7,1,8,4,3,2,6] => ?
=> ? => ? => ? = 3
[9,1,2,3,4,5,6,7,8,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => 1010000000 => ? = 2
[10,1,2,3,4,5,6,7,8,9,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? => ? = 2
[7,3,4,2,1,6,5,8] => ?
=> ? => ? => ? = 4
[6,3,5,2,7,1,8,4] => ?
=> ? => ? => ? = 4
[7,9,6,5,4,3,2,1,8] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? => ? => ? = 2
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000097
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000097: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
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]]
=> [3] => ([],3)
=> 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[1,2],[3]]
=> [3] => ([],3)
=> 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[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]]
=> [4] => ([],4)
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[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]]
=> [5] => ([],5)
=> 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[8,5,4,6,3,7,2,1] => ?
=> ? => ?
=> ? = 3
[7,6,4,5,3,8,2,1] => ?
=> ? => ?
=> ? = 3
[6,4,5,7,3,8,1,2] => ?
=> ? => ?
=> ? = 4
[7,5,6,4,8,1,2,3] => ?
=> ? => ?
=> ? = 4
[7,6,4,5,8,1,2,3] => ?
=> ? => ?
=> ? = 3
[6,5,7,3,4,1,2,8] => ?
=> ? => ?
=> ? = 4
[6,7,4,2,3,1,5,8] => ?
=> ? => ?
=> ? = 3
[6,3,4,5,2,1,7,8] => ?
=> ? => ?
=> ? = 3
[4,3,5,2,8,7,6,1] => ?
=> ? => ?
=> ? = 3
[2,1,4,3,6,7,5,8] => ?
=> ? => ?
=> ? = 4
[1,4,3,5,2,6,7,8] => ?
=> ? => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => ?
=> ? => ?
=> ? = 4
[4,5,1,2,7,3,8,6] => ?
=> ? => ?
=> ? = 3
[2,1,4,3,6,5,8,7,10,9] => [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 2
[5,3,6,7,1,2,4,8] => ?
=> ? => ?
=> ? = 3
[8,6,2,5,7,1,3,4] => ?
=> ? => ?
=> ? = 3
[8,6,3,5,7,1,2,4] => ?
=> ? => ?
=> ? = 3
[6,3,8,2,5,1,4,7] => ?
=> ? => ?
=> ? = 4
[8,4,6,2,5,1,3,7] => ?
=> ? => ?
=> ? = 4
[9,1,2,3,4,5,6,7,8] => [[1,3,4,5,6,7,8,9],[2]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[8,1,2,3,4,5,6,9,7] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[7,1,2,3,4,5,9,6,8] => [[1,3,4,5,6,7,9],[2,8]]
=> ? => ?
=> ? = 3
[9,1,2,3,4,5,8,6,7] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? => ?
=> ? = 3
[7,1,2,3,4,5,8,9,6] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[8,3,9,6,1,7,2,4,5] => [[1,3,6,9],[2,4,8],[5,7]]
=> ? => ?
=> ? = 4
[2,3,4,5,6,7,8,9,1] => [[1,2,3,4,5,6,7,8],[9]]
=> [9] => ([],9)
=> ? = 1
[2,3,4,5,6,7,8,9,10,1] => [[1,2,3,4,5,6,7,8,9],[10]]
=> [10] => ([],10)
=> ? = 1
[2,3,8,1,5,4,7,6] => ?
=> ? => ?
=> ? = 3
[2,4,8,5,1,3,7,6] => ?
=> ? => ?
=> ? = 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,2,2,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 6
[2,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[9,1,2,3,4,5,6,7,10,8] => [[1,3,4,5,6,7,8,9],[2,10]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[7,1,2,3,4,5,6,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[10,1,2,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[4,6,7,1,3,2,5,8] => ?
=> ? => ?
=> ? = 3
[4,5,1,3,7,2,6,8] => ?
=> ? => ?
=> ? = 3
[3,5,1,4,2,7,6,8] => ?
=> ? => ?
=> ? = 4
[6,3,4,1,2,7,5,8] => ?
=> ? => ?
=> ? = 4
[6,4,5,1,7,2,3,8] => ?
=> ? => ?
=> ? = 4
[1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9] => ([],9)
=> ? = 1
[1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10] => ([],10)
=> ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ?
=> ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ?
=> ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ?
=> ? = 1
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St001581
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001581: Graphs ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
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]]
=> [3] => ([],3)
=> 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[2,3,1] => [[1,2],[3]]
=> [3] => ([],3)
=> 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2
[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]]
=> [4] => ([],4)
=> 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => ([],4)
=> 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => ([],4)
=> 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => ([(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
[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]]
=> [5] => ([],5)
=> 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => ([],5)
=> 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => ([],5)
=> 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 2
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[8,5,4,6,3,7,2,1] => ?
=> ? => ?
=> ? = 3
[7,6,4,5,3,8,2,1] => ?
=> ? => ?
=> ? = 3
[6,4,5,7,3,8,1,2] => ?
=> ? => ?
=> ? = 4
[7,5,6,4,8,1,2,3] => ?
=> ? => ?
=> ? = 4
[7,6,4,5,8,1,2,3] => ?
=> ? => ?
=> ? = 3
[6,5,7,3,4,1,2,8] => ?
=> ? => ?
=> ? = 4
[6,7,4,2,3,1,5,8] => ?
=> ? => ?
=> ? = 3
[6,3,4,5,2,1,7,8] => ?
=> ? => ?
=> ? = 3
[4,3,5,2,8,7,6,1] => ?
=> ? => ?
=> ? = 3
[2,1,4,3,6,7,5,8] => ?
=> ? => ?
=> ? = 4
[1,4,3,5,2,6,7,8] => ?
=> ? => ?
=> ? = 3
[4,2,3,1,6,5,7,8] => ?
=> ? => ?
=> ? = 4
[4,5,1,2,7,3,8,6] => ?
=> ? => ?
=> ? = 3
[2,1,4,3,6,5,8,7,10,9] => [[1,3,5,7,9],[2,4,6,8,10]]
=> [2,2,2,2,2] => ([(1,9),(2,8),(2,9),(3,7),(3,8),(3,9),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 5
[9,10,8,7,6,5,4,3,1,2] => [[1,2],[3,10],[4],[5],[6],[7],[8],[9]]
=> ? => ?
=> ? = 2
[5,3,6,7,1,2,4,8] => ?
=> ? => ?
=> ? = 3
[8,6,2,5,7,1,3,4] => ?
=> ? => ?
=> ? = 3
[8,6,3,5,7,1,2,4] => ?
=> ? => ?
=> ? = 3
[6,3,8,2,5,1,4,7] => ?
=> ? => ?
=> ? = 4
[8,4,6,2,5,1,3,7] => ?
=> ? => ?
=> ? = 4
[9,1,2,3,4,5,6,7,8] => [[1,3,4,5,6,7,8,9],[2]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[8,1,2,3,4,5,6,9,7] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[7,1,2,3,4,5,9,6,8] => [[1,3,4,5,6,7,9],[2,8]]
=> ? => ?
=> ? = 3
[9,1,2,3,4,5,8,6,7] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? => ?
=> ? = 3
[7,1,2,3,4,5,8,9,6] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[8,3,9,6,1,7,2,4,5] => [[1,3,6,9],[2,4,8],[5,7]]
=> ? => ?
=> ? = 4
[2,3,4,5,6,7,8,9,1] => [[1,2,3,4,5,6,7,8],[9]]
=> [9] => ([],9)
=> ? = 1
[2,3,4,5,6,7,8,9,10,1] => [[1,2,3,4,5,6,7,8,9],[10]]
=> [10] => ([],10)
=> ? = 1
[2,3,8,1,5,4,7,6] => ?
=> ? => ?
=> ? = 3
[2,4,8,5,1,3,7,6] => ?
=> ? => ?
=> ? = 2
[2,1,4,3,6,5,8,7,10,9,12,11] => [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [2,2,2,2,2,2] => ([(1,11),(2,10),(2,11),(3,9),(3,10),(3,11),(4,8),(4,9),(4,10),(4,11),(5,7),(5,8),(5,9),(5,10),(5,11),(6,7),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11),(8,9),(8,10),(8,11),(9,10),(9,11),(10,11)],12)
=> ? = 6
[2,1,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9],[2]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[9,1,2,3,4,5,6,7,10,8] => [[1,3,4,5,6,7,8,9],[2,10]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[7,1,2,3,4,5,6,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> [2,7] => ([(6,8),(7,8)],9)
=> ? = 2
[2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[8,7,6,5,4,3,2,1,9] => [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2
[9,8,7,6,5,4,3,2,1,10] => [[1,10],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
[10,1,2,3,4,5,6,7,8,9] => [[1,3,4,5,6,7,8,9,10],[2]]
=> [2,8] => ([(7,9),(8,9)],10)
=> ? = 2
[11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ?
=> ? = 2
[4,6,7,1,3,2,5,8] => ?
=> ? => ?
=> ? = 3
[4,5,1,3,7,2,6,8] => ?
=> ? => ?
=> ? = 3
[3,5,1,4,2,7,6,8] => ?
=> ? => ?
=> ? = 4
[6,3,4,1,2,7,5,8] => ?
=> ? => ?
=> ? = 4
[6,4,5,1,7,2,3,8] => ?
=> ? => ?
=> ? = 4
[1,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9] => ([],9)
=> ? = 1
[1,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10] => ([],10)
=> ? = 1
[10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? => ?
=> ? = 2
[1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ?
=> ? = 1
[2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ?
=> ? = 1
Description
The achromatic number of a graph.
This is the maximal number of colours of a proper colouring, such that for any pair of colours there are two adjacent vertices with these colours.
Matching statistic: St000806
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Mp00295: Standard tableaux —valley composition⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 83% ●values known / values provided: 94%●distinct values known / distinct values provided: 83%
Values
[1] => [[1]]
=> [1] => [1] => ? = 1 + 1
[1,2] => [[1,2]]
=> [2] => [1] => ? = 1 + 1
[2,1] => [[1],[2]]
=> [2] => [1] => ? = 1 + 1
[1,2,3] => [[1,2,3]]
=> [3] => [1] => ? = 1 + 1
[1,3,2] => [[1,2],[3]]
=> [3] => [1] => ? = 1 + 1
[2,1,3] => [[1,3],[2]]
=> [2,1] => [1,1] => 3 = 2 + 1
[2,3,1] => [[1,2],[3]]
=> [3] => [1] => ? = 1 + 1
[3,1,2] => [[1,3],[2]]
=> [2,1] => [1,1] => 3 = 2 + 1
[3,2,1] => [[1],[2],[3]]
=> [3] => [1] => ? = 1 + 1
[1,2,3,4] => [[1,2,3,4]]
=> [4] => [1] => ? = 1 + 1
[1,2,4,3] => [[1,2,3],[4]]
=> [4] => [1] => ? = 1 + 1
[1,3,2,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[1,3,4,2] => [[1,2,3],[4]]
=> [4] => [1] => ? = 1 + 1
[1,4,2,3] => [[1,2,4],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[1,4,3,2] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1 + 1
[2,1,3,4] => [[1,3,4],[2]]
=> [2,2] => [2] => 3 = 2 + 1
[2,1,4,3] => [[1,3],[2,4]]
=> [2,2] => [2] => 3 = 2 + 1
[2,3,1,4] => [[1,2,4],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[2,3,4,1] => [[1,2,3],[4]]
=> [4] => [1] => ? = 1 + 1
[2,4,1,3] => [[1,2],[3,4]]
=> [3,1] => [1,1] => 3 = 2 + 1
[2,4,3,1] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1 + 1
[3,1,2,4] => [[1,3,4],[2]]
=> [2,2] => [2] => 3 = 2 + 1
[3,1,4,2] => [[1,3],[2,4]]
=> [2,2] => [2] => 3 = 2 + 1
[3,2,1,4] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[3,2,4,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => 3 = 2 + 1
[3,4,1,2] => [[1,2],[3,4]]
=> [3,1] => [1,1] => 3 = 2 + 1
[3,4,2,1] => [[1,2],[3],[4]]
=> [4] => [1] => ? = 1 + 1
[4,1,2,3] => [[1,3,4],[2]]
=> [2,2] => [2] => 3 = 2 + 1
[4,1,3,2] => [[1,3],[2],[4]]
=> [2,2] => [2] => 3 = 2 + 1
[4,2,1,3] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[4,2,3,1] => [[1,3],[2],[4]]
=> [2,2] => [2] => 3 = 2 + 1
[4,3,1,2] => [[1,4],[2],[3]]
=> [3,1] => [1,1] => 3 = 2 + 1
[4,3,2,1] => [[1],[2],[3],[4]]
=> [4] => [1] => ? = 1 + 1
[1,2,3,4,5] => [[1,2,3,4,5]]
=> [5] => [1] => ? = 1 + 1
[1,2,3,5,4] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,2,4,3,5] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,2,4,5,3] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,2,5,3,4] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,2,5,4,3] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,3,2,4,5] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,3,2,5,4] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,3,4,2,5] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,3,4,5,2] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,3,5,2,4] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,3,5,4,2] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,4,2,3,5] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,4,2,5,3] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,4,3,2,5] => [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,4,3,5,2] => [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,4,5,2,3] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,4,5,3,2] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,5,2,3,4] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,5,2,4,3] => [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,5,3,2,4] => [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,5,3,4,2] => [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[1,5,4,2,3] => [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[2,1,3,4,5] => [[1,3,4,5],[2]]
=> [2,3] => [1,1] => 3 = 2 + 1
[2,1,3,5,4] => [[1,3,4],[2,5]]
=> [2,3] => [1,1] => 3 = 2 + 1
[2,1,4,3,5] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 4 = 3 + 1
[2,1,4,5,3] => [[1,3,4],[2,5]]
=> [2,3] => [1,1] => 3 = 2 + 1
[2,1,5,3,4] => [[1,3,5],[2,4]]
=> [2,2,1] => [2,1] => 4 = 3 + 1
[2,1,5,4,3] => [[1,3],[2,4],[5]]
=> [2,3] => [1,1] => 3 = 2 + 1
[2,3,1,4,5] => [[1,2,4,5],[3]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,3,1,5,4] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,3,4,1,5] => [[1,2,3,5],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[2,3,4,5,1] => [[1,2,3,4],[5]]
=> [5] => [1] => ? = 1 + 1
[2,3,5,1,4] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => 3 = 2 + 1
[2,3,5,4,1] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[2,4,1,3,5] => [[1,2,5],[3,4]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,4,1,5,3] => [[1,2,4],[3,5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,4,3,1,5] => [[1,2,5],[3],[4]]
=> [4,1] => [1,1] => 3 = 2 + 1
[2,4,3,5,1] => [[1,2,4],[3],[5]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,4,5,1,3] => [[1,2,3],[4,5]]
=> [4,1] => [1,1] => 3 = 2 + 1
[2,4,5,3,1] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[2,5,1,3,4] => [[1,2,5],[3,4]]
=> [3,2] => [1,1] => 3 = 2 + 1
[2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[3,4,5,2,1] => [[1,2,3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> [5] => [1] => ? = 1 + 1
[1,2,3,4,5,6] => [[1,2,3,4,5,6]]
=> [6] => [1] => ? = 1 + 1
[1,2,3,4,6,5] => [[1,2,3,4,5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,3,5,6,4] => [[1,2,3,4,5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,3,6,5,4] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,4,5,6,3] => [[1,2,3,4,5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,4,6,5,3] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,5,6,4,3] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,2,6,5,4,3] => [[1,2,3],[4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,3,4,5,6,2] => [[1,2,3,4,5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,3,4,6,5,2] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,3,5,6,4,2] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,3,6,5,4,2] => [[1,2,3],[4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,4,5,6,3,2] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[2,3,4,5,6,1] => [[1,2,3,4,5],[6]]
=> [6] => [1] => ? = 1 + 1
[2,3,4,6,5,1] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
[2,3,5,6,4,1] => [[1,2,3,4],[5],[6]]
=> [6] => [1] => ? = 1 + 1
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
The following 79 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000386The number of factors DDU in a Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St000568The hook number of a binary tree. St001732The number of peaks visible from the left. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St000098The chromatic number of a graph. 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. St000159The number of distinct parts of the integer partition. St000069The number of maximal elements of a poset. St000071The number of maximal chains in a poset. St000527The width of the poset. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000257The number of distinct parts of a partition that occur at least twice. St000068The number of minimal elements in a poset. 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. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000632The jump number of the poset. St000172The Grundy number of a graph. St000528The height of a poset. St000912The number of maximal antichains in a poset. 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. St001343The dimension of the reduced incidence algebra of a poset. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001670The connected partition number of a graph. St001717The largest size of an interval in a poset. 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. 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. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000080The rank of the poset. St000353The number of inner valleys of a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001668The number of points of the poset minus the width of the poset. St000455The second largest eigenvalue of a graph if it is integral. St000256The number of parts from which one can substract 2 and still get an integer partition. St001728The number of invisible descents of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000023The number of inner peaks of a permutation. St000523The number of 2-protected nodes of a rooted tree. St000409The number of pitchforks in a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000822The Hadwiger number of the graph. St000646The number of big ascents of a permutation. St001812The biclique partition number of a graph. St001330The hat guessing number of a graph. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001597The Frobenius rank of a skew partition. St000021The number of descents of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St001487The number of inner corners of a skew partition. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001823The Stasinski-Voll length of a signed permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
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