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Your data matches 59 different statistics following compositions of up to 3 maps.
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Matching statistic: St000291
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 01 => 0
[1,0,1,0]
=> 1010 => 0101 => 1
[1,1,0,0]
=> 1100 => 1001 => 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 2
[1,0,1,1,0,0]
=> 101100 => 011001 => 1
[1,1,0,0,1,0]
=> 110010 => 100101 => 2
[1,1,0,1,0,0]
=> 110100 => 101001 => 2
[1,1,1,0,0,0]
=> 111000 => 110001 => 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 3
[1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 2
[1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 2
[1,0,1,1,0,1,0,0]
=> 10110100 => 01101001 => 2
[1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 3
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 2
[1,1,0,1,0,0,1,0]
=> 11010010 => 10100101 => 3
[1,1,0,1,0,1,0,0]
=> 11010100 => 10101001 => 3
[1,1,0,1,1,0,0,0]
=> 11011000 => 10110001 => 2
[1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 2
[1,1,1,0,0,1,0,0]
=> 11100100 => 11001001 => 2
[1,1,1,0,1,0,0,0]
=> 11101000 => 11010001 => 2
[1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 4
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 3
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 3
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101101001 => 3
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 2
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 3
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 2
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0110100101 => 3
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0110101001 => 3
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0111010001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 4
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001101001 => 3
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1010010101 => 4
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1010100101 => 4
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1010101001 => 4
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1011001001 => 3
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1011010001 => 3
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1100100101 => 3
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1100101001 => 3
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1101000101 => 3
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1101001001 => 3
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1101010001 => 3
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1110010001 => 2
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1110100001 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 010101010101 => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 010101011001 => 4
Description
The number of descents of a binary word.
Matching statistic: St000292
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00134: Standard tableaux —descent word⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> 1 => 0
[1,0,1,0]
=> [[1,3],[2,4]]
=> 101 => 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> 010 => 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 10101 => 2
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 10010 => 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> 01001 => 2
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> 01010 => 2
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> 00100 => 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 1010101 => 3
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> 1010010 => 2
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> 1001001 => 2
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> 1001010 => 2
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> 1000100 => 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> 0100101 => 3
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> 0100010 => 2
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> 0101001 => 3
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> 0101010 => 3
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> 0100100 => 2
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> 0010001 => 2
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0010010 => 2
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> 0010100 => 2
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 0001000 => 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> 101010101 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> 101010010 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> 101001001 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> 101001010 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> 101000100 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> 100100101 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> 100100010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> 100101001 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> 100101010 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> 100010001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> 100010100 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> 100001000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> 010010101 => 4
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> 010001010 => 3
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> 010100101 => 4
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> 010101001 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> 010101010 => 4
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> 010010010 => 3
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> 010010100 => 3
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> 001001001 => 3
[1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> 001001010 => 3
[1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> 001010001 => 3
[1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> 001010010 => 3
[1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> 001010100 => 3
[1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> 000100100 => 2
[1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> 000101000 => 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> 10101010101 => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> 10101010010 => 4
Description
The number of ascents of a binary word.
Matching statistic: St000203
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000203: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [.,[.,.]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [[.,.],.]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [.,[[.,.],.]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [[.,.],[.,.]]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [[.,.],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [[.,[.,.]],.]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [.,[[.,.],[.,.]]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [.,[[.,[.,.]],.]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [[.,.],[[.,.],.]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [[.,.],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [[.,.],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [[.,.],[[.,.],.]]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [[.,[.,.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [[.,[.,.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [[.,[.,[.,.]]],.]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [.,[.,[[.,.],[.,.]]]]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [.,[.,[[.,[.,.]],.]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [.,[[.,.],[.,[.,.]]]]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [.,[[.,[.,[.,.]]],.]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [[.,.],[[.,.],[.,.]]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [[.,.],[.,[.,[.,.]]]]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [[.,.],[[.,.],[.,.]]]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [[.,.],[[.,.],[.,.]]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [[.,[.,.]],[.,[.,.]]]
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [.,[.,[.,[.,[[.,.],.]]]]]
=> 5 = 4 + 1
Description
The number of external nodes of a binary tree.
That is, the number of nodes that can be reached from the root by only left steps or only right steps, plus $1$ for the root node itself. A counting formula for the number of external node in all binary trees of size $n$ can be found in [1].
Matching statistic: St000390
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000390: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 01 => 1 = 0 + 1
[1,0,1,0]
=> 1010 => 0101 => 2 = 1 + 1
[1,1,0,0]
=> 1100 => 1001 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 010101 => 3 = 2 + 1
[1,0,1,1,0,0]
=> 101100 => 011001 => 2 = 1 + 1
[1,1,0,0,1,0]
=> 110010 => 100101 => 3 = 2 + 1
[1,1,0,1,0,0]
=> 110100 => 101001 => 3 = 2 + 1
[1,1,1,0,0,0]
=> 111000 => 110001 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01101001 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 10100101 => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10101001 => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10110001 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 11001001 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11010001 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101101001 => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0110100101 => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0110101001 => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0111010001 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001101001 => 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1010010101 => 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1010100101 => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1010101001 => 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1011001001 => 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1011010001 => 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1100100101 => 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> 1110010100 => 1100101001 => 4 = 3 + 1
[1,1,1,0,1,0,0,0,1,0]
=> 1110100010 => 1101000101 => 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> 1110100100 => 1101001001 => 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1101010001 => 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1110010001 => 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> 1111010000 => 1110100001 => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => 010101010101 => 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => 010101011001 => 5 = 4 + 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000053
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0]
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> 4
Description
The number of valleys of the Dyck path.
Matching statistic: St000552
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000552: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[[]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[[],[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4
Description
The number of cut vertices of a graph.
A cut vertex is one whose deletion increases the number of connected components.
Matching statistic: St001692
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001692: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St001692: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [[]]
=> ([(0,1)],2)
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [[],[]]
=> ([(0,2),(1,2)],3)
=> 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[[]],[]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[],[[]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [[],[],[]]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 3
[1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 3
[1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[[]]]]]]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [[[[[[],[]]]]]]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 4
Description
The number of vertices with higher degree than the average degree in a graph.
Matching statistic: St000010
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00294: Standard tableaux —peak composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1],[2]]
=> [2] => [2]
=> 1 = 0 + 1
[1,0,1,0]
=> [[1,3],[2,4]]
=> [3,1] => [3,1]
=> 2 = 1 + 1
[1,1,0,0]
=> [[1,2],[3,4]]
=> [2,2] => [2,2]
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> [3,2,1] => [3,2,1]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> [4,2] => [4,2]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [[1,2,5],[3,4,6]]
=> [2,3,1] => [3,2,1]
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [[1,2,4],[3,5,6]]
=> [2,2,2] => [2,2,2]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [[1,2,3],[4,5,6]]
=> [3,3] => [3,3]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> [3,2,2,1] => [3,2,2,1]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [[1,3,5,6],[2,4,7,8]]
=> [3,3,2] => [3,3,2]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [[1,3,4,7],[2,5,6,8]]
=> [4,3,1] => [4,3,1]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> [4,2,2] => [4,2,2]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> [5,3] => [5,3]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [[1,2,5,7],[3,4,6,8]]
=> [2,3,2,1] => [3,2,2,1]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> [2,4,2] => [4,2,2]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> [2,2,3,1] => [3,2,2,1]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [[1,2,4,6],[3,5,7,8]]
=> [2,2,2,2] => [2,2,2,2]
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [[1,2,4,5],[3,6,7,8]]
=> [2,3,3] => [3,3,2]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> [3,4,1] => [4,3,1]
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> [3,3,2] => [3,3,2]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [[1,2,3,5],[4,6,7,8]]
=> [3,2,3] => [3,3,2]
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> [4,4] => [4,4]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> [3,2,2,2,1] => [3,2,2,2,1]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> [3,2,3,2] => [3,3,2,2]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,3,5,6,9],[2,4,7,8,10]]
=> [3,3,3,1] => [3,3,3,1]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,6,8],[2,4,7,9,10]]
=> [3,3,2,2] => [3,3,2,2]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> [3,4,3] => [4,3,3]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[1,3,4,7,9],[2,5,6,8,10]]
=> [4,3,2,1] => [4,3,2,1]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,3,4,7,8],[2,5,6,9,10]]
=> [4,4,2] => [4,4,2]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,3,4,6,9],[2,5,7,8,10]]
=> [4,2,3,1] => [4,3,2,1]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,3,4,6,8],[2,5,7,9,10]]
=> [4,2,2,2] => [4,2,2,2]
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[1,3,4,5,9],[2,6,7,8,10]]
=> [5,4,1] => [5,4,1]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,3,4,5,7],[2,6,8,9,10]]
=> [5,2,3] => [5,3,2]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> [6,4] => [6,4]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[1,2,5,7,9],[3,4,6,8,10]]
=> [2,3,2,2,1] => [3,2,2,2,1]
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[1,2,5,6,8],[3,4,7,9,10]]
=> [2,4,2,2] => [4,2,2,2]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[1,2,4,7,9],[3,5,6,8,10]]
=> [2,2,3,2,1] => [3,2,2,2,1]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[1,2,4,6,9],[3,5,7,8,10]]
=> [2,2,2,3,1] => [3,2,2,2,1]
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[1,2,4,6,8],[3,5,7,9,10]]
=> [2,2,2,2,2] => [2,2,2,2,2]
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[1,2,4,5,8],[3,6,7,9,10]]
=> [2,3,3,2] => [3,3,2,2]
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[1,2,4,5,7],[3,6,8,9,10]]
=> [2,3,2,3] => [3,3,2,2]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [[1,2,3,6,9],[4,5,7,8,10]]
=> [3,3,3,1] => [3,3,3,1]
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [[1,2,3,6,8],[4,5,7,9,10]]
=> [3,3,2,2] => [3,3,2,2]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [[1,2,3,5,9],[4,6,7,8,10]]
=> [3,2,4,1] => [4,3,2,1]
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [[1,2,3,5,8],[4,6,7,9,10]]
=> [3,2,3,2] => [3,3,2,2]
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [[1,2,3,5,7],[4,6,8,9,10]]
=> [3,2,2,3] => [3,3,2,2]
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [[1,2,3,4,7],[5,6,8,9,10]]
=> [4,3,3] => [4,3,3]
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [[1,2,3,4,6],[5,7,8,9,10]]
=> [4,2,4] => [4,4,2]
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> [3,2,2,2,2,1] => [3,2,2,2,2,1]
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> [3,2,2,3,2] => [3,3,2,2,2]
=> 5 = 4 + 1
Description
The length of the partition.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000093: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2] => ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [6] => ([],6)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000786
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2] => ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [3] => ([],3)
=> 3 = 2 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [3] => ([],3)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [6] => ([],6)
=> 6 = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 4 + 1
Description
The maximal number of occurrences of a colour in a proper colouring of a graph.
To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions.
For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes, $[2,2,2]$ and $[3,2,1]$. Therefore, the statistic on this graph is $3$.
The following 49 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St000619The number of cyclic descents of a permutation. St000288The number of ones in a binary word. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000991The number of right-to-left minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000691The number of changes of a binary word. St001489The maximum of the number of descents and the number of inverse descents. St000702The number of weak deficiencies of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001649The length of a longest trail in a graph. St001180Number of indecomposable injective modules with projective dimension at most 1. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001960The number of descents of a permutation minus one if its first entry is not one. St000389The number of runs of ones of odd length in a binary word. St000035The number of left outer peaks of a permutation. St000648The number of 2-excedences of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000097The order of the largest clique of the graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001581The achromatic number of a graph. St000386The number of factors DDU in a Dyck path. St000568The hook number of a binary tree. St000659The number of rises of length at least 2 of a Dyck path. St000919The number of maximal left branches of a binary tree. St000098The chromatic number of a graph. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000834The number of right outer peaks of a permutation. St000356The number of occurrences of the pattern 13-2. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St000647The number of big descents of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation.
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