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Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
St000019: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 0
[1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [2,3,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,3,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 2
Description
The cardinality of the support of a permutation. A permutation $\sigma$ may be written as a product $\sigma = s_{i_1}\dots s_{i_k}$ with $k$ minimal, where $s_i = (i,i+1)$ denotes the simple transposition swapping the entries in positions $i$ and $i+1$. The set of indices $\{i_1,\dots,i_k\}$ is the '''support''' of $\sigma$ and independent of the chosen way to write $\sigma$ as such a product. See [2], Definition 1 and Proposition 10. The '''connectivity set''' of $\sigma$ of length $n$ is the set of indices $1 \leq i < n$ such that $\sigma(k) < i$ for all $k < i$. Thus, the connectivity set is the complement of the support.
Matching statistic: St000288
Mp00129: Dyck paths to 321-avoiding permutation (Billey-Jockusch-Stanley)Permutations
Mp00114: Permutations connectivity setBinary words
Mp00105: Binary words complementBinary words
St000288: Binary words ⟶ ℤResult quality: 83% values known / values provided: 97%distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => => => ? = 0
[1,0,1,0]
=> [2,1] => 0 => 1 => 1
[1,1,0,0]
=> [1,2] => 1 => 0 => 0
[1,0,1,0,1,0]
=> [2,3,1] => 00 => 11 => 2
[1,0,1,1,0,0]
=> [2,1,3] => 01 => 10 => 1
[1,1,0,0,1,0]
=> [1,3,2] => 10 => 01 => 1
[1,1,0,1,0,0]
=> [3,1,2] => 00 => 11 => 2
[1,1,1,0,0,0]
=> [1,2,3] => 11 => 00 => 0
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 000 => 111 => 3
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 001 => 110 => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 010 => 101 => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 000 => 111 => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 011 => 100 => 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 100 => 011 => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 101 => 010 => 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 000 => 111 => 3
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 000 => 111 => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 001 => 110 => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 110 => 001 => 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 100 => 011 => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 000 => 111 => 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 111 => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 0000 => 1111 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0001 => 1110 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 0010 => 1101 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 0000 => 1111 => 4
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0011 => 1100 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 0100 => 1011 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0101 => 1010 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 0000 => 1111 => 4
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 0000 => 1111 => 4
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 0001 => 1110 => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0110 => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 0100 => 1011 => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 0000 => 1111 => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0111 => 1000 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1000 => 0111 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1001 => 0110 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 1010 => 0101 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => 1000 => 0111 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1011 => 0100 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => 0000 => 1111 => 4
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => 0001 => 1110 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => 0000 => 1111 => 4
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => 0000 => 1111 => 4
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => 0001 => 1110 => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => 0010 => 1101 => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => 0000 => 1111 => 4
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => 0000 => 1111 => 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => 0011 => 1100 => 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1100 => 0011 => 2
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [4,7,1,2,3,5,6,8] => ? => ? => ? = 6
[1,1,1,1,0,0,1,1,1,0,1,0,0,0,0,0]
=> [1,5,8,2,3,4,6,7] => ? => ? => ? = 6
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [6,8,1,2,3,4,5,7,9] => ? => ? => ? = 7
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => ? => ? => ? = 7
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> [7,1,8,2,3,4,5,6,9] => ? => ? => ? = 7
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => ? => ? => ? = 7
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => ? => ? => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => 0000000000 => 1111111111 => ? = 10
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [8,9,1,2,3,4,5,6,7,10] => ? => ? => ? = 8
[1,1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => ? => ? => ? = 8
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => 0000000000 => 1111111111 => ? = 10
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => 00000000000 => 11111111111 => ? = 11
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [2,8,9,10,11,12,1,3,4,5,6,7] => 00000000000 => 11111111111 => ? = 11
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00229: Dyck paths Delest-ViennotDyck paths
Mp00023: Dyck paths to non-crossing permutationPermutations
Mp00127: Permutations left-to-right-maxima to Dyck pathDyck paths
St000394: Dyck paths ⟶ ℤResult quality: 91% values known / values provided: 91%distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,2] => [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [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]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [2,4,3,1,7,6,8,5] => [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> ? = 6
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,8,4,7,5,6,3,1] => ?
=> ? = 7
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [2,8,6,5,4,3,7,1] => ?
=> ? = 7
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [2,6,3,4,5,1,7,8] => [1,1,0,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? = 5
[1,1,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,0,1,0]
=> [1,3,2,7,5,6,4,8] => ?
=> ? = 4
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,2,8,5,6,7,4] => ?
=> ? = 5
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5,8,7] => [1,1,1,0,0,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 5
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [3,2,4,1,6,8,7,5] => [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> ? = 6
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> [3,2,4,1,7,6,8,5] => [1,1,1,0,0,1,0,0,1,1,1,0,0,1,0,0]
=> ? = 6
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,0,0,1,0,0,0]
=> [8,3,4,6,5,2,7,1] => ?
=> ? = 7
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [8,3,4,7,5,6,2,1] => ?
=> ? = 7
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,5,4,6,1,8,7] => [1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> [3,2,1,7,6,5,8,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,1,0,0]
=> ? = 6
[1,1,0,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,2,1,4,5,8,7,6] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 4
[1,1,1,0,0,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,0,1,0]
=> [1,4,3,7,5,6,2,8] => ?
=> ? = 5
[1,1,1,0,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,1,0,0]
=> [5,2,4,3,6,1,8,7] => [1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> ? = 6
[1,1,1,0,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> [7,4,3,2,5,6,1,8] => ?
=> ? = 6
[1,1,1,1,0,0,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [1,6,3,5,4,7,2,8] => ?
=> ? = 5
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [1,5,3,4,2,7,6,8] => ?
=> ? = 4
[1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0,1,1,0,0]
=> [5,2,3,4,1,6,8,7] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> ? = 5
[1,1,1,1,1,0,0,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,6,3,4,5,7,2,8] => ?
=> ? = 5
[1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,8,3,6,5,7,4,2] => ?
=> ? = 6
[1,1,1,1,1,1,0,0,1,0,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,8,3,4,5,7,6,2] => ?
=> ? = 6
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,8,3,4,5,6,7,9,2] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 7
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0,1,0]
=> [8,6,3,4,5,2,7,1,9] => ?
=> ? = 7
[1,1,1,1,1,1,0,0,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> [1,9,7,4,5,6,3,8,2] => ?
=> ? = 7
[1,1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,2,7,4,5,6,3,1,9] => ?
=> ? = 7
[1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [8,7,3,4,5,6,2,1,9] => ?
=> ? = 7
[1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> [1,0,1,1,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,9,3,8,5,6,7,4,2] => ?
=> ? = 7
[1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [9,8,3,4,5,6,7,2,1] => ?
=> ? = 8
[1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [10,2,3,4,5,6,7,8,9,1,11] => ?
=> ? = 9
[1,1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [9,8,3,4,5,6,7,2,1,10] => ?
=> ? = 8
[1,1,1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,11,3,4,5,6,7,8,9,10,2] => ?
=> ? = 9
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> [2,10,4,8,6,5,7,3,9,1] => ?
=> ? = 9
[1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,1,0,1,1,0,0,0,1,0,0,0,1,0,0,0]
=> [2,12,4,10,6,8,7,5,9,3,11,1] => ?
=> ? = 11
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [10,9,3,4,5,6,7,8,2,1] => ?
=> ? = 9
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [10,9,8,4,5,6,7,3,2,1] => ?
=> ? = 9
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0,0,0]
=> [10,8,7,5,6,4,3,2,9,1] => ?
=> ? = 9
[1,1,0,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? => ?
=> ? = 9
[1,1,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,1,0,0,0,0,0]
=> [10,9,4,5,6,3,7,8,2,1] => ?
=> ? = 9
[1,1,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0,0,0]
=> [10,9,4,5,8,6,7,3,2,1] => ?
=> ? = 9
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 6
[1,0,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 9
[1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ?
=> ? => ?
=> ? = 11
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00118: Dyck paths swap returns and last descentDyck paths
Mp00099: Dyck paths bounce pathDyck paths
St000476: Dyck paths ⟶ ℤResult quality: 67% values known / values provided: 81%distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> ? = 6
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 7
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 7
[1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 4
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,1,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,0,0,1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 7
[1,1,0,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 7
[1,1,1,0,0,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,0,0,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,0,0,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,1,1,0,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 6
[1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 8
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 7
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> ? = 9
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1
[1,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
[1,1,1,0,1,0,1,0,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,1,0,0]
=> ?
=> ? = 7
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 8
[1,1,1,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 7
Description
The sum of the semi-lengths of tunnels before a valley of a Dyck path. For each valley $v$ in a Dyck path $D$ there is a corresponding tunnel, which is the factor $T_v = s_i\dots s_j$ of $D$ where $s_i$ is the step after the first intersection of $D$ with the line $y = ht(v)$ to the left of $s_j$. This statistic is $$ \sum_v (j_v-i_v)/2. $$
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00101: Dyck paths decomposition reverseDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000010: Integer partitions ⟶ ℤResult quality: 78% values known / values provided: 78%distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,2,1]
=> ? = 6
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1,1,1]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,3,2,1]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? = 6
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [5,5,3,3,2]
=> ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,2,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,1]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,2,1]
=> ? = 7
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 7
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,5,3,2,2,2,1]
=> ? = 7
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [7,5,5,3,3,1,1]
=> ? = 7
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,1,1]
=> ? = 6
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,5,2,2,2,1]
=> ? = 7
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,0,1,0]
=> [7,4,3,3,1]
=> ? = 5
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1]
=> ? = 5
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [7,4,2,2,2,1]
=> ? = 6
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,3,1,1]
=> ? = 7
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,1,1,1]
=> ? = 7
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [7,4,4,4,2,2,1]
=> ? = 7
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,2,1]
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [7,3,2,2,2,2,1]
=> ? = 7
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3,2,1]
=> ? = 7
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,3,3,3]
=> ? = 5
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,2,2,2,2,1]
=> ? = 7
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,2]
=> ? = 5
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1]
=> ? = 5
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,5,3,1,1]
=> ? = 6
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> [6,5,5,2,1,1]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? = 7
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,1]
=> ? = 5
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,6,5,3,3,1,1]
=> ? = 7
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,1]
=> ? = 6
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [6,6,5,2,2,2,1]
=> ? = 7
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1]
=> ? = 6
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,1,1,1]
=> ? = 7
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,1]
=> ? = 6
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [6,6,4,4,3,1,1]
=> ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [6,4,2,2,2,2,1]
=> ? = 7
Description
The length of the partition.
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00030: Dyck paths zeta mapDyck paths
Mp00027: Dyck paths to partitionInteger partitions
St000147: Integer partitions ⟶ ℤResult quality: 75% values known / values provided: 75%distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,2,1]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,1]
=> ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? = 6
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,4,2,2]
=> ? = 5
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [5,5,4,2,2]
=> ? = 5
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> ? = 6
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,3,2,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,3,2,1]
=> ? = 7
[1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,3,3,3,2,1]
=> ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? = 6
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,2,1]
=> ? = 7
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [5,4,3,2,2,2]
=> ? = 5
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> ? = 7
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,1]
=> ? = 7
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,1,1]
=> ? = 7
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [7,6,3,1,1,1]
=> ? = 7
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,2,1]
=> ? = 6
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,1,1]
=> ? = 6
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,1]
=> ? = 7
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4,1,1]
=> ? = 7
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,2,2,1]
=> ? = 6
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,1,1]
=> ? = 6
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3]
=> ? = 6
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,1,1]
=> ? = 7
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5,1]
=> ? = 7
[1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,2,1]
=> ? = 5
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,6,3,1,1,1]
=> ? = 6
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [5,5,4,2,2,2]
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2]
=> ? = 7
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,5,3,3,2,2]
=> ? = 7
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,6,3,2,2,2]
=> ? = 7
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [7,4,4,2,2,1]
=> ? = 7
[1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [6,5,4,1,1,1]
=> ? = 6
[1,1,0,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0,1,0]
=> [7,5,4,4,2]
=> ? = 7
[1,1,0,1,1,0,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,2,1]
=> ? = 7
[1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2,2]
=> ? = 7
[1,1,0,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,1,1]
=> ? = 6
[1,1,0,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [7,4,4,3,1,1,1]
=> ? = 7
[1,1,0,1,1,1,0,1,0,0,1,0,1,0,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,0,1,0]
=> [7,6,4,4,1]
=> ? = 7
[1,1,0,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> [7,6,4,2]
=> ? = 7
[1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2]
=> ? = 7
[1,1,1,0,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [5,5,4,4,3,2]
=> ? = 5
[1,1,1,0,0,1,1,0,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [5,5,2,2,2,1]
=> ? = 5
Description
The largest part of an integer partition.
Matching statistic: St000093
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000093: Graphs ⟶ ℤResult quality: 58% values known / values provided: 64%distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(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]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(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]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 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
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00100: Dyck paths touch compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000786: Graphs ⟶ ℤResult quality: 58% values known / values provided: 64%distinct values known / distinct values provided: 58%
Values
[1,0]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => ([],3)
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => ([],4)
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(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]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(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]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [5] => ([],5)
=> 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [3,1,1,3] => ([(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2,2] => ([(1,7),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,2,4] => ([(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,0,1,0,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> [1,1,0,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [2,4,2] => ([(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,1,5] => ([(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,6] => ([(5,7),(6,7)],8)
=> ? = 6 + 1
[1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,2,3] => ([(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 4 + 1
[1,1,0,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,1,0,0,1,0,1,0,0]
=> [1,7] => ([(6,7)],8)
=> ? = 6 + 1
[1,1,0,0,1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,1,1,3] => ([(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
[1,1,0,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,5] => ([(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,0,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,1,0,0,1,1,0,0]
=> [4,2,2] => ([(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 5 + 1
[1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [4,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,1,0,1,0,0,0]
=> [8] => ([],8)
=> ? = 7 + 1
[1,1,0,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? = 6 + 1
[1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [8] => ([],8)
=> ? = 7 + 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 $3$.
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00030: Dyck paths zeta mapDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000054: Permutations ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 3 = 2 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 4 = 3 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 4 = 3 + 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 5 = 4 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 5 = 4 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 5 = 4 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 5 = 4 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 5 = 4 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 5 = 4 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 5 = 4 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 4 = 3 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 4 = 3 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 5 = 4 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 5 = 4 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 3 = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => ? = 4 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => ? = 6 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 3 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => ? = 6 + 1
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => ? = 3 + 1
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,1,4,6] => ? = 6 + 1
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => ? = 4 + 1
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4 + 1
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 6 + 1
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,3,4,5] => ? = 6 + 1
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,2,3,1,4,5,7] => ? = 5 + 1
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => ? = 3 + 1
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,6,1,2,3,7] => ? = 4 + 1
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5 + 1
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => ? = 5 + 1
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? = 3 + 1
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,1,2,4] => ? = 5 + 1
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,4,1,2,5,7] => ? = 5 + 1
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => ? = 3 + 1
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,1,2,4] => ? = 5 + 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => ? = 5 + 1
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,1,2,3] => ? = 5 + 1
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,1,2,7] => ? = 5 + 1
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => ? = 6 + 1
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [6,4,5,1,2,3,7] => ? = 5 + 1
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => ? = 4 + 1
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ? = 5 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 5 + 1
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,2,1,3,4,7] => ? = 4 + 1
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 5 + 1
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 6 + 1
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,1,2] => ? = 5 + 1
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => ? = 6 + 1
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? = 6 + 1
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [6,5,7,1,2,3,4] => ? = 5 + 1
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5 + 1
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ? = 3 + 1
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ? = 4 + 1
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ? = 5 + 1
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ? = 5 + 1
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 5 + 1
[1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [7,2,3,4,5,6,1] => ? = 6 + 1
[1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,1,2] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [8,6,5,7,4,3,2,1] => ? = 7 + 1
Description
The first entry of the permutation. This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1]. This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals $$ \frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1). $$
Mp00120: Dyck paths Lalanne-Kreweras involutionDyck paths
Mp00030: Dyck paths zeta mapDyck paths
Mp00025: Dyck paths to 132-avoiding permutationPermutations
St000141: Permutations ⟶ ℤResult quality: 55% values known / values provided: 55%distinct values known / distinct values provided: 92%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [2,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [3,2,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 4
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 4
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 4
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 4
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,3,2,1] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [7,5,4,6,3,2,1] => ? = 6
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,6,7,3,2,1] => ? = 4
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [7,5,4,3,2,6,1] => ? = 6
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3,5,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [5,4,6,3,2,7,1] => ? = 4
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [7,4,3,5,6,2,1] => ? = 6
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,2,1] => ? = 3
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2,4,1] => ? = 6
[1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,6,1,7] => ? = 3
[1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [7,5,2,3,1,4,6] => ? = 6
[1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> [5,2,3,4,1,6,7] => ? = 4
[1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> [5,4,1,2,3,6,7] => ? = 4
[1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [7,5,4,1,2,3,6] => ? = 6
[1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [7,6,2,1,3,4,5] => ? = 6
[1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> [6,2,3,1,4,5,7] => ? = 5
[1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,1,2,7] => ? = 3
[1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,1,0,0,0]
=> [5,4,6,1,2,3,7] => ? = 4
[1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [6,5,1,2,3,4,7] => ? = 5
[1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [6,7,2,1,3,4,5] => ? = 5
[1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,1,0,0,0,0]
=> [4,3,5,6,7,1,2] => ? = 3
[1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,1,2,4] => ? = 5
[1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,1,1,0,0,0,1,0,0]
=> [6,3,4,1,2,5,7] => ? = 5
[1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [4,2,3,5,1,6,7] => ? = 3
[1,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,1,2,4] => ? = 5
[1,1,1,0,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,1,0,0]
=> [6,7,3,1,2,4,5] => ? = 5
[1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [6,5,7,3,4,1,2] => ? = 5
[1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [6,5,7,4,1,2,3] => ? = 5
[1,1,1,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,1,2,7] => ? = 5
[1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [7,5,2,1,3,4,6] => ? = 6
[1,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,0]
=> [6,4,5,1,2,3,7] => ? = 5
[1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,1,0,0,0]
=> [5,2,3,1,4,6,7] => ? = 4
[1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [6,7,5,3,4,1,2] => ? = 5
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [6,7,5,4,1,2,3] => ? = 5
[1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [5,6,2,1,3,4,7] => ? = 4
[1,1,1,1,0,0,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,1,2,5] => ? = 5
[1,1,1,1,0,0,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> [6,7,4,1,2,3,5] => ? = 5
[1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [7,3,4,5,6,2,1] => ? = 6
[1,1,1,1,0,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,4,5,7,1,2] => ? = 5
[1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [7,5,6,2,3,4,1] => ? = 6
[1,1,1,1,0,1,0,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,1,0,0]
=> [6,4,5,7,1,2,3] => ? = 5
[1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [7,6,2,3,4,5,1] => ? = 6
[1,1,1,1,0,1,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,1,0,0]
=> [6,5,7,1,2,3,4] => ? = 5
[1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,1,0,0]
=> [6,2,1,3,4,5,7] => ? = 5
[1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,5,6,7,2,3,1] => ? = 3
[1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [5,6,7,2,3,4,1] => ? = 4
[1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [6,7,2,3,4,5,1] => ? = 5
[1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [6,7,3,4,5,1,2] => ? = 5
[1,1,1,1,1,0,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [6,7,5,1,2,3,4] => ? = 5
Description
The maximum drop size of a permutation. The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
The following 61 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000024The number of double up and double down steps of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000067The inversion number of the alternating sign matrix. St000653The last descent of a permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St000957The number of Bruhat lower covers of a permutation. St000809The reduced reflection length of the permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St000316The number of non-left-to-right-maxima of a permutation. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000240The number of indices that are not small excedances. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St000216The absolute length of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000356The number of occurrences of the pattern 13-2. St001082The number of boxed occurrences of 123 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St000840The number of closers smaller than the largest opener in a perfect matching. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St000673The number of non-fixed points of a permutation. St000242The number of indices that are not cyclical small weak excedances. St000358The number of occurrences of the pattern 31-2. St000672The number of minimal elements in Bruhat order not less than the permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000238The number of indices that are not small weak excedances. St000235The number of indices that are not cyclical small weak excedances. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000327The number of cover relations in a poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000066The column of the unique '1' in the first row of the alternating sign matrix. St001861The number of Bruhat lower covers of a permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001645The pebbling number of a connected graph. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000133The "bounce" of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St001960The number of descents of a permutation minus one if its first entry is not one. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001555The order of a signed permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.