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Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—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]
=> [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 1
[1,1,0,0,1,0]
=> [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,1,0,0,0]
=> [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 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]
=> [4,5,3,2,1] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 1
Description
The staircase size of the code of a permutation. The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$. The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$. This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,1]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [2]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,1,1]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [2,2]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [3,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [3,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [3,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => [2,2,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => [3,1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [3,1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => [2,2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => [2,2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [3,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => [2,2,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => [2,2,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [3,1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => [3,2]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [3,1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [3,1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [4,1]
=> 1
[]
=> [] => [] => []
=> ? = 0
Description
The size of a partition minus its first part. This is the number of boxes in its diagram that are not in the first row.
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 98% ā—values known / values provided: 98%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => 11 => 2
[1,0,1,1,0,0]
=> [2,3,1] => [1,3,2] => 01 => 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => 10 => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [1,4,3,2] => 011 => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => 101 => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 101 => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,4,3] => 001 => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,4,2,1] => 110 => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 110 => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,2,4,1] => 110 => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [1,3,2,4] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,3,1,4] => 100 => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => 1111 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [1,5,4,3,2] => 0111 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [2,5,4,3,1] => 1011 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [2,1,5,4,3] => 1011 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [1,2,5,4,3] => 0011 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [3,5,4,2,1] => 1101 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,5,4,2] => 1101 => 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => 1101 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [3,2,1,5,4] => 1101 => 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [1,3,2,5,4] => 0101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => 1001 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,3,1,5,4] => 1001 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [2,1,3,5,4] => 1001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,5,3,2,1] => 1110 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => 1110 => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [4,2,5,3,1] => 1110 => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,2,1,5,3] => 1110 => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [1,4,2,5,3] => 0110 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,3,5,2,1] => 1110 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,3,1,5,2] => 1110 => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,3,2,5,1] => 1110 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => 1110 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [1,4,3,2,5] => 0110 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,4,3,5,1] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => 1010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [2,1,4,3,5] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,4,5,2,1] => 1100 => 2
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [7,6,4,5,3,2,1,8] => [3,7,6,5,4,2,1,8] => ? => ? = 5
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [7,5,4,3,6,2,1,8] => [4,3,2,7,6,5,1,8] => ? => ? = 5
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,10,9] => 100000001 => ? = 2
[]
=> [] => [] => => ? = 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [8,7,5,4,6,3,2,1,9] => [4,3,8,7,6,5,2,1,9] => ? => ? = 6
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [8,9,7,6,5,4,3,2,1,10] => [1,9,8,7,6,5,4,3,2,10] => 011111110 => ? = 7
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [9,8,7,5,6,4,3,2,1,10] => [4,9,8,7,6,5,3,2,1,10] => ? => ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [8,9,1,2,3,4,5,6,7,10] => [3,4,5,6,7,8,1,9,2,10] => ? => ? = 2
Description
The number of ones in a binary word. This is also known as the Hamming weight of the word.
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 96% ā—values known / values provided: 96%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [[1,2],[3]]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [[1,3],[2],[4]]
=> 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [[1],[2],[3],[4],[5]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [[1,2],[3],[4],[5]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [[1,2],[3],[4],[5]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [[1,3],[2],[4],[5]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [[1,2,3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [[1,2],[3],[4],[5]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => [[1,3],[2,4],[5]]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => [[1,3],[2,4],[5]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [[1,4],[2],[3],[5]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [[1,2,3],[4],[5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [[1,3,4],[2],[5]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => [[1,4],[2,5],[3]]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => [[1,3],[2,4],[5]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [[1,5],[2],[3],[4]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => [[1,2,4],[3,5]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [[1,3,5],[2],[4]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [[1,2,3,5],[4]]
=> 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [9,2,1,3,4,5,6,7,8] => [2,1,3,4,5,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,1] => [3,2,4,5,6,7,8,9,1] => [[1,3,4,5,6,7,8],[2],[9]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [10,2,1,3,4,5,6,7,8,9] => [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [9,3,1,2,4,5,6,7,8] => [1,3,2,4,5,6,7,9,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [9,2,3,1,4,5,6,7,8] => [2,3,1,4,5,6,7,9,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [4,2,3,5,6,7,8,9,1] => [2,4,3,5,6,7,8,9,1] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [3,4,2,5,6,7,8,9,1] => [3,4,2,5,6,7,8,9,1] => [[1,2,4,5,6,7,8],[3],[9]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [3,2,4,5,6,7,8,9,10,1] => [[1,3,4,5,6,7,8,9],[2],[10]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [7,6,8,5,4,3,2,1,9] => [7,6,8,5,4,3,2,1,9] => [[1,3,9],[2],[4],[5],[6],[7],[8]]
=> ? = 6
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [8,7,5,4,6,3,2,1,9] => [5,4,8,7,6,3,2,1,9] => [[1,3,9],[2,4],[5],[6],[7],[8]]
=> ? = 6
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1,9] => [2,3,4,5,6,7,8,1,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1,8,9] => [2,3,4,5,6,7,1,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [7,8,1,2,3,4,5,6,9] => [1,2,3,4,7,5,8,6,9] => [[1,2,3,4,5,7,9],[6,8]]
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,7,9] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [7,1,2,3,4,5,6,8,9] => [1,2,3,4,5,7,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 9
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [1,2,3,4,5,6,7,8,10,9,11] => [[1,2,3,4,5,6,7,8,9,11],[10]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10,11] => [2,1,3,4,5,6,7,8,9,10,11] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
Description
The number of descents of a standard tableau. Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000228
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 91% ā—values known / values provided: 91%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> []
=> 0
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> []
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> []
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [1,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 6
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => ?
=> ?
=> ? = 5
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => ?
=> ?
=> ? = 4
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [6,3,4,2,5,7,1,8] => ?
=> ?
=> ? = 4
[]
=> [] => []
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,9,1] => ?
=> ?
=> ? = 6
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,4,7,5,6,8,9,1] => ?
=> ?
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,9,8,7,6,1] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [9,4,3,6,5,8,7,2,1] => ?
=> ?
=> ? = 4
[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] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [9,8,4,5,6,7,3,2,1] => ?
=> ?
=> ? = 3
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [9,8,5,4,7,6,3,2,1] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ?
=> ? = 7
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9] => ?
=> ?
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8,9] => ?
=> ?
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4,9,8,7] => ?
=> ?
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,8,7,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [4,3,2,1,7,6,5,8,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [4,3,2,1,7,6,5,9,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,8,7,6,5,9] => ?
=> ?
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,4,3,2,1,7,6,8,9] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [5,4,3,2,1,8,7,6,9] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [6,5,4,3,2,1,8,7,9] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ?
=> ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9,10] => ?
=> ?
=> ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ?
=> ? = 8
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9,10] => ?
=> ?
=> ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7,10,9] => ?
=> ?
=> ? = 7
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5,10,9,8] => ?
=> ?
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,8,7,6,5,9,10] => ?
=> ?
=> ? = 6
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,9,8,10] => ?
=> ?
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [5,4,3,2,1,9,8,7,6,10] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [6,5,4,3,2,1,9,8,7,10] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [7,6,5,4,3,2,1,9,8,10] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,6,8,9,10,1] => ?
=> ?
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [10,9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3
[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] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [10,9,8,5,6,7,4,3,2,1] => ?
=> ?
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [10,9,4,5,6,7,8,3,2,1] => ?
=> ?
=> ? = 4
[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] => ?
=> ?
=> ? = 3
[1,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3,2,7,6,5,10,9,8,1] => ?
=> ?
=> ? = 6
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 90% ā—values known / values provided: 90%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> 0
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 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]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[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
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[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]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[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]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? = 5
[1,1,1,0,0,0,1,1,1,0,0,1,1,0,0,0]
=> [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]
=> ? = 5
[1,1,1,0,0,1,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ? = 5
[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]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
[]
=> []
=> []
=> ? = 0
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ?
=> ? = 6
[1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 6
[1,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> ? = 1
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 3
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ?
=> ?
=> ? = 7
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 7
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? = 6
[1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ?
=> ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> ?
=> ? = 3
[1,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 3
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [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]
=> ? = 4
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ? = 2
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 6
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000377
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 90% ā—values known / values provided: 90%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [1]
=> 0
[1,0,1,0]
=> [1,2] => [2]
=> [1,1]
=> 1
[1,1,0,0]
=> [2,1] => [1,1]
=> [2]
=> 0
[1,0,1,0,1,0]
=> [1,2,3] => [3]
=> [1,1,1]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [3]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [3]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [3]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1]
=> [2,1]
=> 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4]
=> [1,1,1,1]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [2,1,1]
=> 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [2,2]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [4]
=> 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,1]
=> [2,1,1]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,1,1]
=> [2,2]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [2,2]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,1]
=> [2,2]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,1,1]
=> [2,2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1]
=> [3,1]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5]
=> [1,1,1,1,1]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [4,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,2]
=> [5]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [2,1,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [4,1]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [4,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [4,1]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [4,1]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2]
=> [5]
=> 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2]
=> [5]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [5]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,2,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [5]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [2,1,1,1]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,1]
=> [2,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [4,1]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [4,1]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [4,1]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [4,1]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,1,1,1]
=> [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 5
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 6
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,6,4,5,7,8,1] => ?
=> ?
=> ? = 5
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => ?
=> ?
=> ? = 5
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8] => ?
=> ?
=> ? = 5
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => ?
=> ?
=> ? = 5
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [6,2,3,4,5,7,1,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8] => ?
=> ?
=> ? = 4
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => ?
=> ?
=> ? = 4
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [6,3,4,2,5,7,1,8] => ?
=> ?
=> ? = 4
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,9,1] => ?
=> ?
=> ? = 6
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,4,7,5,6,8,9,1] => ?
=> ?
=> ? = 6
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,9,8,7,6,1] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [9,4,3,6,5,8,7,2,1] => ?
=> ?
=> ? = 4
[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] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [9,8,4,5,6,7,3,2,1] => ?
=> ?
=> ? = 3
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [7,6,5,4,3,2,9,8,1] => ?
=> ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [9,8,5,4,7,6,3,2,1] => ?
=> ?
=> ? = 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ?
=> ? = 7
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9] => ?
=> ?
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8,9] => ?
=> ?
=> ? = 6
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4,9,8,7] => ?
=> ?
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,8,7,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [4,3,2,1,7,6,5,8,9] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [4,3,2,1,7,6,5,9,8] => ?
=> ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,8,7,6,5,9] => ?
=> ?
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,4,3,2,1,7,6,8,9] => ?
=> ?
=> ? = 4
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [5,4,3,2,1,8,7,6,9] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [6,5,4,3,2,1,8,7,9] => ?
=> ?
=> ? = 3
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ?
=> ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9,10] => ?
=> ?
=> ? = 8
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ?
=> ? = 8
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9,10] => ?
=> ?
=> ? = 7
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7,10,9] => ?
=> ?
=> ? = 7
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5,10,9,8] => ?
=> ?
=> ? = 6
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,8,7,6,5,9,10] => ?
=> ?
=> ? = 6
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,9,8,10] => ?
=> ?
=> ? = 5
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [5,4,3,2,1,9,8,7,6,10] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [6,5,4,3,2,1,9,8,7,10] => ?
=> ?
=> ? = 4
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [7,6,5,4,3,2,1,9,8,10] => ?
=> ?
=> ? = 3
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,6,8,9,10,1] => ?
=> ?
=> ? = 7
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [10,9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3
[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] => ?
=> ?
=> ? = 5
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [10,9,8,5,6,7,4,3,2,1] => ?
=> ?
=> ? = 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [10,9,4,5,6,7,8,3,2,1] => ?
=> ?
=> ? = 4
[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] => ?
=> ?
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,0]
=> [8,7,6,5,4,3,2,10,9,1] => ?
=> ?
=> ? = 2
Description
The dinv defect of an integer partition. This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000738
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 90% ā—values known / values provided: 90%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,0,0]
=> [2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1]
=> [[1,3],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [[1,2],[3,4]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1]
=> [[1,3,4],[2]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [3,2]
=> [[1,2,5],[3,4]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,1]
=> [[1,3,4,5],[2]]
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,2,5,4,3,8,7,6] => ?
=> ?
=> ? = 5 + 1
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,8,4,6,7,5,2] => ?
=> ?
=> ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,5,4,6,7,8] => ?
=> ?
=> ? = 6 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8] => ?
=> ?
=> ? = 5 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [4,3,2,1,7,6,5,8] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [6,3,4,2,5,7,1,8] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,9,8,7,6,5,4,3,2,1,11] => [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,10,9,8,7,6,5,4,3,2] => [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[]
=> [] => []
=> []
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,6,5,7,8,9,1] => ?
=> ?
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [2,3,4,7,5,6,8,9,1] => ?
=> ?
=> ? = 6 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,9,8,7,6,1] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [9,4,3,6,5,8,7,2,1] => ?
=> ?
=> ? = 4 + 1
[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] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [9,8,4,5,6,7,3,2,1] => ?
=> ?
=> ? = 3 + 1
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [9,8,5,4,7,6,3,2,1] => ?
=> ?
=> ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,8,9,10,11,1] => [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 9 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9] => ?
=> ?
=> ? = 7 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9] => ?
=> ?
=> ? = 6 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,6,5,4,7,8,9] => ?
=> ?
=> ? = 6 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4,9,8,7] => ?
=> ?
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [4,3,2,1,6,5,7,8,9] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [4,3,2,1,6,5,8,7,9] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [4,3,2,1,7,6,5,8,9] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [4,3,2,1,7,6,5,9,8] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,8,7,6,5,9] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0,1,0]
=> [5,4,3,2,1,7,6,8,9] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [5,4,3,2,1,8,7,6,9] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> [6,5,4,3,2,1,8,7,9] => ?
=> ?
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5,6,7,8,9,10] => ?
=> ?
=> ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,6,5,7,8,9,10] => ?
=> ?
=> ? = 8 + 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,5,8,7,9,10] => ?
=> ?
=> ? = 8 + 1
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [3,2,1,5,4,6,7,8,9,10] => ?
=> ?
=> ? = 7 + 1
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,2,1,6,5,4,8,7,10,9] => ?
=> ?
=> ? = 7 + 1
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [4,3,2,1,7,6,5,10,9,8] => ?
=> ?
=> ? = 6 + 1
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,8,7,6,5,9,10] => ?
=> ?
=> ? = 6 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [5,4,3,2,1,7,6,9,8,10] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [5,4,3,2,1,9,8,7,6,10] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0,1,0]
=> [6,5,4,3,2,1,9,8,7,10] => ?
=> ?
=> ? = 4 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0,1,0]
=> [7,6,5,4,3,2,1,9,8,10] => ?
=> ?
=> ? = 3 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> [10,9,8,7,6,5,4,3,2,11,1] => [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> [11,10,9,8,6,7,5,4,3,2,1] => [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> [2,3,4,5,7,6,8,9,10,1] => ?
=> ?
=> ? = 7 + 1
[1,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0,0]
=> [10,9,5,4,3,8,7,6,2,1] => ?
=> ?
=> ? = 3 + 1
[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] => ?
=> ?
=> ? = 5 + 1
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [10,9,8,5,6,7,4,3,2,1] => ?
=> ?
=> ? = 2 + 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [10,9,4,5,6,7,8,3,2,1] => ?
=> ?
=> ? = 4 + 1
[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] => ?
=> ?
=> ? = 3 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000439
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 80% ā—values known / values provided: 80%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 2 = 0 + 2
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 3 = 1 + 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 3 = 1 + 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 0 + 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 6 = 4 + 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[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]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[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]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 2 + 2
[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]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3 = 1 + 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5 = 3 + 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 4 = 2 + 2
[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]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 4 = 2 + 2
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 1 + 2
[1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 5 + 2
[1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 2
[1,0,1,1,1,1,0,1,0,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,1,0,0,1,0,0,0]
=> ? = 4 + 2
[1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 + 2
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 + 2
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 5 + 2
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 4 + 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 5 + 2
[1,1,1,0,1,0,0,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 5 + 2
[1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,1,0,0]
=> ? = 5 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? = 5 + 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0]
=> ? = 4 + 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,1,0,0,0,1,0,0,0]
=> ? = 4 + 2
[1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,1,0,0,1,0,0,0]
=> ? = 4 + 2
[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]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 2
[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]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0]
=> ? = 4 + 2
[1,1,1,1,0,1,0,0,1,0,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 4 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 2
[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]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 3 + 2
[1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,1,0,0,0]
=> ? = 3 + 2
[1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,1,0,0,1,0,0]
=> ? = 3 + 2
[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]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3 + 2
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 2 + 2
[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]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 2
[]
=> []
=> []
=> []
=> ? = 0 + 2
[1,1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> ?
=> ?
=> ? = 6 + 2
[1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 6 + 2
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? = 6 + 2
[1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> ? = 5 + 2
[1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 4 + 2
[1,1,1,1,1,0,0,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 4 + 2
[1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 4 + 2
[1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 3 + 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> ?
=> ?
=> ? = 2 + 2
[1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 2 + 2
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0]
=> ? = 7 + 2
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0]
=> ? = 7 + 2
[1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,1,0,0,0,0,0,0]
=> ? = 6 + 2
[1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0]
=> ? = 6 + 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0,0]
=> ? = 6 + 2
[1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,1,0,0,0]
=> ? = 6 + 2
[1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5 + 2
[1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,1,0,0,0,0,0]
=> ? = 5 + 2
Description
The position of the first down step of a Dyck path.
Mp00030: Dyck paths —zeta map⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001733: Dyck paths ⟶ ℤResult quality: 71% ā—values known / values provided: 71%ā—distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[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]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[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]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[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]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[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]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,0,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 4 + 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[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]
=> [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 4 + 1
[1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,0,0,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0]
=> ? = 3 + 1
[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]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 3 + 1
[1,1,1,1,1,1,0,0,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 2 + 1
[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]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 2 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[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]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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]
=> ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [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 + 1
[1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [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 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 2 + 1
[]
=> []
=> []
=> []
=> ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7 + 1
Description
The number of weak left to right maxima of a Dyck path. A weak left to right maximum is a peak whose height is larger than or equal to the height of all peaks to its left.
The following 129 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000783The side length of the largest staircase partition fitting into a partition. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000306The bounce count of a Dyck path. St000245The number of ascents of a permutation. St000441The number of successions of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000214The number of adjacencies of a permutation. St000019The cardinality of the support of a permutation. St000018The number of inversions of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000632The jump number of the poset. St001432The order dimension of the partition. St000502The number of successions of a set partitions. St000678The number of up steps after the last double rise of a Dyck path. St001809The index of the step at the first peak of maximal height in a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000874The position of the last double rise in a Dyck path. St000444The length of the maximal rise of a Dyck path. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000536The pathwidth of a graph. 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. St001580The acyclic chromatic number of a graph. St000053The number of valleys of the Dyck path. St000211The rank of the set partition. St000234The number of global ascents of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001298The number of repeated entries in the Lehmer code of a permutation. St001963The tree-depth of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000527The width of the poset. St000862The number of parts of the shifted shape of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001489The maximum of the number of descents and the number of inverse descents. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000470The number of runs in a permutation. St000354The number of recoils of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000957The number of Bruhat lower covers of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001461The number of topologically connected components of the chord diagram of a permutation. St000702The number of weak deficiencies of a permutation. St000067The inversion number of the alternating sign matrix. St000332The positive inversions of an alternating sign matrix. St000021The number of descents of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000316The number of non-left-to-right-maxima of a permutation. St000336The leg major index of a standard tableau. St000325The width of the tree associated to a permutation. St000443The number of long tunnels of a Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000168The number of internal nodes of an ordered tree. St000238The number of indices that are not small weak excedances. St000331The number of upper interactions of a Dyck path. St000653The last descent of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001175The size of a partition minus the hook length of the base cell. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St000015The number of peaks of a Dyck path. St000056The decomposition (or block) number of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000240The number of indices that are not small excedances. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000991The number of right-to-left minima of a permutation. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001480The number of simple summands of the module J^2/J^3. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000822The Hadwiger number of the graph. St001812The biclique partition number of a graph. St001427The number of descents of a signed permutation. St001331The size of the minimal feedback vertex set. St001741The largest integer such that all patterns of this size are contained in the permutation. St000731The number of double exceedences of a permutation. St000619The number of cyclic descents of a permutation. St000039The number of crossings of a permutation. St000185The weighted size of a partition. St000314The number of left-to-right-maxima of a permutation. St001214The aft of an integer partition. St000145The Dyson rank of a partition. St001668The number of points of the poset minus the width of the poset. St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001863The number of weak excedances of a signed permutation. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000144The pyramid weight of the Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000493The los statistic of a set partition. St000497The lcb statistic of a set partition. St001626The number of maximal proper sublattices of a lattice.