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Your data matches 70 different statistics following compositions of up to 3 maps.
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Matching statistic: St000662
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000662: 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]
=> [1,2] => 0
[1,1,0,0]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3
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.
Matching statistic: St000288
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => 0 => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1 => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 00 => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 10 => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 10 => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 01 => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 11 => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 100 => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => 100 => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 010 => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 110 => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 010 => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => 010 => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => 001 => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => 101 => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 110 => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 101 => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => 011 => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 111 => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1000 => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 1000 => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => 0100 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 1100 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 1000 => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => 0100 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 0100 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 0010 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => 1010 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => 1100 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => 1010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => 0110 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 1110 => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 0100 => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 0100 => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 0010 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => 1010 => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => 0100 => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => 0010 => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 0010 => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 0001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => 1001 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => 1010 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => 1001 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => 0101 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => 1101 => 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => 1100 => 2
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [2,4,7,1,3,5,6,8] => ? => ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => [8,7,3,6,5,4,1,2] => ? => ? = 5
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,8,4,3,5,7,6,1] => [3,8,4,5,7,1,6,2] => ? => ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,2,6,5,4,8,7,1] => [4,1,3,8,2,5,7,6] => ? => ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => [4,3,2,7,6,5,1,8] => ? => ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [3,7,6,5,4,2,1,8] => ? => ? = 5
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => [4,9,8,7,6,5,3,2,1,10] => ? => ? = 7
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => [4,3,8,7,6,5,2,1,9] => ? => ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => [8,3,7,6,5,4,2,1,9] => ? => ? = 6
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => [5,4,9,8,7,6,3,1,2] => ? => ? = 6
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9,8,5,7,6,4,3,2] => [9,4,8,7,6,5,3,1,2] => ? => ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,6,7,5,4,3,2] => [5,10,9,8,7,6,4,3,1,2] => ? => ? = 7
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [7,6,5,4,3,2,1,10,8,9] => ? => ? = 7
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,10,9,8,7,6,5,4,3] => [10,9,8,7,6,5,1,4,2,3] => ? => ? = 7
[]
=> [] => [] => ? => ? = 0
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,10,9] => 100000001 => ? = 2
[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] => [1,3,5,7,10,2,4,6,8,9] => ? => ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,10,9,8,7] => [10,9,1,3,5,8,2,4,6,7] => ? => ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3,10,9,8,7] => [3,10,4,9,1,5,8,2,6,7] => ? => ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,1,8,7,6,5,4,3,10,9] => [7,6,5,4,1,3,10,2,8,9] => ? => ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,1,5,7,10,4,6,8,9] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,6,5,10,9,8,7] => [1,10,2,9,3,5,8,4,6,7] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,8,7,6,5,10,9] => [1,7,2,6,3,5,10,4,8,9] => ? => ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,2,1,10,9,8,7,6,5] => [10,9,1,8,2,7,3,6,4,5] => ? => ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,5,4,3,2,1,10,9,8,7] => [3,2,1,10,4,9,5,8,6,7] => ? => ? = 5
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [10,9,8,5,4,7,6,3,2,1] => [4,10,5,9,8,7,6,3,2,1] => ? => ? = 7
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [10,5,4,3,2,9,8,7,6,1] => [2,10,3,9,4,8,5,7,6,1] => ? => ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0]
=> [10,7,6,5,4,3,2,9,8,1] => [6,5,4,3,2,10,7,9,8,1] => ? => ? = 7
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [10,3,2,9,8,7,6,5,4,1] => [10,9,8,7,2,6,3,5,4,1] => ? => ? = 7
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,3,2,1,10,7,6,9,8,5] => [1,6,10,2,7,9,3,8,4,5] => ? => ? = 3
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [10,3,2,5,4,7,6,9,8,1] => [2,4,6,10,3,5,7,9,8,1] => ? => ? = 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [9,8,7,6,5,4,3,2,1,12,10,11] => ? => ? = 9
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
=> [10,9,6,5,4,3,8,7,2,1] => [5,4,3,10,6,9,8,7,2,1] => ? => ? = 7
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,2,5,4,1,8,7,10,9] => [2,5,3,4,1,7,10,6,8,9] => ? => ? = 3
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 91% ●values known / values provided: 91%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [[1]]
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => [[1,2]]
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => [[1],[2]]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [[1,3],[2]]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [[1],[2],[3]]
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [[1,3,4],[2]]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [[1,2],[3,4]]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [[1,4],[2],[3]]
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [[1,3,4],[2]]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [[1,2,4],[3]]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [[1,3],[2],[4]]
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [[1,4],[2],[3]]
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [[1,3],[2,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [[1,2],[3],[4]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [[1],[2],[3],[4]]
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [[1,2,5],[3,4]]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [[1,3,4,5],[2]]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [[1,2],[3,5],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [[1,5],[2],[3],[4]]
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [[1,2,3],[4,5]]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [[1,3,5],[2],[4]]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [[1,2,4,5],[3]]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [[1,3,4],[2],[5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [[1,3,5],[2],[4]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [[1,2],[3,4],[5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [[1,4],[2],[3],[5]]
=> 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,7,6,8] => [2,4,7,1,3,5,6,8] => ?
=> ? = 1
[1,0,1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [1,8,4,7,6,5,3,2] => [8,7,3,6,5,4,1,2] => ?
=> ? = 5
[1,1,0,1,1,1,0,0,1,0,1,1,0,0,0,0]
=> [2,8,4,3,5,7,6,1] => [3,8,4,5,7,1,6,2] => ?
=> ? = 3
[1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> [3,2,6,5,4,8,7,1] => [4,1,3,8,2,5,7,6] => ?
=> ? = 3
[1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [7,5,4,3,6,2,1,8] => [4,3,2,7,6,5,1,8] => ?
=> ? = 5
[1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,6,4,5,3,2,1,8] => [3,7,6,5,4,2,1,8] => ?
=> ? = 5
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,9,8,7,5,6,4,3,2] => [5,9,8,7,6,4,3,1,2] => [[1,2],[3,9],[4],[5],[6],[7],[8]]
=> ? = 6
[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] => [10,9,8,7,6,5,4,3,2,1,11] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 9
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [9,8,7,5,6,4,3,2,1,10] => [4,9,8,7,6,5,3,2,1,10] => ?
=> ? = 7
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [8,7,5,4,6,3,2,1,9] => [4,3,8,7,6,5,2,1,9] => ?
=> ? = 6
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [8,7,4,6,5,3,2,1,9] => [8,3,7,6,5,4,2,1,9] => ?
=> ? = 6
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,9,8,6,5,7,4,3,2] => [5,4,9,8,7,6,3,1,2] => ?
=> ? = 6
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,9,8,5,7,6,4,3,2] => [9,4,8,7,6,5,3,1,2] => ?
=> ? = 6
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,10,9,8,6,7,5,4,3,2] => [5,10,9,8,7,6,4,3,1,2] => ?
=> ? = 7
[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] => [11,10,9,8,7,6,5,4,3,1,2] => [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 9
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,0,0]
=> [8,7,6,5,4,3,2,1,10,9] => [7,6,5,4,3,2,1,10,8,9] => ?
=> ? = 7
[1,1,0,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,10,9,8,7,6,5,4,3] => [10,9,8,7,6,5,1,4,2,3] => ?
=> ? = 7
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,1] => [2,1,3,4,5,6,7,9,8] => [[1,3,4,5,6,7,8],[2,9]]
=> ? = 2
[1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,8,9,1] => [2,3,1,4,5,6,7,9,8] => [[1,2,4,5,6,7,8],[3,9]]
=> ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,8,9,11,10] => [11,1,2,3,4,5,6,7,8,9,10] => [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,8,9,10,1] => [2,1,3,4,5,6,7,8,10,9] => [[1,3,4,5,6,7,8,9],[2,10]]
=> ? = 2
[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] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9] => [1,3,5,7,10,2,4,6,8,9] => ?
=> ? = 1
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,6,5,10,9,8,7] => [10,9,1,3,5,8,2,4,6,7] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3,10,9,8,7] => [3,10,4,9,1,5,8,2,6,7] => ?
=> ? = 3
[1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> [2,1,8,7,6,5,4,3,10,9] => [7,6,5,4,1,3,10,2,8,9] => ?
=> ? = 5
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,3,2,1,6,5,8,7,10,9] => [3,2,1,5,7,10,4,6,8,9] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [4,3,2,1,6,5,10,9,8,7] => [1,10,2,9,3,5,8,4,6,7] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,8,7,6,5,10,9] => [1,7,2,6,3,5,10,4,8,9] => ?
=> ? = 3
[1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [4,3,2,1,10,9,8,7,6,5] => [10,9,1,8,2,7,3,6,4,5] => ?
=> ? = 5
[1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> [6,5,4,3,2,1,10,9,8,7] => [3,2,1,10,4,9,5,8,6,7] => ?
=> ? = 5
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [2,3,4,5,6,7,1,8,9] => [1,2,3,4,5,7,6,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [2,3,4,5,6,7,8,1,9] => [1,2,3,4,5,6,8,7,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[1,1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,0]
=> [10,9,8,5,4,7,6,3,2,1] => [4,10,5,9,8,7,6,3,2,1] => ?
=> ? = 7
[1,1,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0]
=> [10,5,4,3,2,9,8,7,6,1] => [2,10,3,9,4,8,5,7,6,1] => ?
=> ? = 5
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,0,0]
=> [10,7,6,5,4,3,2,9,8,1] => [6,5,4,3,2,10,7,9,8,1] => ?
=> ? = 7
[1,1,1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [10,3,2,9,8,7,6,5,4,1] => [10,9,8,7,2,6,3,5,4,1] => ?
=> ? = 7
[1,1,1,1,0,0,0,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,3,2,1,10,7,6,9,8,5] => [1,6,10,2,7,9,3,8,4,5] => ?
=> ? = 3
[1,1,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,0,0]
=> [10,3,2,5,4,7,6,9,8,1] => [2,4,6,10,3,5,7,9,8,1] => ?
=> ? = 3
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0]
=> [10,9,8,7,6,5,4,3,2,1,12,11] => [9,8,7,6,5,4,3,2,1,12,10,11] => ?
=> ? = 9
[1,1,1,1,1,1,1,1,0,0,0,0,1,1,0,0,0,0,0,0]
=> [10,9,6,5,4,3,8,7,2,1] => [5,4,3,10,6,9,8,7,2,1] => ?
=> ? = 7
[1,1,1,1,0,0,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [6,3,2,5,4,1,8,7,10,9] => [2,5,3,4,1,7,10,6,8,9] => ?
=> ? = 3
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: St000068
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000068: Posets ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => ([],2)
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([],3)
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => ([(2,3)],4)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([],4)
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
=> 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => ([(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [6,2,3,5,1,4] => ([(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 2 + 1
[1,1,1,0,0,0,1,1,0,1,0,0]
=> [3,2,1,5,6,4] => [3,2,5,6,1,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> ? = 2 + 1
[1,1,1,0,0,1,0,0,1,1,0,0]
=> [3,2,4,1,6,5] => [3,2,4,6,1,5] => ([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> ? = 2 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,4,2,6,5,1] => [3,4,6,2,5,1] => ([(1,5),(2,3),(3,4),(3,5)],6)
=> ? = 3 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [5,6,7,1,2,3,4] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [6,5,7,1,2,3,4] => ([(0,3),(1,6),(2,6),(3,5),(5,4)],7)
=> ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ([(3,4),(4,6),(6,5)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [4,7,1,2,3,5,6] => ([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,4,6,1,2,3,5] => ([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [4,5,6,7,1,2,3] => ([(0,5),(1,6),(4,3),(5,4),(6,2)],7)
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [6,4,5,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,5,3] => [6,7,4,5,1,2,3] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [5,4,7,1,2,3,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(3,4),(4,6)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [5,4,6,7,1,2,3] => ([(0,6),(1,6),(2,3),(3,5),(6,4)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [5,7,4,6,1,2,3] => ([(0,6),(1,4),(2,3),(2,6),(4,5)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,4,7,3] => [5,6,4,7,1,2,3] => ([(0,6),(1,3),(2,4),(3,5),(4,6)],7)
=> ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [6,5,4,7,1,2,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5)],7)
=> ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,5,7,4,3] => [6,5,7,4,1,2,3] => ([(1,6),(2,6),(3,4),(4,5)],7)
=> ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [3,7,1,2,4,5,6] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [3,6,7,1,2,4,5] => ([(0,4),(1,5),(1,6),(4,6),(5,2),(6,3)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,3,6,1,2,4,5] => ([(1,4),(2,3),(2,6),(4,6),(6,5)],7)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [3,5,1,2,4,6,7] => ([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [3,5,6,1,2,4,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,2),(6,5)],7)
=> ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [7,3,5,6,1,2,4] => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [6,3,5,1,2,4,7] => ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,5,4] => [6,7,3,5,1,2,4] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [7,6,3,5,1,2,4] => ([(2,5),(3,4),(3,6),(5,6)],7)
=> ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [3,4,7,1,2,5,6] => ([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [3,4,6,1,2,5,7] => ([(0,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [3,4,6,7,1,2,5] => ([(0,5),(1,3),(3,6),(4,2),(5,4),(5,6)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,5,2] => [6,7,3,4,5,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [5,3,4,7,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [5,3,4,6,7,1,2] => ([(0,6),(1,3),(2,4),(4,6),(6,5)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [5,7,3,4,6,1,2] => ([(0,3),(1,5),(2,4),(2,6),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,4,7,2] => [5,6,3,4,7,1,2] => ([(0,5),(1,4),(2,3),(4,6),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,4,2] => [5,6,7,3,4,1,2] => ([(0,5),(1,4),(2,6),(6,3)],7)
=> ? = 2 + 1
[1,0,1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,3,6,5,4,7,2] => [6,5,3,4,7,1,2] => ([(0,6),(1,6),(2,4),(3,5),(5,6)],7)
=> ? = 3 + 1
[1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,3,6,5,7,4,2] => [6,5,7,3,4,1,2] => ([(0,6),(1,6),(2,5),(3,4)],7)
=> ? = 3 + 1
[1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,4,3,2,5,7,6] => [4,3,7,1,2,5,6] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(3,6),(6,4)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,4,3,2,6,5,7] => [4,3,6,1,2,5,7] => ([(0,4),(0,6),(1,4),(1,6),(2,3),(3,6),(4,5),(6,5)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,1,0,0]
=> [1,4,3,2,6,7,5] => [4,3,6,7,1,2,5] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(3,6),(5,4)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,4,3,2,7,6,5] => [4,7,3,6,1,2,5] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(2,6),(4,6)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0]
=> [1,4,3,5,2,7,6] => [4,3,5,7,1,2,6] => ([(0,5),(1,5),(2,3),(3,6),(5,4),(5,6)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => [4,3,5,6,7,1,2] => ([(0,6),(1,6),(2,3),(4,5),(6,4)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,3,5,7,6,2] => [4,7,3,5,6,1,2] => ([(0,6),(1,3),(2,4),(2,6),(6,5)],7)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,4,3,6,5,2,7] => [4,6,3,5,1,2,7] => ([(0,5),(1,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2 + 1
Description
The number of minimal elements in a poset.
Matching statistic: St000703
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 54% ●values known / values provided: 54%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,1,2] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [2,4,1,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [3,4,2,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [4,2,1,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [2,4,3,5,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [2,3,5,1,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [3,2,4,5,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [3,2,4,1,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [4,2,3,5,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [3,4,5,1,2] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [2,4,1,3,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [4,3,2,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [2,5,3,1,4] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [2,5,1,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [1,4,3,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [4,3,5,2,1] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [2,4,5,3,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [4,5,3,2,1] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [3,5,2,1,4] => 3
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [2,3,4,6,5,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [2,3,5,4,6,7,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [2,3,5,4,6,1,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [2,3,6,4,5,7,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [2,3,4,6,1,5,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => [2,3,6,5,4,7,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [2,3,4,7,5,1,6] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [2,4,3,5,6,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [2,4,3,5,6,1,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => [2,4,3,6,5,7,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => [2,4,6,5,7,1,3] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [2,4,3,5,1,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => [2,5,3,4,6,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [2,5,3,4,6,1,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [2,6,3,4,5,7,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [2,5,4,6,7,1,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => [2,4,5,6,1,3,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [2,6,4,5,3,7,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [2,4,6,7,5,1,3] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => [2,4,5,7,1,3,6] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => [2,3,5,1,4,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [4,3,7,1,2,5,6] => [2,5,4,3,6,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [2,5,4,3,6,1,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [2,6,4,3,5,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => [2,5,3,6,7,1,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [2,3,6,4,1,5,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [2,6,5,4,3,7,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [2,3,7,4,5,1,6] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => [2,3,7,6,1,5,4] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => [2,3,6,1,4,5,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => [2,6,5,3,4,7,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => [2,3,7,5,4,1,6] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => [2,3,7,4,1,5,6] => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => [2,3,7,1,4,5,6] => ? = 4
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => [3,2,4,5,6,1,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => [3,2,4,6,5,7,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => [3,6,4,5,7,1,2] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => [3,2,4,5,1,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => [3,2,5,4,6,7,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => [3,2,5,4,6,1,7] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => [3,2,6,4,5,7,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [6,2,5,1,3,4,7] => [3,5,4,6,1,2,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [5,2,4,7,1,3,6] => [3,4,6,5,2,7,1] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,7,5,6,4] => [5,7,2,6,1,3,4] => [3,6,4,7,5,1,2] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [7,6,2,5,1,3,4] => [3,5,4,7,1,2,6] => ? = 3
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => [3,2,4,1,5,6,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => [4,2,3,5,6,1,7] => ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => [4,2,3,6,5,7,1] => ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [7,2,3,6,1,4,5] => [4,3,6,5,7,1,2] => ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => [4,2,3,5,1,6,7] => ? = 1
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000245
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [3,2,1,4] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [4,2,1,3] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [3,2,4,1] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [3,4,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [2,3,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [1,3,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [5,3,2,1,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [4,2,1,5,3] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [5,4,2,1,3] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [4,3,1,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [5,3,1,4,2] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [4,1,5,3,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [3,1,4,2,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [5,2,1,3,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [4,1,5,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [2,1,4,5,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [5,4,3,1,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [4,3,2,5,1] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [5,3,2,4,1] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [4,2,5,3,1] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [3,2,4,1,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [5,4,2,3,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [4,3,5,2,1] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [5,3,4,2,1] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [4,5,3,2,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [3,4,2,1,5] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [5,2,3,1,4] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [4,5,2,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [2,4,1,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [2,3,1,4,5] => 3
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [7,5,4,3,2,1,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [6,4,3,2,1,7,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [7,6,4,3,2,1,5] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [6,5,3,2,1,7,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [7,5,3,2,1,6,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [6,3,2,1,7,5,4] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [5,3,2,1,6,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [7,4,3,2,1,5,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => [6,3,2,1,7,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [4,3,2,1,6,7,5] => ? = 2
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [7,6,5,3,2,1,4] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [6,5,4,2,1,7,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [7,5,4,2,1,6,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => [6,4,2,1,7,5,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => [5,4,2,1,6,3,7] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [7,6,4,2,1,5,3] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => [6,5,2,1,7,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [7,5,2,1,6,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [6,2,1,7,5,4,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [5,2,1,6,4,3,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => [7,4,2,1,5,3,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [6,2,1,7,4,3,5] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [4,2,1,6,3,7,5] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => [4,2,1,5,3,6,7] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => [7,6,3,2,1,4,5] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [4,3,7,1,2,5,6] => [6,5,2,1,7,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [7,5,2,1,6,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [6,2,1,7,5,3,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => [5,2,1,6,4,7,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [7,3,2,1,5,6,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [6,2,1,7,3,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [3,2,1,6,7,5,4] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => [3,2,1,5,6,4,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => [6,2,1,7,3,4,5] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => [3,2,1,6,7,4,5] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => [3,2,1,5,6,7,4] => ? = 3
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => [7,6,5,4,2,1,3] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => [6,5,4,3,1,7,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => [7,5,4,3,1,6,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => [6,4,3,1,7,5,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => [5,4,3,1,6,2,7] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => [7,6,4,3,1,5,2] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => [6,5,3,1,7,4,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => [7,5,3,1,6,4,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => [6,3,1,7,5,4,2] => ? = 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [7,2,4,6,1,3,5] => [5,3,1,6,4,2,7] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [6,2,5,1,3,4,7] => [7,4,3,1,5,2,6] => ? = 2
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [5,2,4,7,1,3,6] => [6,3,1,7,4,2,5] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,7,5,6,4] => [5,7,2,6,1,3,4] => [4,3,1,6,2,7,5] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [7,6,2,5,1,3,4] => [4,3,1,5,2,6,7] => ? = 3
Description
The number of ascents of a permutation.
Matching statistic: St000632
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000632: Posets ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => [1] => ([],1)
=> 0
[1,0,1,0]
=> [1,2] => [1,2] => ([(0,1)],2)
=> 0
[1,1,0,0]
=> [2,1] => [2,1] => ([],2)
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => ([],3)
=> 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => ([(2,3)],4)
=> 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => ([],4)
=> 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => ([(0,3),(1,2),(1,4),(3,4)],5)
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => ([(2,3),(3,4)],5)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => ([(0,4),(1,2),(1,4),(4,3)],5)
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => ([(0,4),(1,2),(2,3),(2,4)],5)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => ([(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => ([(0,4),(1,2),(1,3)],5)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => ([(3,4)],5)
=> 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => ([(1,3),(1,4),(4,2)],5)
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => ([(1,4),(4,2),(4,3)],5)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => ([(0,3),(0,4),(1,2),(1,4)],5)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => ([(2,3),(2,4)],5)
=> 3
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ([(1,6),(3,5),(4,3),(5,2),(6,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ([(0,5),(1,3),(1,6),(2,6),(4,2),(5,4)],7)
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ([(2,6),(4,5),(5,3),(6,4)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ([(0,5),(1,4),(1,6),(2,6),(5,2),(6,3)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ([(0,3),(0,6),(1,4),(2,6),(3,5),(4,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ([(0,5),(1,4),(3,6),(4,3),(5,2),(5,6)],7)
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ([(1,4),(2,3),(2,6),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => ([(0,6),(1,3),(1,4),(5,2),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ([(0,4),(1,3),(1,6),(4,6),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ([(0,3),(1,4),(1,6),(2,5),(3,6),(4,5),(6,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => ([(1,4),(2,3),(2,6),(4,6),(6,5)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ([(0,4),(1,5),(4,6),(5,2),(5,6),(6,3)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ([(0,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ([(0,5),(1,3),(3,6),(4,2),(4,6),(5,4)],7)
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => ([(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => ([(0,5),(0,6),(1,4),(2,3),(2,5),(4,6)],7)
=> ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => ([(2,5),(3,4),(3,6),(5,6)],7)
=> ? = 3
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => ([(0,4),(1,3),(1,5),(4,6),(5,2),(5,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ([(0,5),(1,3),(1,4),(2,6),(3,6),(4,6),(5,2)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => ([(0,5),(0,6),(1,3),(2,4),(3,6),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => ([(0,6),(1,5),(5,4),(6,2),(6,3)],7)
=> ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => ([(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => ([(0,6),(1,3),(1,4),(1,5),(6,2)],7)
=> ? = 3
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ([(0,6),(1,3),(1,6),(4,2),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(4,2),(6,4)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ([(0,6),(1,4),(1,6),(3,5),(4,2),(4,5),(6,3)],7)
=> ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => ([(1,6),(2,3),(2,6),(4,5),(6,4)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ([(0,6),(1,3),(1,6),(2,5),(3,5),(5,4),(6,2)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ([(0,6),(1,3),(1,6),(2,4),(3,2),(3,5),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [5,2,4,7,1,3,6] => ([(0,5),(0,6),(1,4),(2,3),(2,4),(3,5),(3,6),(4,6)],7)
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,7,5,6,4] => [5,7,2,6,1,3,4] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(6,3)],7)
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => ([(0,6),(1,3),(1,6),(3,5),(4,2),(5,4),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => ([(0,6),(1,4),(4,3),(4,6),(5,2),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => ([(0,6),(1,4),(2,5),(3,5),(4,3),(4,6),(6,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => ([(0,6),(1,3),(3,4),(3,6),(4,2),(4,5),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [7,2,3,6,1,4,5] => ([(1,6),(2,3),(3,5),(3,6),(6,4)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => ([(0,5),(1,4),(3,6),(4,3),(4,5),(5,6),(6,2)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => ([(0,6),(1,4),(4,5),(5,2),(5,6),(6,3)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => ([(0,6),(1,4),(2,5),(3,2),(3,6),(4,3),(6,5)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => ([(0,6),(1,4),(3,5),(4,3),(5,2),(5,6)],7)
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => [7,2,3,4,6,1,5] => ([(1,3),(2,6),(3,5),(5,4),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [6,2,3,5,1,4,7] => ([(0,6),(1,5),(2,3),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [5,2,3,4,7,1,6] => ([(0,6),(1,5),(1,6),(2,3),(3,4),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => [5,7,2,3,6,1,4] => ([(0,6),(1,4),(2,3),(2,5),(4,5),(4,6)],7)
=> ? = 2
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [7,6,2,3,5,1,4] => ([(2,6),(3,4),(4,5),(4,6)],7)
=> ? = 3
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [5,2,4,1,3,6,7] => ([(0,6),(1,5),(2,3),(2,5),(3,6),(5,6),(6,4)],7)
=> ? = 2
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St000470
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 60%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 46% ●values known / values provided: 46%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [2,4,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,3,4,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [4,1,3,2] => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,4,2,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [4,2,1,3] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [1,3,5,2,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,3,4,5,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [2,3,5,1,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [2,4,5,1,3] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [2,5,1,3,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [2,5,1,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,4,5,3,2] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [3,5,2,4,1] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [3,5,4,1,2] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,5,4,3,2] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,3,4,5,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [3,5,1,2,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [1,5,2,4,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [5,1,2,3,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [5,1,2,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,4,3,5,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [3,1,2,5,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [5,3,4,1,2] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [5,1,4,3,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => [3,4,5,7,1,2,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [3,4,6,7,1,2,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [3,4,7,1,2,5,6] => ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [3,4,7,1,2,6,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [3,5,7,1,4,6,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [1,5,7,3,6,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [1,4,6,7,3,5,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [4,5,7,1,3,6,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => [3,4,7,1,6,2,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [1,4,6,7,5,2,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [4,5,7,1,3,2,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [4,5,7,6,1,2,3] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => [1,4,7,6,2,5,3] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => [1,5,7,4,6,3,2] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => [1,5,7,4,2,6,3] => ? = 3 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => [1,4,7,6,5,2,3] => ? = 3 + 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => [2,3,4,6,7,1,5] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => [2,3,5,7,1,4,6] => ? = 1 + 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => [2,3,4,7,1,6,5] => ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => [2,3,5,6,7,1,4] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => [2,4,5,7,1,3,6] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => [2,4,6,7,1,3,5] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => [2,4,7,1,3,5,6] => ? = 1 + 1
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,6,4] => [7,2,4,6,1,3,5] => [2,4,7,1,3,6,5] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,5,4,7] => [6,2,5,1,3,4,7] => [2,3,6,7,1,5,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,5,7,4] => [5,2,4,7,1,3,6] => [2,5,7,1,4,6,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,7,5,6,4] => [5,7,2,6,1,3,4] => [2,5,7,3,6,1,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,6,5,4] => [7,6,2,5,1,3,4] => [2,3,7,1,6,5,4] => ? = 3 + 1
[1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => [2,4,5,6,7,1,3] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => [2,3,4,7,1,5,6] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => [2,3,4,6,1,5,7] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => [2,5,7,1,3,4,6] => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,3,4,2,7,6,5] => [7,2,3,6,1,4,5] => [2,3,7,1,4,6,5] => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => [2,3,5,6,1,4,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => [2,3,7,1,4,5,6] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => [2,3,6,1,4,5,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,3,4,5,7,6,2] => [7,2,3,4,6,1,5] => [2,7,1,3,4,6,5] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,5,2,7] => [6,2,3,5,1,4,7] => [2,3,6,1,5,7,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,5,7,2] => [5,2,3,4,7,1,6] => [5,7,2,3,4,6,1] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,7,5,6,2] => [5,7,2,3,6,1,4] => [5,7,2,3,6,1,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,6,5,2] => [7,6,2,3,5,1,4] => [2,7,1,3,6,5,4] => ? = 3 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,4,2,6,7] => [5,2,4,1,3,6,7] => [2,5,6,7,1,4,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,4,2,7,6] => [4,2,3,7,1,5,6] => [2,4,7,3,5,6,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,4,6,2,7] => [4,2,3,6,1,5,7] => [2,4,6,3,5,7,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,4,6,7,2] => [4,2,3,5,7,1,6] => [4,7,2,3,5,6,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,4,7,6,2] => [3,7,2,4,6,1,5] => [3,7,2,4,6,1,5] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,6,4,5,2,7] => [4,6,2,5,1,3,7] => [4,6,7,2,5,1,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,6,4,5,7,2] => [4,5,2,3,7,1,6] => [4,5,2,3,7,1,6] => ? = 2 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
Matching statistic: St000354
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 60%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 41% ●values known / values provided: 41%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => [1] => [1] => ? = 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [2,3,1] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [2,3,4,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [2,3,1,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,1,4,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [3,4,2,1] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => [4,1,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [3,4,1,5,2] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [3,4,5,2,1] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [3,1,4,5,2] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [3,1,4,2,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [4,1,2,5,3] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [4,2,5,3,1] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [3,4,2,1,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [4,2,1,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => [4,5,1,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [4,5,3,2,1] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,3,4,2,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [1,4,2,5,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [2,4,5,3,1] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [2,3,1,5,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => [3,5,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [3,5,4,2,1] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => [2,3,4,5,6,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => [2,3,4,5,6,1,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => [3,4,5,6,1,7,2] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => [3,4,5,6,7,2,1] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => [2,3,4,5,1,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => [3,4,5,1,6,7,2] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => [3,4,5,1,6,2,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => [4,5,6,1,2,7,3] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => [4,5,6,2,7,3,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => [3,4,5,6,2,1,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => [4,5,6,2,1,7,3] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => [4,5,6,7,1,3,2] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => [4,5,6,7,3,2,1] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => [2,3,4,1,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => [3,4,1,5,6,7,2] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => [3,4,1,5,6,2,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => [4,5,1,6,2,7,3] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => [4,5,2,6,7,3,1] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => [3,4,1,5,2,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => [4,5,1,2,6,7,3] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => [4,5,1,2,6,3,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => [5,6,1,2,3,7,4] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => [5,6,2,3,7,4,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => [4,5,2,6,3,1,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => [5,6,2,3,1,7,4] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => [5,6,3,7,1,4,2] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => [5,6,3,7,4,2,1] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => [3,4,5,2,1,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [4,3,7,1,2,5,6] => [4,5,2,1,6,7,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => [4,5,2,1,6,3,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => [5,6,2,1,3,7,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => [5,6,1,3,7,4,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => [4,5,6,1,3,2,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => [5,6,3,1,2,7,4] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => [5,6,7,1,2,4,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => [5,6,7,2,4,3,1] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => [4,5,6,3,2,1,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => [5,6,3,2,1,7,4] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => [5,6,7,2,1,4,3] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => [5,6,7,1,4,3,2] => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => [5,6,7,4,3,2,1] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => [2,3,1,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => [3,1,4,5,6,7,2] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => [3,1,4,5,6,2,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => [4,1,5,6,2,7,3] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => [4,2,5,6,7,3,1] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => [3,1,4,5,2,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => [4,1,5,2,6,7,3] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => [4,1,5,2,6,3,7] => ? = 1
Description
The number of recoils of a permutation.
A '''recoil''', or '''inverse descent''' of a permutation $\pi$ is a value $i$ such that $i+1$ appears to the left of $i$ in $\pi_1,\pi_2,\dots,\pi_n$.
In other words, this is the number of descents of the inverse permutation. It can be also be described as the number of occurrences of the mesh pattern $([2,1], {(0,1),(1,1),(2,1)})$, i.e., the middle row is shaded.
Matching statistic: St000021
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 60%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 60%
Values
[1,0]
=> [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,4,1,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [3,5,1,4,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => 3
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => [7,6,1,2,3,4,5] => ? = 2
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,6,4] => [7,4,6,1,2,3,5] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,5,4,7] => [6,5,1,2,3,4,7] => ? = 2
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => [5,4,7,1,2,3,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => [5,7,6,1,2,3,4] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,6,5,4] => [7,6,5,1,2,3,4] => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,6,5] => [7,3,6,1,2,4,5] => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,6,3] => [7,3,4,6,1,2,5] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,5,3,7] => [6,3,5,1,2,4,7] => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,5,7,3] => [5,3,4,7,1,2,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,7,5,6,3] => [5,7,3,6,1,2,4] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,6,5,3] => [7,6,3,5,1,2,4] => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,4,3,6,7] => [5,4,1,2,3,6,7] => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,4,3,7,6] => [4,3,7,1,2,5,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,4,6,3,7] => [4,3,6,1,2,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => [4,3,5,7,1,2,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,4,7,6,3] => [3,7,4,6,1,2,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,6,4,5,3,7] => [4,6,5,1,2,3,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,6,4,5,7,3] => [4,5,3,7,1,2,6] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,7,4,5,6,3] => [4,5,7,6,1,2,3] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,7,4,6,5,3] => [7,4,6,5,1,2,3] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,5,4,3,7] => [6,5,4,1,2,3,7] => ? = 3
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,5,4,7,3] => [5,4,3,7,1,2,6] => ? = 3
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,7,5,4,6,3] => [5,4,7,6,1,2,3] => ? = 3
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,7,5,6,4,3] => [4,7,6,5,1,2,3] => ? = 3
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,6,5,4,3] => [7,6,5,4,1,2,3] => ? = 4
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,6,5] => [7,2,6,1,3,4,5] => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ? = 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
The following 60 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St001427The number of descents of a signed permutation. St000013The height of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000652The maximal difference between successive positions of a permutation. St000141The maximum drop size of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000877The depth of the binary word interpreted as a path. St000308The height of the tree associated to a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000317The cycle descent number of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000358The number of occurrences of the pattern 31-2. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000442The maximal area to the right of an up step of a Dyck path. St000991The number of right-to-left minima of a permutation. St000094The depth of an ordered tree. St001589The nesting number of a perfect matching. St001590The crossing number of a perfect matching. St000732The number of double deficiencies of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000702The number of weak deficiencies of a permutation. St000914The sum of the values of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St001720The minimal length of a chain of small intervals in a lattice. St001890The maximum magnitude of the Möbius function of a poset. 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. St000983The length of the longest alternating subword. St000381The largest part of an integer composition. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St000097The order of the largest clique of the graph. St001712The number of natural descents of a standard Young tableau. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000942The number of critical left to right maxima of the parking functions. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001624The breadth of a lattice.
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