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Your data matches 27 different statistics following compositions of up to 3 maps.
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Matching statistic: St000662
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Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ 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]
=> [3,1,2] => 1
[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]
=> [1,4,2,3] => 1
[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]
=> [2,4,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 1
[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]
=> [1,2,5,3,4] => 1
[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]
=> [1,3,5,2,4] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 1
[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]
=> [2,1,5,3,4] => 1
[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]
=> [2,3,5,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 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.
Matching statistic: St000288
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(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00131: Permutations —descent bottoms⟶ 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
Mp00131: Permutations —descent bottoms⟶ 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]
=> [3,1,2] => [2,3,1] => 10 => 1
[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] => 100 => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,4,1,2] => 100 => 1
[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]
=> [2,4,1,3] => [1,3,4,2] => 010 => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 100 => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 110 => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => 110 => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 100 => 1
[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] => 1000 => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,5,1,2,3] => 1000 => 1
[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] => 1000 => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 1000 => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 1000 => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => 1000 => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => 1000 => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => 1010 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => 1100 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => 1000 => 1
[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] => 0100 => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => 0100 => 1
[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]
=> [2,3,5,1,4] => [1,2,4,5,3] => 0010 => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => 0100 => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 1100 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 1010 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 0100 => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => 1000 => 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => [5,6,1,2,3,4,7,8] => ? => ? = 1
[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
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,4,5,7,8] => [2,5,6,1,3,4,7,8] => ? => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,4,7,5,8] => [7,2,5,1,3,4,6,8] => ? => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,4,5,6,8] => [2,5,6,7,1,3,4,8] => ? => ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6,8] => [4,2,7,1,3,5,6,8] => ? => ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,8,6,7] => [4,2,7,8,1,3,5,6] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,4,2,6,3,5,7,8] => [5,2,6,1,3,4,7,8] => ? => ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,2,6,3,7,5,8] => [2,7,5,1,3,4,6,8] => ? => ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,6,3,4,7,8] => [5,6,3,1,2,4,7,8] => ? => ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,6,2,3,4,7,5,8] => [7,3,4,5,1,2,6,8] => ? => ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [4,5,6,2,7,8,1,3] => ? => ? = 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,1,3,4,6] => [5,2,6,7,1,3,8,4] => ? => ? = 3
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [4,5,7,8,1,2,3,6] => [1,5,2,6,7,3,8,4] => ? => ? = 3
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [4,6,7,8,1,2,3,5] => [1,5,6,2,7,3,8,4] => ? => ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [6,7,1,2,3,4,8,5] => [8,3,4,5,1,6,2,7] => ? => ? = 3
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [4,5,6,7,8,2,9,1,3] => ? => ? = 2
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [3,4,5,6,7,8,1,9,2,10] => ? => ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [4,5,6,7,8,2,1,3,9] => ? => ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [3,4,5,6,1,7,8,2,9] => ? => ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [5,6,7,8,9,3,1,2,4] => ? => ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [4,5,6,7,2,8,9,1,3] => ? => ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [4,5,6,7,8,9,2,10,1,3] => ? => ? = 2
[]
=> [] => [] => => ? = 0
[1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,8,7,10,9] => [1,3,5,7,10,2,4,6,8,9] => ? => ? = 1
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,8,10,1,2,3,4,9] => [1,6,2,7,3,8,4,9,10,5] => ? => ? = 4
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [2,7,8,9,10,1,3,4,5,6] => [6,7,2,8,3,9,1,4,10,5] => ? => ? = 4
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,7,8,9,10,1,2,3,5,6] => [6,7,1,2,8,3,9,4,10,5] => 101110000 => ? = 4
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
=> [5,6,7,9,10,1,2,3,4,8] => [1,6,2,7,3,8,9,4,10,5] => ? => ? = 4
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> [5,6,8,9,10,1,2,3,4,7] => [1,6,2,7,8,3,9,4,10,5] => ? => ? = 4
[1,1,1,1,1,0,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [5,7,8,9,10,1,2,3,4,6] => [1,6,7,2,8,3,9,4,10,5] => ? => ? = 4
[1,1,1,0,1,1,0,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> [3,5,7,9,10,1,2,4,6,8] => [2,6,7,1,3,8,9,4,10,5] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [2,4,6,9,10,1,3,5,7,8] => [6,2,3,7,8,9,1,4,10,5] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0]
=> [2,4,6,8,10,1,3,5,7,9] => [6,2,3,7,8,1,4,9,10,5] => ? => ? = 3
[1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,0,0,0,0]
=> [2,4,6,8,10,12,1,3,5,7,9,11] => [3,7,8,2,4,9,10,1,5,11,12,6] => ? => ? = 3
[1,1,1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [4,5,6,9,10,1,2,3,7,8] => [1,2,6,3,7,8,9,4,10,5] => ? => ? = 3
[1,1,1,1,0,1,0,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> [4,5,8,9,10,1,2,3,6,7] => [1,2,6,7,8,3,9,4,10,5] => ? => ? = 3
[1,1,1,1,0,1,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [4,6,7,8,10,1,2,3,5,9] => [6,1,2,7,3,8,4,9,10,5] => ? => ? = 4
[1,1,1,1,1,1,0,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0,0]
=> [6,7,8,9,10,12,1,2,3,4,5,11] => [1,7,2,8,3,9,4,10,5,11,12,6] => ? => ? = 5
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 90%●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: 90% ●values known / values provided: 90%●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]
=> [3,1,2] => [2,3,1] => [[1,2],[3]]
=> 1
[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]
=> [1,4,2,3] => [3,4,1,2] => [[1,2],[3,4]]
=> 1
[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]
=> [2,4,1,3] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [[1,4],[2],[3]]
=> 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [[1,3],[2,4]]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [[1,2,3],[4]]
=> 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]]
=> 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]
=> [1,2,5,3,4] => [4,5,1,2,3] => [[1,2,5],[3,4]]
=> 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]]
=> 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]
=> [1,3,5,2,4] => [2,4,5,1,3] => [[1,2,3],[4,5]]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [[1,2,5],[3,4]]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [[1,2,3],[4,5]]
=> 1
[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]
=> [2,1,5,3,4] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[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]
=> [2,3,5,1,4] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [[1,2,3,5],[4]]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [[1,3,5],[2],[4]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [[1,3,4],[2,5]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,3,6,4,5,7,8] => [5,6,1,2,3,4,7,8] => ?
=> ? = 1
[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
[1,0,1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,3,2,6,4,5,7,8] => [2,5,6,1,3,4,7,8] => ?
=> ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0,1,0]
=> [1,3,2,6,4,7,5,8] => [7,2,5,1,3,4,6,8] => ?
=> ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,3,2,7,4,5,6,8] => [2,5,6,7,1,3,4,8] => ?
=> ? = 1
[1,0,1,1,1,0,0,1,0,0,1,1,0,0,1,0]
=> [1,4,2,5,3,7,6,8] => [4,2,7,1,3,5,6,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> [1,4,2,5,3,8,6,7] => [4,2,7,8,1,3,5,6] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0,1,0,1,0]
=> [1,4,2,6,3,5,7,8] => [5,2,6,1,3,4,7,8] => ?
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,1,0,0,1,0]
=> [1,4,2,6,3,7,5,8] => [2,7,5,1,3,4,6,8] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,5,2,6,3,4,7,8] => [5,6,3,1,2,4,7,8] => ?
=> ? = 2
[1,0,1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,8,3,4,6,7] => [5,6,2,7,8,1,3,4] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,6,2,3,4,7,5,8] => [7,3,4,5,1,2,6,8] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,7,3,4,5,8] => [5,6,7,3,1,2,4,8] => ?
=> ? = 2
[1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,6,8,2,3,4,5,7] => [4,5,6,2,7,8,1,3] => ?
=> ? = 2
[1,1,0,1,1,1,0,1,1,0,1,0,0,0,0,0]
=> [2,5,7,8,1,3,4,6] => [5,2,6,7,1,3,8,4] => ?
=> ? = 3
[1,1,1,1,0,1,0,1,1,0,1,0,0,0,0,0]
=> [4,5,7,8,1,2,3,6] => [1,5,2,6,7,3,8,4] => ?
=> ? = 3
[1,1,1,1,0,1,1,0,1,0,1,0,0,0,0,0]
=> [4,6,7,8,1,2,3,5] => [1,5,6,2,7,3,8,4] => ?
=> ? = 3
[1,1,1,1,1,1,0,1,0,0,0,0,0,1,0,0]
=> [6,7,1,2,3,4,8,5] => [8,3,4,5,1,6,2,7] => ?
=> ? = 3
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,7,9] => [2,3,4,5,6,7,8,1,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,8,1,2,3,4,5,6,9] => [3,4,5,6,7,1,8,2,9] => ?
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,8,9,2,3,4,5,6,7] => [4,5,6,7,8,2,9,1,3] => ?
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1,2,3,4,5,6,7,8,9,11] => [2,3,4,5,6,7,8,9,10,1,11] => [[1,2,3,4,5,6,7,8,9,11],[10]]
=> ? = 1
[1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,9,1,2,3,4,5,6,7,10] => [3,4,5,6,7,8,1,9,2,10] => ?
=> ? = 2
[1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,8,2,3,4,5,6,9] => [4,5,6,7,8,2,1,3,9] => ?
=> ? = 2
[1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,8,1,2,3,4,5,7,9] => [3,4,5,6,1,7,8,2,9] => ?
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,8,2,9,3,4,5,6,7] => [5,6,7,8,9,3,1,2,4] => ?
=> ? = 2
[1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,7,9,2,3,4,5,6,8] => [4,5,6,7,2,8,9,1,3] => ?
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,9,10,2,3,4,5,6,7,8] => [4,5,6,7,8,9,2,10,1,3] => ?
=> ? = 2
[1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,11,2,3,4,5,6,7,8,9,10] => [3,4,5,6,7,8,9,10,11,1,2] => [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 1
[1,1,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,8,1,3,4,5,6,7,9] => [1,3,4,5,6,7,8,2,9] => [[1,2,3,4,5,6,7,9],[8]]
=> ? = 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> [7,1,2,3,4,5,6,9,8] => [9,2,3,4,5,6,1,7,8] => [[1,3,4,5,6,8,9],[2],[7]]
=> ? = 2
[1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [1,8,2,3,4,5,6,9,7] => [9,3,4,5,6,7,1,2,8] => [[1,3,4,5,6,9],[2,8],[7]]
=> ? = 2
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [7,1,2,3,4,5,6,8,9] => [2,3,4,5,6,7,1,8,9] => [[1,2,3,4,5,6,8,9],[7]]
=> ? = 1
[1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [6,7,8,9,1,2,3,4,5] => [5,6,1,7,2,8,3,9,4] => ?
=> ? = 4
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [8,1,2,3,4,5,6,9,7] => [9,2,3,4,5,6,7,1,8] => [[1,3,4,5,6,7,9],[2],[8]]
=> ? = 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
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [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
[1,1,0,1,0,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [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
[1,1,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0,0,0]
=> [5,6,7,8,10,1,2,3,4,9] => [1,6,2,7,3,8,4,9,10,5] => ?
=> ? = 4
[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
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [9,1,2,3,4,5,6,7,10,8] => [10,2,3,4,5,6,7,8,1,9] => [[1,3,4,5,6,7,8,10],[2],[9]]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [2,3,4,5,6,7,8,9,10,11,1] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [2,7,8,9,10,1,3,4,5,6] => [6,7,2,8,3,9,1,4,10,5] => ?
=> ? = 4
[1,1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,0]
=> [8,9,10,1,2,3,4,5,6,7] => [4,5,6,7,8,1,9,2,10,3] => ?
=> ? = 3
[1,1,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0,0]
=> [7,8,9,10,1,2,3,4,5,6] => [5,6,7,1,8,2,9,3,10,4] => ?
=> ? = 4
[1,1,1,1,0,1,1,1,0,1,0,1,0,1,0,0,0,0,0,0]
=> [4,7,8,9,10,1,2,3,5,6] => [6,7,1,2,8,3,9,4,10,5] => ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,0,1,1,0,1,0,0,0,0,0,0]
=> [5,6,7,9,10,1,2,3,4,8] => [1,6,2,7,3,8,9,4,10,5] => ?
=> ? = 4
[1,1,1,1,1,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0]
=> [5,6,8,9,10,1,2,3,4,7] => [1,6,2,7,8,3,9,4,10,5] => ?
=> ? = 4
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: St000703
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 44% ●values known / values provided: 44%●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: 44% ●values known / values provided: 44%●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]
=> [3,1,2] => [2,3,1] => [3,2,1] => 1
[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]
=> [1,4,2,3] => [3,4,1,2] => [2,4,3,1] => 1
[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]
=> [2,4,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [3,1,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 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] => 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]
=> [1,2,5,3,4] => [4,5,1,2,3] => [2,3,5,4,1] => 1
[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]
=> [1,3,5,2,4] => [2,4,5,1,3] => [3,2,5,4,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [2,4,3,1,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [2,4,1,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [3,4,5,2,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [2,5,3,4,1] => 1
[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]
=> [2,1,5,3,4] => [1,4,5,2,3] => [1,3,5,4,2] => 1
[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]
=> [2,3,5,1,4] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [3,4,2,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [2,4,5,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => 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] => [2,3,4,6,5,7,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [2,3,4,5,7,6,1] => ? = 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
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [2,3,5,4,7,6,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [2,3,4,6,5,1,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => [2,3,5,6,7,4,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [2,3,4,7,5,6,1] => ? = 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] => [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]
=> [1,2,4,3,7,5,6] => [3,6,7,1,2,4,5] => [2,4,3,5,7,6,1] => ? = 1
[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]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => [2,5,3,4,7,6,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => [2,4,3,6,5,1,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => [2,4,5,6,1,7,3] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => [2,5,4,6,7,3,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => [2,4,3,7,5,6,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [2,3,5,4,1,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,4,1,2,3,5,6] => [2,3,5,1,6,7,4] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => [2,3,5,1,6,4,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [5,3,7,1,2,4,6] => [2,4,5,6,3,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [6,3,7,1,2,4,5] => [2,4,6,5,7,3,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => [2,4,5,6,3,1,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [3,7,5,1,2,4,6] => [2,4,3,6,1,7,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [2,5,3,6,7,4,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => [2,4,5,7,3,6,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [2,3,6,4,5,1,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,4,5,1,2,3,6] => [2,3,6,5,1,7,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => [2,3,5,1,7,6,4] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => [2,4,6,7,5,3,1] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => [2,3,7,4,5,6,1] => ? = 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] => [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]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => [3,2,4,5,7,6,1] => ? = 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] => [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
[1,0,1,1,0,0,1,1,0,1,1,0,0,0]
=> [1,3,2,5,7,4,6] => [2,4,6,7,1,3,5] => [3,2,5,4,7,6,1] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [2,5,6,1,3,4,7] => [3,2,4,6,5,1,7] => ? = 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [7,2,5,1,3,4,6] => [3,5,4,6,1,7,2] => ? = 2
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => [6,2,4,7,1,3,5] => [3,4,5,6,7,2,1] => ? = 2
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => [2,5,6,7,1,3,4] => [3,2,4,7,5,6,1] => ? = 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] => [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
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)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 83%
Mp00069: Permutations —complement⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 83%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [2,1] => [2,1] => 0
[1,1,0,0]
=> [2,1] => [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => [3,2,1] => 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] => [2,3,1] => [1,3,2] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => [4,3,2,1] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => [3,4,2,1] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => [2,4,3,1] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => [3,2,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [4,1,3,2] => [4,2,3,1] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => [1,4,3,2] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => [3,1,4,2] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => [2,1,4,3] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [3,1,4,2] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,4,3,1] => [4,1,3,2] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [2,4,1,3] => [1,3,4,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,1,4,3] => [1,4,2,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,4,3,2] => [4,3,1,2] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,4,3,1,2] => [4,5,3,2,1] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [5,4,2,3,1] => [3,5,4,2,1] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [5,4,2,1,3] => [4,3,5,2,1] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [5,4,1,3,2] => [5,3,4,2,1] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [5,3,4,2,1] => [2,5,4,3,1] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [5,3,4,1,2] => [4,2,5,3,1] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [5,3,2,4,1] => [3,2,5,4,1] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [5,3,2,1,4] => [4,3,2,5,1] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [5,3,1,4,2] => [5,3,2,4,1] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [5,2,4,3,1] => [5,2,4,3,1] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,2,4,1,3] => [2,4,5,3,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [5,2,1,4,3] => [2,5,3,4,1] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [5,1,4,3,2] => [5,4,2,3,1] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [4,5,3,2,1] => [1,5,4,3,2] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [4,5,3,1,2] => [4,1,5,3,2] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [4,5,2,3,1] => [3,1,5,4,2] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [4,5,2,1,3] => [4,3,1,5,2] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [4,5,1,3,2] => [5,3,1,4,2] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [4,3,5,2,1] => [2,1,5,4,3] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,3,5,1,2] => [4,2,1,5,3] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,5,1] => [3,2,1,5,4] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,3,1,5,2] => [5,3,2,1,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,5,3,1] => [5,2,1,4,3] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,4,1,5,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,2,1,5,3] => [2,5,3,1,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [4,1,5,3,2] => [5,4,2,1,3] => 1
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => [7,6,5,4,3,1,2] => [6,7,5,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,4,6,5,7] => [7,6,5,4,2,3,1] => [5,7,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => [7,6,5,4,2,1,3] => [6,5,7,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [7,6,5,4,1,3,2] => [7,5,6,4,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,3,5,4,6,7] => [7,6,5,3,4,2,1] => [4,7,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,3,5,4,7,6] => [7,6,5,3,4,1,2] => [6,4,7,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,3,5,6,4,7] => [7,6,5,3,2,4,1] => [5,4,7,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => [7,6,5,3,2,1,4] => [6,5,4,7,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [7,6,5,3,1,4,2] => [7,5,4,6,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [7,6,5,2,4,3,1] => [7,4,6,5,3,2,1] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [7,6,5,2,4,1,3] => [4,6,7,5,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [7,6,5,2,1,4,3] => [4,7,5,6,3,2,1] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [7,6,5,1,4,3,2] => [7,6,4,5,3,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,2,4,3,5,6,7] => [7,6,4,5,3,2,1] => [3,7,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,2,4,3,5,7,6] => [7,6,4,5,3,1,2] => [6,3,7,5,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,2,4,3,6,5,7] => [7,6,4,5,2,3,1] => [5,3,7,6,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,2,4,3,6,7,5] => [7,6,4,5,2,1,3] => [6,5,3,7,4,2,1] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [7,6,4,5,1,3,2] => [7,5,3,6,4,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,2,4,5,3,6,7] => [7,6,4,3,5,2,1] => [4,3,7,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,2,4,5,3,7,6] => [7,6,4,3,5,1,2] => [6,4,3,7,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,4,5,6,3,7] => [7,6,4,3,2,5,1] => [5,4,3,7,6,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => [7,6,4,3,2,1,5] => [6,5,4,3,7,2,1] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [7,6,4,3,1,5,2] => [7,5,4,3,6,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [7,6,4,2,5,3,1] => [7,4,3,6,5,2,1] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,6,4,2,5,1,3] => [4,6,3,7,5,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [7,6,4,2,1,5,3] => [4,7,5,3,6,2,1] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [7,6,4,1,5,3,2] => [7,6,4,3,5,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [7,6,3,5,4,2,1] => [7,3,6,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,6,3,5,4,1,2] => [3,6,7,5,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [7,6,3,5,2,4,1] => [3,5,7,6,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [7,6,3,5,2,1,4] => [3,6,5,7,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [7,6,3,5,1,4,2] => [3,7,5,6,4,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [7,6,3,2,5,4,1] => [3,7,4,6,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [7,6,3,2,5,1,4] => [4,3,6,7,5,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [7,6,3,2,1,5,4] => [4,3,7,5,6,2,1] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [7,6,3,1,5,4,2] => [3,7,6,4,5,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [7,6,2,5,4,3,1] => [7,6,3,5,4,2,1] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,6,2,5,4,1,3] => [6,7,3,5,4,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [7,6,2,5,1,4,3] => [7,3,5,6,4,2,1] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [7,6,2,1,5,4,3] => [7,3,6,4,5,2,1] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [7,6,1,5,4,3,2] => [7,6,5,3,4,2,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6,7] => [7,5,6,4,3,2,1] => [2,7,6,5,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,2,4,5,7,6] => [7,5,6,4,3,1,2] => [6,2,7,5,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,2,4,6,5,7] => [7,5,6,4,2,3,1] => [5,2,7,6,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,3,2,4,6,7,5] => [7,5,6,4,2,1,3] => [6,5,2,7,4,3,1] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [7,5,6,4,1,3,2] => [7,5,2,6,4,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,3,2,5,4,6,7] => [7,5,6,3,4,2,1] => [4,2,7,6,5,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4,7,6] => [7,5,6,3,4,1,2] => [6,4,2,7,5,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,3,2,5,6,4,7] => [7,5,6,3,2,4,1] => [5,4,2,7,6,3,1] => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,3,2,5,6,7,4] => [7,5,6,3,2,1,4] => [6,5,4,2,7,3,1] => ? = 1
Description
The number of ascents of a permutation.
Matching statistic: St000470
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 67%
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]
=> [3,1,2] => [2,3,1] => [2,1,3] => 2 = 1 + 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]
=> [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 2 = 1 + 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]
=> [2,4,1,3] => [1,3,4,2] => [3,1,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [2,4,3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [2,1,3,4] => 2 = 1 + 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]
=> [1,2,5,3,4] => [4,5,1,2,3] => [1,4,5,2,3] => 2 = 1 + 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]
=> [1,3,5,2,4] => [2,4,5,1,3] => [2,4,1,3,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [3,4,5,1,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [1,3,5,4,2] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [4,5,2,3,1] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [3,4,1,2,5] => 2 = 1 + 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]
=> [2,1,5,3,4] => [1,4,5,2,3] => [4,5,1,2,3] => 2 = 1 + 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]
=> [2,3,5,1,4] => [1,2,4,5,3] => [4,1,2,3,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [1,5,3,4,2] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [4,1,2,5,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [3,1,2,4,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [1,4,6,7,2,3,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [1,5,6,7,2,3,4] => ? = 1 + 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]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => [3,4,6,1,2,5,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => [3,5,6,7,1,2,4] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => [3,6,7,1,4,5,2] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => [3,5,6,1,2,4,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => [1,5,6,7,3,4,2] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [3,6,7,1,4,2,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => [5,6,7,1,3,4,2] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [4,5,6,7,1,2,3] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => [5,6,7,1,3,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => [4,5,6,1,2,3,7] => ? = 1 + 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]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => [2,3,6,7,1,4,5] => ? = 1 + 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]
=> [1,3,2,5,7,4,6] => [2,4,6,7,1,3,5] => [2,4,6,1,3,5,7] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,3,2,6,4,5,7] => [2,5,6,1,3,4,7] => [2,5,6,7,1,3,4] => ? = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,1,0,0]
=> [1,3,2,6,4,7,5] => [7,2,5,1,3,4,6] => [2,3,5,7,1,6,4] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,1,0,0,0]
=> [1,3,2,6,7,4,5] => [6,2,4,7,1,3,5] => [2,6,7,1,4,5,3] => ? = 2 + 1
[1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,7,4,5,6] => [2,5,6,7,1,3,4] => [2,5,6,1,3,4,7] => ? = 1 + 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]
=> [1,3,4,2,7,5,6] => [2,3,6,7,1,4,5] => [2,6,7,1,3,4,5] => ? = 1 + 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]
=> [1,3,4,5,7,2,6] => [2,3,4,6,7,1,5] => [2,6,1,3,4,5,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0,1,0]
=> [1,3,4,6,2,5,7] => [2,3,5,6,1,4,7] => [2,3,5,1,4,6,7] => ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,3,4,6,2,7,5] => [7,2,3,5,1,4,6] => [2,3,7,1,5,6,4] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,3,4,6,7,2,5] => [6,2,3,4,7,1,5] => [6,7,2,3,4,5,1] => ? = 2 + 1
[1,0,1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,3,4,7,2,5,6] => [2,3,5,6,7,1,4] => [2,5,1,3,4,6,7] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,3,5,2,4,6,7] => [2,4,5,1,3,6,7] => [2,4,5,6,1,3,7] => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,1,0,0]
=> [1,3,5,2,4,7,6] => [7,2,4,1,3,5,6] => [2,4,5,7,1,6,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> [1,3,5,2,6,4,7] => [6,2,4,1,3,5,7] => [2,4,6,7,1,5,3] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,3,5,2,6,7,4] => [5,2,3,7,1,4,6] => [2,5,7,3,4,6,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,3,5,2,7,4,6] => [6,2,3,7,1,4,5] => [2,6,7,3,4,5,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,3,5,6,2,4,7] => [5,2,3,6,1,4,7] => [2,5,6,3,4,7,1] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,3,5,6,2,7,4] => [3,7,2,5,1,4,6] => [3,5,7,2,6,1,4] => ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,3,5,6,7,2,4] => [3,6,2,4,7,1,5] => [3,6,2,4,7,1,5] => ? = 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
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
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]
=> [3,1,2] => [2,3,1] => [3,1,2] => 1
[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]
=> [1,4,2,3] => [3,4,1,2] => [3,4,1,2] => 1
[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]
=> [2,4,1,3] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [2,4,1,3] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,1,2,3] => 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] => 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]
=> [1,2,5,3,4] => [4,5,1,2,3] => [3,4,5,1,2] => 1
[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]
=> [1,3,5,2,4] => [2,4,5,1,3] => [4,1,5,2,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [3,4,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [3,4,2,5,1] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [4,2,5,1,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [4,5,1,2,3] => 1
[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]
=> [2,1,5,3,4] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[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]
=> [2,3,5,1,4] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [2,3,5,1,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1
[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]
=> [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [3,4,5,6,7,1,2] => ? = 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] => [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]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [4,5,6,1,7,2,3] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [3,4,5,6,1,2,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [7,5,1,2,3,4,6] => [3,4,5,6,2,7,1] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => [4,5,6,2,7,1,3] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [4,5,6,7,1,2,3] => ? = 1
[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]
=> [1,2,4,3,7,5,6] => [3,6,7,1,2,4,5] => [4,5,1,6,7,2,3] => ? = 1
[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]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => [5,6,1,2,7,3,4] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => [4,5,1,6,2,3,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => [4,5,2,6,3,7,1] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => [5,6,2,3,7,1,4] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => [5,6,1,7,2,3,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [3,4,5,1,2,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,4,1,2,3,5,6] => [3,4,5,2,6,7,1] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => [3,4,5,2,6,1,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [5,3,7,1,2,4,6] => [4,5,2,6,1,7,3] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [6,3,7,1,2,4,5] => [4,5,2,6,7,1,3] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => [4,5,2,6,1,3,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [3,7,5,1,2,4,6] => [4,5,1,6,3,7,2] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [5,6,1,3,7,2,4] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => [5,6,2,7,1,3,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [4,5,6,1,2,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,4,5,1,2,3,6] => [4,5,6,2,3,7,1] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => [4,5,6,3,7,1,2] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => [5,6,3,7,1,2,4] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => [5,6,7,1,2,3,4] => ? = 1
[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]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => [4,1,5,6,7,2,3] => ? = 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] => [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: St001269
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St001269: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 67%
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] => [3,1,2] => 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]
=> [3,1,2] => [2,3,1] => [3,2,1] => 1
[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] => [4,1,2,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,4,1,2] => [4,1,3,2] => 1
[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] => [1,4,2,3] => 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]
=> [2,4,1,3] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => [3,4,1,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 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] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [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] => [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] => [5,1,3,2,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,5,1,2,3] => [5,1,2,4,3] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [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] => [5,2,1,3,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [4,2,1,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [5,2,3,1,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => [5,2,1,4,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => [4,1,3,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => [3,1,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => [5,4,1,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => [5,1,3,4,2] => 1
[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] => [1,5,2,3,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [1,5,3,2,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => [1,5,2,4,3] => 1
[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] => [1,2,5,3,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,2,4,3,5] => 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]
=> [2,3,5,1,4] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => [5,3,1,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => [4,1,5,2,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[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] => [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] => [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] => [7,1,2,3,5,4,6] => ? = 1
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => [7,1,2,3,4,6,5] => ? = 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] => [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] => [7,1,2,4,3,5,6] => ? = 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] => [6,1,2,4,3,5,7] => ? = 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] => [7,1,2,4,5,3,6] => ? = 1
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => [7,1,2,4,3,6,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => [6,1,2,3,5,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [7,5,1,2,3,4,6] => [5,1,2,3,7,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => [7,1,2,6,3,4,5] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => [7,1,2,3,5,6,4] => ? = 1
[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] => [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] => [7,1,3,2,4,5,6] => ? = 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] => [6,1,3,2,4,5,7] => ? = 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] => [7,1,3,2,5,4,6] => ? = 1
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [3,6,7,1,2,4,5] => [7,1,3,2,4,6,5] => ? = 1
[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] => [5,1,3,2,4,6,7] => ? = 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] => [7,1,3,4,2,5,6] => ? = 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] => [6,1,3,4,2,5,7] => ? = 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] => [7,1,3,4,5,2,6] => ? = 1
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => [7,1,3,4,2,6,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => [6,1,3,2,5,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => [5,1,7,2,3,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => [7,1,6,3,2,4,5] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => [7,1,3,2,5,6,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => [5,1,2,4,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,4,1,2,3,5,6] => [4,1,2,7,3,5,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => [4,1,2,6,3,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [5,3,7,1,2,4,6] => [7,1,5,2,3,4,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [6,3,7,1,2,4,5] => [7,1,6,2,4,3,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => [6,1,5,2,3,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [3,7,5,1,2,4,6] => [5,1,3,2,7,4,6] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => [7,1,3,6,2,4,5] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => [7,1,5,2,3,6,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => [6,1,2,4,5,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,4,5,1,2,3,6] => [5,1,2,7,4,3,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => [4,1,2,7,3,6,5] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => [7,1,6,2,5,3,4] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => [7,1,2,4,5,6,3] => ? = 1
[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] => [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] => [7,2,1,3,4,5,6] => ? = 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] => [6,2,1,3,4,5,7] => ? = 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] => [7,2,1,3,5,4,6] => ? = 1
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => [7,2,1,3,4,6,5] => ? = 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] => [5,2,1,3,4,6,7] => ? = 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] => [7,2,1,4,3,5,6] => ? = 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] => [6,2,1,4,3,5,7] => ? = 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] => [7,2,1,4,5,3,6] => ? = 1
Description
The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation.
Matching statistic: St000021
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
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]
=> [3,1,2] => [2,3,1] => 1
[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]
=> [1,4,2,3] => [3,4,1,2] => 1
[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]
=> [2,4,1,3] => [1,3,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[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]
=> [1,2,5,3,4] => [4,5,1,2,3] => 1
[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]
=> [1,3,5,2,4] => [2,4,5,1,3] => 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => 1
[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]
=> [2,1,5,3,4] => [1,4,5,2,3] => 1
[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]
=> [2,3,5,1,4] => [1,2,4,5,3] => 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 1
[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]
=> [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 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] => ? = 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]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [7,5,1,2,3,4,6] => ? = 2
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 2
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1
[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]
=> [1,2,4,3,7,5,6] => [3,6,7,1,2,4,5] => ? = 1
[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]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => ? = 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => ? = 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,4,1,2,3,5,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [5,3,7,1,2,4,6] => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [6,3,7,1,2,4,5] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [3,7,5,1,2,4,6] => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => ? = 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,4,5,1,2,3,6] => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1
[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]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => ? = 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] => ? = 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].
Matching statistic: St000325
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,2] => [1,2] => 1 = 0 + 1
[1,1,0,0]
=> [2,1] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3,1,2] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [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] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,2,1,3] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [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] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [4,5,1,2,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [2,4,5,1,3] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,4,1,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [5,3,1,2,4] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [4,2,5,1,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [3,4,5,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [1,4,5,2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [1,3,4,2,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,1,3,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[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
[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
[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
[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
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,5,6] => [6,7,1,2,3,4,5] => ? = 1 + 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] => ? = 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] => ? = 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] => ? = 1 + 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
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,3,5,7,4,6] => [4,6,7,1,2,3,5] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,3,6,4,5,7] => [5,6,1,2,3,4,7] => ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,4,7,5] => [7,5,1,2,3,4,6] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,6,7,4,5] => [6,4,7,1,2,3,5] => ? = 2 + 1
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,3,7,4,5,6] => [5,6,7,1,2,3,4] => ? = 1 + 1
[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
[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
[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
[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
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,2,4,3,7,5,6] => [3,6,7,1,2,4,5] => ? = 1 + 1
[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
[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
[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
[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
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,4,5,7,3,6] => [3,4,6,7,1,2,5] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,2,4,6,3,5,7] => [3,5,6,1,2,4,7] => ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,2,4,6,3,7,5] => [7,3,5,1,2,4,6] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,4,6,7,3,5] => [6,3,4,7,1,2,5] => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,4,7,3,5,6] => [3,5,6,7,1,2,4] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,2,5,3,4,6,7] => [4,5,1,2,3,6,7] => ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,2,5,3,4,7,6] => [7,4,1,2,3,5,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4,7] => [6,4,1,2,3,5,7] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,3,6,7,4] => [5,3,7,1,2,4,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,2,5,3,7,4,6] => [6,3,7,1,2,4,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,2,5,6,3,4,7] => [5,3,6,1,2,4,7] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,2,5,6,3,7,4] => [3,7,5,1,2,4,6] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,2,5,6,7,3,4] => [3,6,4,7,1,2,5] => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,2,5,7,3,4,6] => [5,3,6,7,1,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,2,6,3,4,5,7] => [4,5,6,1,2,3,7] => ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,2,6,3,4,7,5] => [7,4,5,1,2,3,6] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,2,6,3,7,4,5] => [6,7,4,1,2,3,5] => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,2,6,7,3,4,5] => [5,6,3,7,1,2,4] => ? = 2 + 1
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,2,7,3,4,5,6] => [4,5,6,7,1,2,3] => ? = 1 + 1
[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
[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
[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
[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
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,2,4,7,5,6] => [2,6,7,1,3,4,5] => ? = 1 + 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] => ? = 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] => ? = 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] => ? = 1 + 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
The following 17 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000155The number of exceedances (also excedences) of a permutation. St001427The number of descents of a signed permutation. St000834The number of right outer peaks of a permutation. St000092The number of outer peaks of a permutation. St000353The number of inner valleys of a permutation. St001890The maximum magnitude of the Möbius function of a poset. St000914The sum of the values of the Möbius function of a poset. St001896The number of right descents of a signed permutations. St000455The second largest eigenvalue of a graph if it is integral. St001597The Frobenius rank of a skew partition. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St001624The breadth of a lattice. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000640The rank of the largest boolean interval in a poset.
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