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Your data matches 122 different statistics following compositions of up to 3 maps.
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Matching statistic: St000374
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,5,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,2,5,3] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,2,5,3,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,4,2,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,2,6,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,4,2,5,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,6,3,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,2,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,2,6,3,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,5,2,6,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,2,6,5,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,4,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,6,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,2,5,6,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,4,5,2] => 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000703
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,2,3] => 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,3,2] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,5,4,2,3] => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,3,4,2] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,2,4] => 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,2,5,3] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,4,2] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,2,3,5,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,2,4,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,2,5,3,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,3,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,4,2,3,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,2,6,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,4,2,5,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,3,4,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,6,3,5,2,4] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,4,5,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,3,4,5,2] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,5,3,4] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,4,5,3] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,2,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,2,5,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,2,6,3,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,5,2,6,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,2,6,5,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,4,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,6,5,2,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,2,5,6,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,4,5,2] => 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: St000288
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => 0 => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => 01 => 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => 00 => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 011 => 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => 001 => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => 001 => 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => 010 => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 000 => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0111 => 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => 0011 => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => 0011 => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => 0101 => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => 0011 => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => 0001 => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => 0110 => 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => 0110 => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => 0010 => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => 0001 => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => 0010 => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => 0100 => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0000 => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => 01111 => 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => 00111 => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => 00111 => 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => 01011 => 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => 00011 => 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => 00111 => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => 00011 => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => 01101 => 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => 01100 => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => 00101 => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => 00011 => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => 00101 => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => 01001 => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => 00001 => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => 00111 => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => 00011 => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => 00011 => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => 00101 => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => 00001 => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => 01110 => 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => 00110 => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => 01110 => 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => 01101 => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => 00110 => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => 00110 => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => 01010 => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => 01100 => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => 00010 => 1
[]
=> [1] => [1] => => ? = 0
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 90% ●values known / values provided: 98%●distinct values known / distinct values provided: 90%
Values
[1,0]
=> [2,1] => [1,2] => [[1,2]]
=> 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [[1,2],[3]]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [[1,2,3]]
=> 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [[1,2],[3],[4]]
=> 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [[1,2,3],[4]]
=> 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [[1,2,3],[4]]
=> 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [[1,2,4],[3]]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [[1,2],[3],[4],[5]]
=> 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [[1,2,3],[4],[5]]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [[1,2,3],[4],[5]]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [[1,2,4],[3],[5]]
=> 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [[1,2,3,4],[5]]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [[1,2,3],[4],[5]]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [[1,2,3,4],[5]]
=> 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [[1,2,5],[3],[4]]
=> 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [[1,2,3],[4,5]]
=> 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [[1,2,3,4],[5]]
=> 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [[1,2,3,5],[4]]
=> 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [[1,2,4,5],[3]]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [[1,2,3,4,5]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [[1,2],[3],[4],[5],[6]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [[1,2,3],[4],[5],[6]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [[1,2,3],[4],[5],[6]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [[1,2,4],[3],[5],[6]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [[1,2,3,4],[5],[6]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [[1,2,3],[4],[5],[6]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [[1,2,3,4],[5],[6]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [[1,2,5],[3],[4],[6]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [[1,2,5,6],[3],[4]]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [[1,2,3],[4,5],[6]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [[1,2,3,4],[5],[6]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [[1,2,3,5],[4],[6]]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [[1,2,4,5],[3],[6]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [[1,2,3,4,5],[6]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [[1,2,3],[4],[5],[6]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [[1,2,3,4],[5],[6]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [[1,2,3,4],[5],[6]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [[1,2,3,5],[4],[6]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [[1,2,3,4,5],[6]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [[1,2,6],[3],[4],[5]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [[1,2,3],[4,6],[5]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [[1,2,6],[3],[4],[5]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [[1,2,5],[3,6],[4]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [[1,2,3],[4,6],[5]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [[1,2,3],[4,6],[5]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [[1,2,4],[3,6],[5]]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [[1,2,5,6],[3],[4]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [[1,2,3,4],[5,6]]
=> 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [1,2,3,4,5,6,7,8,9,11,10] => [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => [1,10,8,6,4,2,9,7,5,3] => [[1,2,7],[3,8],[4,9],[5,10],[6]]
=> ? = 7
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [11,1,2,3,4,5,6,7,8,9,10] => [1,11,10,9,8,7,6,5,4,3,2] => [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
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: St001644
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001644: Graphs ⟶ ℤResult quality: 80% ●values known / values provided: 81%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 0
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> 0
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> 0
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 2
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 4
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 6
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8] => ([],8)
=> ?
=> ? = 0
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ?
=> ? = 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ?
=> ? = 1
[]
=> [] => ?
=> ?
=> ? = 0
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 7
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9] => ([],9)
=> ?
=> ? = 0
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 8
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ?
=> ? = 9
Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
Matching statistic: St000470
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 74%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,2,4] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,4,2,5,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,4,3,2,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,3,2,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,3,2,6,4] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,2,4,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,2,3] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,2,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,2,6,5,3] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,6,4,2] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,5,2,3,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,2,3,6,4] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,2,4,6,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,3,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,3,6,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,5,4,2] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,4,2,5] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,6,5,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,6,4,5,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,6,3,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,5,6,3,2] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,4,6,7,5,3,2] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [1,7,6,5,3,2,4] => [1,4,7,6,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,4,7,5,3,2,6] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [1,5,3,2,7,6,4] => [1,7,6,4,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,6,4,7,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => ? = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,4,7,3,2,5,6] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,6,8,7,5,4,3,2] => [1,8,7,5,4,3,2,6] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [1,8,6,7,5,4,3,2] => [1,7,5,4,3,2,8,6] => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,4,8,7,6,5,3,2] => [1,8,7,6,5,3,2,4] => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,4,6,8,7,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [1,4,8,6,7,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => [1,8,7,6,4,5,3,2] => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [1,8,7,5,3,2,4,6] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,7,5,3,2,4,8,6] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [1,6,8,7,4,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => [1,8,6,7,4,5,3,2] => [1,5,3,2,7,4,8,6] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,5,8,7,6,3,4,2] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,8,7,6,5,3,2,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => [1,6,8,7,5,3,2,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => [1,8,6,7,5,3,2,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => [1,5,3,2,8,7,6,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => [1,8,7,5,3,2,6,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,8,6,4,2,7,5,3] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => [1,5,3,2,6,8,7,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => [1,5,3,2,8,6,7,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => [1,8,5,3,2,6,7,4] => ? => ? = 4 + 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,8,1,5,2,3,4,7] => [1,6,3,2,8,7,4,5] => ? => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => [1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => [1,2,3,6,8,7,5,4] => [1,2,3,8,7,5,4,6] => ? = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,8,1,4,7,5,6] => [1,2,3,8,6,7,5,4] => [1,2,3,7,5,4,8,6] => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,5,4,1,2,3,6,7] => [1,8,7,6,3,4,2,5] => ? => ? = 4 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [2,3,8,6,1,4,5,7] => [1,2,3,8,7,5,4,6] => ? => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,3,7,8,1,4,5,6] => [1,2,3,7,5,4,8,6] => ? => ? = 3 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => [1,2,7,4,6,3,8,5] => ? => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => [1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => [1,2,3,8,4,5,6,7] => [1,2,3,5,6,7,8,4] => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,8,6,4,2,9,7,5,3] => ? => ? = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => ? = 1 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => [1,10,8,6,4,2,9,7,5,3] => ? => ? = 7 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,10,1] => [1,2,3,4,5,6,7,8,9,10] => [1,2,3,4,5,6,7,8,9,10] => ? = 0 + 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: St000619
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 73%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 73%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 1 = 0 + 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,3,2,4] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,4,2,3] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,4,2,5,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,5,3,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,3,5,2,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,4,3,2,5] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,3,2,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,5,3,2,6,4] => 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,3,2,4,5] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,5,4,2,3] => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,4,2,3,5] => 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,4,2,6,5,3] => 4 = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,3,5,6,4,2] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,4,2,6,3,5] => 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,5,2,3,4] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,5,2,3,6,4] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,5,2,4,6,3] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,4,3,5] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,5,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,5,3,6,4] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,3,6,5,4,2] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,3,6,4,2,5] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,4,6,5,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,6,4,5,3,2] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,4,6,3,2,5] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,3,6,5,2,4] => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,3,5,2,6,4] => 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,5,6,3,2] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,3,6,2,4,5] => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,4,6,7,5,3,2] => ? = 3 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,4,6,7,5,2,3] => ? = 2 + 1
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [1,7,6,5,3,2,4] => [1,4,7,6,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,4,7,5,3,2,6] => ? = 3 + 1
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,7,1,2,3,4,6] => [1,5,3,2,7,6,4] => [1,7,6,4,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,6,4,7,5,3,2] => ? = 4 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,4,7,6,3,2,5] => ? = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,4,7,3,2,5,6] => ? = 2 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,6,8,7,5,4,3,2] => [1,8,7,5,4,3,2,6] => ? = 5 + 1
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [1,8,6,7,5,4,3,2] => [1,7,5,4,3,2,8,6] => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,4,8,7,6,5,3,2] => [1,8,7,6,5,3,2,4] => ? = 5 + 1
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,4,6,8,7,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [1,4,8,6,7,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => [1,8,7,6,4,5,3,2] => ? => ? = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [1,8,7,5,3,2,4,6] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,7,5,3,2,4,8,6] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [1,6,8,7,4,5,3,2] => ? => ? = 4 + 1
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => [1,8,6,7,4,5,3,2] => [1,5,3,2,7,4,8,6] => ? = 4 + 1
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,5,8,7,6,3,4,2] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,8,7,6,5,3,2,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => [1,6,8,7,5,3,2,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => [1,8,6,7,5,3,2,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,0,0,1,0,1,0]
=> [5,8,1,2,3,4,6,7] => [1,5,3,2,8,7,6,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [8,6,1,2,3,4,5,7] => [1,8,7,5,3,2,6,4] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => [1,8,6,4,2,7,5,3] => ? => ? = 5 + 1
[1,1,0,1,0,1,0,1,1,0,0,0,1,0]
=> [5,6,1,2,3,8,4,7] => [1,5,3,2,6,8,7,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> [5,8,1,2,3,7,4,6] => [1,5,3,2,8,6,7,4] => ? => ? = 4 + 1
[1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> [8,6,1,2,3,7,4,5] => [1,8,5,3,2,6,7,4] => ? => ? = 4 + 1
[1,1,0,1,1,0,1,0,1,0,0,0,1,0]
=> [6,8,1,5,2,3,4,7] => [1,6,3,2,8,7,4,5] => ? => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [2,3,8,1,4,5,6,7] => [1,2,3,8,7,6,5,4] => [1,2,3,8,7,6,5,4] => ? = 4 + 1
[1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [2,3,6,1,4,8,5,7] => [1,2,3,6,8,7,5,4] => [1,2,3,8,7,5,4,6] => ? = 3 + 1
[1,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> [2,3,8,1,4,7,5,6] => [1,2,3,8,6,7,5,4] => [1,2,3,7,5,4,8,6] => ? = 3 + 1
[1,1,1,0,1,0,1,0,0,0,1,0,1,0]
=> [8,5,4,1,2,3,6,7] => [1,8,7,6,3,4,2,5] => ? => ? = 4 + 1
[1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [2,3,8,6,1,4,5,7] => [1,2,3,8,7,5,4,6] => ? => ? = 3 + 1
[1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [2,3,7,8,1,4,5,6] => [1,2,3,7,5,4,8,6] => ? => ? = 3 + 1
[1,1,1,1,0,0,1,0,1,0,1,0,0,0]
=> [2,7,8,6,1,3,4,5] => [1,2,7,4,6,3,8,5] => ? => ? = 3 + 1
[1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [2,3,4,5,8,1,6,7] => [1,2,3,4,5,8,7,6] => [1,2,3,4,5,8,7,6] => ? = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,8,1,7] => [1,2,3,4,5,6,8,7] => [1,2,3,4,5,6,8,7] => ? = 1 + 1
[1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [2,3,8,5,6,7,1,4] => [1,2,3,8,4,5,6,7] => [1,2,3,5,6,7,8,4] => ? = 1 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,7,8] => [1,2,3,4,5,6,7,8] => ? = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,1,2,3,4,5,6,7,8] => [1,9,8,7,6,5,4,3,2] => [1,9,8,7,6,5,4,3,2] => ? = 7 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,9,1,2,3,4,5,6,7] => [1,8,6,4,2,9,7,5,3] => ? => ? = 6 + 1
[1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,9,1,8] => [1,2,3,4,5,6,7,9,8] => [1,2,3,4,5,6,7,9,8] => ? = 1 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,3,4,5,6,7,8,9,1] => [1,2,3,4,5,6,7,8,9] => [1,2,3,4,5,6,7,8,9] => ? = 0 + 1
[1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,10,1,9] => [1,2,3,4,5,6,7,8,10,9] => [1,2,3,4,5,6,7,8,10,9] => ? = 1 + 1
[1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [2,3,4,5,6,7,8,9,11,1,10] => [1,2,3,4,5,6,7,8,9,11,10] => [1,2,3,4,5,6,7,8,9,11,10] => ? = 1 + 1
[]
=> [1] => [1] => [1] => ? = 0 + 1
[1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [10,9,1,2,3,4,5,6,7,8] => [1,10,8,6,4,2,9,7,5,3] => ? => ? = 7 + 1
Description
The number of cyclic descents of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is given by the number of indices $1 \leq i \leq n$ such that $\pi(i) > \pi(i+1)$ where we set $\pi(n+1) = \pi(1)$.
Matching statistic: St001489
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,4,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,2,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,5,4,2,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,5,3,4,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,5,2,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,2,5,3,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,4,5,3,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,5,3,6,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,4,5,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,2,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,6,2,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,5,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,5,6,4,3,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,5,4,6,3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,4,3,6,2,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,5,4,6,2,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,6,2,4,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,6,3,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,3,5,6,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,5,6,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,5,4,6,3,7,2] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,5,2,6,4,7,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [1,6,7,5,4,3,2] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [1,6,7,5,4,2,3] => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [1,6,7,5,2,4,3] => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [1,6,7,5,3,4,2] => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [1,6,7,5,2,3,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [1,7,6,5,3,2,4] => [1,6,5,7,4,3,2] => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,6,5,7,4,2,3] => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,5,4,7,3,6,2] => ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,6,5,7,2,4,3] => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [1,6,5,7,3,4,2] => ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,6,5,7,2,3,4] => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [1,6,7,2,5,4,3] => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [1,6,7,2,5,3,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [1,6,7,4,5,3,2] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [1,6,7,4,5,2,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [1,5,4,6,7,3,2] => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [1,5,4,6,7,2,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [1,6,7,2,3,5,4] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [1,6,7,2,4,5,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [1,6,7,3,4,5,2] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [1,7,6,5,2,3,4] => [1,5,6,7,4,3,2] => ? = 3
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [1,6,7,5,2,3,4] => [1,5,6,7,4,2,3] => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [1,5,7,6,2,3,4] => [1,5,6,7,2,4,3] => ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,6,8,7,5,4,3,2] => [1,8,7,6,5,2,4,3] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [1,8,6,7,5,4,3,2] => [1,8,7,6,5,3,4,2] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,4,8,7,6,5,3,2] => [1,8,7,2,6,5,4,3] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,4,6,8,7,5,3,2] => [1,8,7,2,6,3,5,4] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [1,4,8,6,7,5,3,2] => [1,8,7,2,6,4,5,3] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => [1,8,7,6,4,5,3,2] => [1,8,7,5,6,4,3,2] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [1,8,7,5,3,2,4,6] => [1,6,5,7,4,8,3,2] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,7,5,3,2,4,8,6] => [1,5,4,6,3,8,2,7] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [1,6,8,7,4,5,3,2] => [1,8,7,5,6,2,4,3] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => [1,8,6,7,4,5,3,2] => [1,8,7,5,6,3,4,2] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,5,8,7,6,3,4,2] => [1,8,6,7,2,5,4,3] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,8,7,6,5,3,2,4] => [1,7,6,8,5,4,3,2] => ? = 5
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => [1,6,8,7,5,3,2,4] => [1,7,6,8,5,2,4,3] => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => [1,8,6,7,5,3,2,4] => [1,7,6,8,5,3,4,2] => ? = 4
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000354
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000354: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [2,1] => [1,2] => [1,2] => 0
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,2,3] => 0
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => [1,3,4,2] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => [1,5,4,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,3,5,4,2] => [1,5,2,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [1,5,2,3,4] => 1
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [1,4,5,3,2] => 2
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [1,4,3,5,2] => 2
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,5,2,3] => [1,4,5,2,3] => 1
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [1,2,4,5,3] => 1
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [1,3,4,5,2] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [6,1,2,3,4,5] => [1,6,5,4,3,2] => [1,6,5,4,3,2] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [5,1,2,3,6,4] => [1,5,6,4,3,2] => [1,6,5,4,2,3] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [4,1,2,6,3,5] => [1,4,6,5,3,2] => [1,6,5,2,4,3] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [6,1,2,5,3,4] => [1,6,4,5,3,2] => [1,6,5,3,4,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => [1,4,5,6,3,2] => [1,6,5,2,3,4] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [3,1,6,2,4,5] => [1,3,6,5,4,2] => [1,6,2,5,4,3] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [3,1,5,2,6,4] => [1,3,5,6,4,2] => [1,6,2,5,3,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [6,1,4,2,3,5] => [1,6,5,3,4,2] => [1,6,4,5,3,2] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => [1,6,4,2,3,5] => [1,4,5,3,6,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [5,1,4,2,6,3] => [1,5,6,3,4,2] => [1,6,4,5,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => [1,3,4,6,5,2] => [1,6,2,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,6,5,2,4] => [1,3,6,4,5,2] => [1,6,2,4,5,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [6,1,4,5,2,3] => [1,6,3,4,5,2] => [1,6,3,4,5,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => [1,3,4,5,6,2] => [1,6,2,3,4,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,6,1,3,4,5] => [1,2,6,5,4,3] => [1,2,6,5,4,3] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [1,2,6,5,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,4,1,6,3,5] => [1,2,4,6,5,3] => [1,2,6,3,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,6,1,5,3,4] => [1,2,6,4,5,3] => [1,2,6,4,5,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,4,1,5,6,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1
[1,1,0,1,0,0,1,0,1,0]
=> [6,3,1,2,4,5] => [1,6,5,4,2,3] => [1,5,6,4,3,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [1,5,6,4,2,3] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [6,4,1,2,3,5] => [1,6,5,3,2,4] => [1,5,4,6,3,2] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => [1,5,3,2,6,4] => [1,4,3,6,2,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [1,5,4,6,2,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [4,3,1,6,2,5] => [1,4,6,5,2,3] => [1,5,6,2,4,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [6,3,1,5,2,4] => [1,6,4,5,2,3] => [1,5,6,3,4,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [6,4,1,5,2,3] => [1,6,3,2,4,5] => [1,4,3,5,6,2] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [4,3,1,5,6,2] => [1,4,5,6,2,3] => [1,5,6,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,1,2,3,4,5,6] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => [1,7,5,3,2,4,6] => [1,5,4,6,3,7,2] => ? = 3
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [7,1,5,2,3,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 3
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [6,1,5,2,3,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [3,1,7,6,2,4,5] => [1,3,7,5,2,4,6] => [1,5,2,6,4,7,3] => ? = 2
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [7,1,6,5,2,3,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 2
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [7,3,1,2,4,5,6] => [1,7,6,5,4,2,3] => [1,6,7,5,4,3,2] => ? = 4
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [6,3,1,2,4,7,5] => [1,6,7,5,4,2,3] => [1,6,7,5,4,2,3] => ? = 3
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [5,3,1,2,7,4,6] => [1,5,7,6,4,2,3] => [1,6,7,5,2,4,3] => ? = 3
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [7,3,1,2,6,4,5] => [1,7,5,6,4,2,3] => [1,6,7,5,3,4,2] => ? = 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [1,6,7,5,2,3,4] => ? = 2
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [7,4,1,2,3,5,6] => [1,7,6,5,3,2,4] => [1,6,5,7,4,3,2] => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,0]
=> [6,4,1,2,3,7,5] => [1,6,7,5,3,2,4] => [1,6,5,7,4,2,3] => ? = 3
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => [1,7,5,3,2,6,4] => [1,5,4,7,3,6,2] => ? = 4
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,4,1,2,7,3,6] => [1,5,7,6,3,2,4] => [1,6,5,7,2,4,3] => ? = 3
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [7,4,1,2,6,3,5] => [1,7,5,6,3,2,4] => [1,6,5,7,3,4,2] => ? = 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [1,6,5,7,2,3,4] => ? = 2
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [4,3,1,7,2,5,6] => [1,4,7,6,5,2,3] => [1,6,7,2,5,4,3] => ? = 3
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [4,3,1,6,2,7,5] => [1,4,6,7,5,2,3] => [1,6,7,2,5,3,4] => ? = 2
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [7,3,1,5,2,4,6] => [1,7,6,4,5,2,3] => [1,6,7,4,5,3,2] => ? = 3
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [6,3,1,5,2,7,4] => [1,6,7,4,5,2,3] => [1,6,7,4,5,2,3] => ? = 2
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [7,4,1,5,2,3,6] => [1,7,6,3,2,4,5] => [1,5,4,6,7,3,2] => ? = 3
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [6,4,1,5,2,7,3] => [1,6,7,3,2,4,5] => [1,5,4,6,7,2,3] => ? = 2
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [4,3,1,5,7,2,6] => [1,4,5,7,6,2,3] => [1,6,7,2,3,5,4] => ? = 2
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,3,1,7,6,2,5] => [1,4,7,5,6,2,3] => [1,6,7,2,4,5,3] => ? = 2
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [7,3,1,5,6,2,4] => [1,7,4,5,6,2,3] => [1,6,7,3,4,5,2] => ? = 2
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [4,3,1,5,6,7,2] => [1,4,5,6,7,2,3] => [1,6,7,2,3,4,5] => ? = 1
[1,1,1,0,1,0,0,0,1,0,1,0]
=> [7,3,4,1,2,5,6] => [1,7,6,5,2,3,4] => [1,5,6,7,4,3,2] => ? = 3
[1,1,1,0,1,0,0,0,1,1,0,0]
=> [6,3,4,1,2,7,5] => [1,6,7,5,2,3,4] => [1,5,6,7,4,2,3] => ? = 2
[1,1,1,0,1,0,0,1,0,0,1,0]
=> [7,3,5,1,2,4,6] => [1,7,6,4,2,3,5] => [1,5,6,4,7,3,2] => ? = 3
[1,1,1,0,1,0,0,1,1,0,0,0]
=> [6,3,5,1,2,7,4] => [1,6,7,4,2,3,5] => [1,5,6,4,7,2,3] => ? = 2
[1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,3,4,1,7,2,6] => [1,5,7,6,2,3,4] => [1,5,6,7,2,4,3] => ? = 2
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [7,3,4,1,6,2,5] => [1,7,5,6,2,3,4] => [1,5,6,7,3,4,2] => ? = 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => [1,5,6,7,2,3,4] => ? = 1
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [7,3,6,5,1,2,4] => [1,7,4,5,2,3,6] => [1,5,6,3,4,7,2] => ? = 2
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,1,2,3,4,5,6,7] => [1,8,7,6,5,4,3,2] => [1,8,7,6,5,4,3,2] => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,1,2,3,4,8,5,7] => [1,6,8,7,5,4,3,2] => [1,8,7,6,5,2,4,3] => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,1,2,3,4,7,5,6] => [1,8,6,7,5,4,3,2] => [1,8,7,6,5,3,4,2] => ? = 5
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [4,1,2,8,3,5,6,7] => [1,4,8,7,6,5,3,2] => [1,8,7,2,6,5,4,3] => ? = 5
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [4,1,2,6,3,8,5,7] => [1,4,6,8,7,5,3,2] => [1,8,7,2,6,3,5,4] => ? = 4
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,1,2,8,3,7,5,6] => [1,4,8,6,7,5,3,2] => [1,8,7,2,6,4,5,3] => ? = 4
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [8,1,2,5,3,4,6,7] => [1,8,7,6,4,5,3,2] => [1,8,7,5,6,4,3,2] => ? = 5
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [8,1,2,6,3,4,5,7] => [1,8,7,5,3,2,4,6] => [1,6,5,7,4,8,3,2] => ? = 4
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => [1,7,5,3,2,4,8,6] => [1,5,4,6,3,8,2,7] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [6,1,2,5,3,8,4,7] => [1,6,8,7,4,5,3,2] => [1,8,7,5,6,2,4,3] => ? = 4
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> [8,1,2,5,3,7,4,6] => [1,8,6,7,4,5,3,2] => [1,8,7,5,6,3,4,2] => ? = 4
[1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [5,1,4,2,8,3,6,7] => [1,5,8,7,6,3,4,2] => [1,8,6,7,2,5,4,3] => ? = 4
[1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [8,4,1,2,3,5,6,7] => [1,8,7,6,5,3,2,4] => [1,7,6,8,5,4,3,2] => ? = 5
[1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,1,2,3,8,5,7] => [1,6,8,7,5,3,2,4] => [1,7,6,8,5,2,4,3] => ? = 4
[1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [8,4,1,2,3,7,5,6] => [1,8,6,7,5,3,2,4] => [1,7,6,8,5,3,4,2] => ? = 4
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: St001330
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> [1] => ([],1)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,0]
=> [2] => ([],2)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => ([],3)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ([],1)
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 3 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,0,1,1,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,0,0,1,1,0,1,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,0,1,0,0,0,1,0]
=> [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 2 + 1
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
The following 112 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000454The largest eigenvalue of a graph if it is integral. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000662The staircase size of the code of a permutation. St000054The first entry of the permutation. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St000155The number of exceedances (also excedences) of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001812The biclique partition number of a graph. St000052The number of valleys of a Dyck path not on the x-axis. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001427The number of descents of a signed permutation. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000996The number of exclusive left-to-right maxima of a permutation. St001487The number of inner corners of a skew partition. St001712The number of natural descents of a standard Young tableau. 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 $eA$. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001769The reflection length of a signed permutation. St001896The number of right descents of a signed permutations. St000507The number of ascents of a standard tableau. St001863The number of weak excedances of a signed permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000007The number of saliances of the permutation. St000451The length of the longest pattern of the form k 1 2. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000647The number of big descents of a permutation. St000884The number of isolated descents of a permutation. St000711The number of big exceedences of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St000292The number of ascents of a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000864The number of circled entries of the shifted recording tableau of a permutation. St000201The number of leaf nodes in a binary tree. St000214The number of adjacencies of a permutation. St000542The number of left-to-right-minima of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000015The number of peaks of a Dyck path. St000702The number of weak deficiencies of a permutation. St000991The number of right-to-left minima of a permutation. St000053The number of valleys of the Dyck path. St000083The number of left oriented leafs of a binary tree except the first one. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000710The number of big deficiencies of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001152The number of pairs with even minimum in a perfect matching. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001874Lusztig's a-function for the symmetric group. St001964The interval resolution global dimension of a poset. St000062The length of the longest increasing subsequence of the permutation. St000093The cardinality of a maximal independent set of vertices of a graph. St000164The number of short pairs. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000443The number of long tunnels of a Dyck path. St000746The number of pairs with odd minimum in a perfect matching. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001589The nesting number of a perfect matching. St000236The number of cyclical small weak excedances. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. 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. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000386The number of factors DDU in a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St000528The height of a poset. St000912The number of maximal antichains in a poset. St001343The dimension of the reduced incidence algebra of a poset. 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. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St000097The order of the largest clique of the graph. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001581The achromatic number of a graph. St001877Number of indecomposable injective modules with projective dimension 2.
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