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Your data matches 175 different statistics following compositions of up to 3 maps.
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Matching statistic: St000374
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ 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]
=> [1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [2,3,1] => 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [2,4,1,3] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [2,4,5,1,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [4,3,5,2,1] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [2,3,5,1,4] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [3,2,5,1,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [2,5,4,1,3] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[]
=> []
=> [1] => [1] => 0
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: St000996
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00237: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> [2,1] => [2,1] => 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [3,1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [3,2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [4,1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [3,1,4,2] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,5,2,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [5,4,2,1,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [4,1,2,5,3] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [4,2,1,5,3] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [4,1,5,3,2] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[]
=> []
=> [1] => [1] => 0
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001673
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St001673: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0]
=> 10 => [1,2] => 1
[2]
=> [1,0,1,0]
=> 1010 => [1,2,2] => 1
[1,1]
=> [1,1,0,0]
=> 1100 => [1,1,3] => 1
[3]
=> [1,0,1,0,1,0]
=> 101010 => [1,2,2,2] => 1
[2,1]
=> [1,0,1,1,0,0]
=> 101100 => [1,2,1,3] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => [1,1,2,3] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => [1,2,2,2,2] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => [1,2,2,1,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> 111000 => [1,1,1,4] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => [1,2,1,2,3] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => [1,1,2,2,3] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => [1,2,1,1,4] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => [1,1,1,3,3] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => [1,1,1,2,4] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => [1,1,1,1,5] => 1
[]
=> []
=> => [1] => 0
Description
The degree of asymmetry of an integer composition.
This is the number of pairs of symmetrically positioned distinct entries.
Matching statistic: St000767
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000767: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [1,1] => [2] => 1
[1,1]
=> [1,1,0,0]
=> [2] => [1,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1] => [3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,1] => [1,2] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [2,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,3] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,2] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,2] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[]
=> []
=> [] => [] => ? = 0
Description
The number of runs in an integer composition.
Writing the composition as $c_1^{e_1} \dots c_\ell^{e_\ell}$, where $c_i \neq c_{i+1}$ for all $i$, the number of runs is $\ell$, see [def.2.8, 1].
It turns out that the total number of runs in all compositions of $n$ equals the total number of odd parts in all these compositions, see [1].
Matching statistic: St000903
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000903: Integer compositions ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [1] => 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2] => 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1
[]
=> []
=> []
=> [] => ? = 0
Description
The number of different parts of an integer composition.
Matching statistic: St001359
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1] => [1] => 1
[2]
=> [1,0,1,0]
=> [1,2] => [1,2] => 1
[1,1]
=> [1,1,0,0]
=> [2,1] => [2,1] => 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => [1,3,2] => 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [3,1,4,2] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => 1
[]
=> []
=> [] => [] => ? = 0
Description
The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles.
In other words, this is $2^k$ where $k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition.
The generating function for the number of equivalence classes, $f(n)$, is
$$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$$
Matching statistic: St000486
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000486: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [2,1] => [1,2] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [1,2,3] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,4,3,2] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,2,3] => 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,5,4,3,2] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,4,5,3,2] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,3,4,2] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => 1 = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => 0 = 1 - 1
[]
=> []
=> [1] => [1] => ? = 0 - 1
Description
The number of cycles of length at least 3 of a permutation.
Matching statistic: St000630
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St000630: Binary words ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> 10 => 2 = 1 + 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1100 => 2 = 1 + 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1010 => 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 111000 => 2 = 1 + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 110100 => 3 = 2 + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 101100 => 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 2 = 1 + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 101010 => 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => 3 = 2 + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => 3 = 2 + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 10110010 => 3 = 2 + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => 3 = 2 + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 2 = 1 + 1
[]
=> []
=> []
=> => ? = 0 + 1
Description
The length of the shortest palindromic decomposition of a binary word.
A palindromic decomposition (paldec for short) of a word $w=a_1,\dots,a_n$ is any list of factors $p_1,\dots,p_k$ such that $w=p_1\dots p_k$ and each $p_i$ is a palindrome, i.e. coincides with itself read backwards.
Matching statistic: St000650
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
St000650: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 94%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [2,1] => [2,1] => 0 = 1 - 1
[2]
=> [1,0,1,0]
=> [3,1,2] => [1,3,2] => 0 = 1 - 1
[1,1]
=> [1,1,0,0]
=> [2,3,1] => [2,1,3] => 0 = 1 - 1
[3]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => 0 = 1 - 1
[2,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => [3,1,4,2] => 1 = 2 - 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,3,2] => 1 = 2 - 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => 0 = 1 - 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [4,1,2,5,3] => 1 = 2 - 1
[2,2]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => 0 = 1 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => 0 = 1 - 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => 1 = 2 - 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [3,1,4,5,2] => 1 = 2 - 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [2,5,4,1,3] => 1 = 2 - 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,5,1,4,2] => 1 = 2 - 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,3,4,5] => 0 = 1 - 1
[]
=> []
=> [1] => [1] => ? = 0 - 1
Description
The number of 3-rises of a permutation.
A 3-rise of a permutation $\pi$ is an index $i$ such that $\pi(i)+3 = \pi(i+1)$.
For 1-rises, or successions, see [[St000441]], for 2-rises see [[St000534]].
Matching statistic: St000568
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00140: Dyck paths —logarithmic height to pruning number⟶ Binary trees
St000568: Binary trees ⟶ ℤResult quality: 67% ●values known / values provided: 88%●distinct values known / distinct values provided: 67%
Values
[1]
=> [1,0]
=> [1,0]
=> [.,.]
=> ? = 1
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> [[.,.],.]
=> 1
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [.,[.,.]]
=> 1
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [[.,.],[.,.]]
=> 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [.,[[.,.],.]]
=> 2
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 2
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [[[.,.],.],[.,.]]
=> 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [.,[[.,.],[.,.]]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [.,[.,[.,.]]]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [[.,.],[.,[.,.]]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],.]
=> 2
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [.,[.,[[.,.],.]]]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],.],.]
=> 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [[.,[.,[.,.]]],.]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [.,[.,[.,[.,.]]]]
=> 1
[]
=> []
=> []
=> ?
=> ? = 0
Description
The hook number of a binary tree.
A hook of a binary tree is a vertex together with is left- and its right-most branch. Then there is a unique decomposition of the tree into hooks and the hook number is the number of hooks in this decomposition.
The following 165 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000886The number of permutations with the same antidiagonal sums. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000455The second largest eigenvalue of a graph if it is integral. St001946The number of descents in a parking function. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000091The descent variation of a composition. St000260The radius of a connected graph. St001488The number of corners of a skew partition. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001568The smallest positive integer that does not appear twice in the partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St000444The length of the maximal rise of a Dyck path. St000871The number of very big ascents of a permutation. St001712The number of natural descents of a standard Young tableau. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000478Another weight of a partition according to Alladi. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St000934The 2-degree of an integer partition. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001624The breadth of a lattice. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000234The number of global ascents of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000353The number of inner valleys of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St001344The neighbouring number of a permutation. St001388The number of non-attacking neighbors of a permutation. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001840The number of descents of a set partition. St000056The decomposition (or block) number of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000354The number of recoils of a permutation. St000502The number of successions of a set partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000991The number of right-to-left minima of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. 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 $eA$. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001801Half the number of preimage-image pairs of different parity in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001928The number of non-overlapping descents in a permutation. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000824The sum of the number of descents and the number of recoils of a permutation. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000632The jump number of the poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000907The number of maximal antichains of minimal length in a poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000717The number of ordinal summands of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000782The indicator function of whether a given perfect matching is an L & P matching. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001545The second Elser number of a connected graph. St001569The maximal modular displacement of a permutation. St000102The charge of a semistandard tableau. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St001060The distinguishing index of a graph. St001556The number of inversions of the third entry of a permutation.
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