searching the database
Your data matches 387 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
(click to perform a complete search on your data)
Matching statistic: St000141
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => 2 = 1 + 1
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000375
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => 0 = 1 - 1
Description
The number of non weak exceedences of a permutation that are 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 exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000649
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000649: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3,2,1] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 1 = 2 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => 1 = 2 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 2 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 1 = 2 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 0 = 1 - 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => 0 = 1 - 1
Description
The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.
Matching statistic: St001761
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001761: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => 3 = 2 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => 3 = 2 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => 3 = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 3 = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => 2 = 1 + 1
Description
The maximal multiplicity of a letter in a reduced word of a permutation.
For example, the permutation $3421$ has the reduced word $s_2 s_1 s_2 s_3 s_2$, where $s_2$ appears three times.
Matching statistic: St000171
Values
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000271
Values
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
Description
The chromatic index of a graph.
This is the minimal number of colours needed such that no two adjacent edges have the same colour.
Matching statistic: St000535
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(0,6),(1,2),(1,4),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1
Description
The rank-width of a graph.
Matching statistic: St001031
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001031: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,0,0,0]
=> [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,3,5,4,2,6] => [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [1,3,5,4,6,2] => [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,4,3,5,2,6] => [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [1,1,0,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> [2,1,5,4,6,3] => [1,1,0,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [1,1,0,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> [2,3,5,4,1,6] => [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,4,6,1] => [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> [2,4,3,1,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> [2,4,3,1,6,5] => [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001111
Values
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
Description
The weak 2-dynamic chromatic number of a graph.
A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-weak-dynamic number of a graph.
Matching statistic: St001112
Values
[1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 1
[1,0,1,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> 1
[1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,1,1,1,0,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(4,5)],6)
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,0,0,1,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> ([(4,5)],6)
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,0,1,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ([(3,6),(4,5),(5,6)],7)
=> 2
[1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,0,1,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,0,1,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,0,1,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
[1,1,0,1,1,0,0,0,1,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> ([(5,6)],7)
=> 1
Description
The 3-weak dynamic number of a graph.
A $k$-weak-dynamic coloring of a graph $G$ is a (non-proper) coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-weak-dynamic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $3$-weak-dynamic number of a graph.
The following 377 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001118The acyclic chromatic index of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001694The number of maximal dissociation sets in a graph. St001716The 1-improper chromatic number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001820The size of the image of the pop stack sorting operator. St000028The number of stack-sorts needed to sort a permutation. St000172The Grundy number of a graph. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000388The number of orbits of vertices of a graph under automorphisms. St000442The maximal area to the right of an up step of a Dyck path. St000537The cutwidth of a graph. St000709The number of occurrences of 14-2-3 or 14-3-2. St000730The maximal arc length of a set partition. St000872The number of very big descents of a permutation. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001306The number of induced paths on four vertices in a graph. St001350Half of the Albertson index of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001394The genus of a permutation. St001513The number of nested exceedences of a permutation. St001581The achromatic number of a graph. St001623The number of doubly irreducible elements of a lattice. St001670The connected partition number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001846The number of elements which do not have a complement in the lattice. St001963The tree-depth of a graph. St000013The height of a Dyck path. St000637The length of the longest cycle in a graph. St001315The dissociation number of a graph. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000225Difference between largest and smallest parts in a partition. St000254The nesting number of a set partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000409The number of pitchforks in a binary tree. St000654The first descent of a permutation. St000729The minimal arc length of a set partition. St000873The aix statistic of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001571The Cartan determinant of the integer partition. St001665The number of pure excedances of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000010The length of the partition. St000039The number of crossings of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000147The largest part of an integer partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000233The number of nestings of a set partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000306The bounce count of a Dyck path. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000444The length of the maximal rise of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000527The width of the poset. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000636The hull number of a graph. St000650The number of 3-rises of a permutation. St000651The maximal size of a rise in a permutation. St000652The maximal difference between successive positions of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000661The number of rises of length 3 of a Dyck path. St000662The staircase size of the code of a permutation. St000663The number of right floats of a permutation. St000668The least common multiple of the parts of the partition. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000779The tier of a permutation. St000871The number of very big ascents of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000982The length of the longest constant subword. St001029The size of the core of a graph. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001083The number of boxed occurrences of 132 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001128The exponens consonantiae of a partition. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001781The interlacing number of a set partition. St001810The number of fixed points of a permutation smaller than its largest moved point. St001847The number of occurrences of the pattern 1432 in a permutation. St001962The proper pathwidth of a graph. St000166The depth minus 1 of an ordered tree. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001093The detour number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001717The largest size of an interval in a poset. St000094The depth of an ordered tree. St001638The book thickness of a graph. St001644The dimension of a graph. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001964The interval resolution global dimension of a poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000836The number of descents of distance 2 of a permutation. St000837The number of ascents of distance 2 of a permutation. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St000058The order of a permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000100The number of linear extensions of a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000635The number of strictly order preserving maps of a poset into itself. St000640The rank of the largest boolean interval in a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000260The radius of a connected graph. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001651The Frankl number of a lattice. St001330The hat guessing number of a graph. St000035The number of left outer peaks of a permutation. St000647The number of big descents of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000648The number of 2-excedences of a permutation. St000264The girth of a graph, which is not a tree. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000079The number of alternating sign matrices for a given Dyck path. St000710The number of big deficiencies of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000711The number of big exceedences of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000099The number of valleys of a permutation, including the boundary. St000325The width of the tree associated to a permutation. St000628The balance of a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001811The Castelnuovo-Mumford regularity of a permutation. St000638The number of up-down runs of a permutation. St000877The depth of the binary word interpreted as a path. St001589The nesting number of a perfect matching. St001060The distinguishing index of a graph. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001877Number of indecomposable injective modules with projective dimension 2. St000741The Colin de Verdière graph invariant. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001645The pebbling number of a connected graph. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001117The game chromatic index of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001742The difference of the maximal and the minimal degree in a graph. St001812The biclique partition number of a graph. St000447The number of pairs of vertices of a graph with distance 3. St000450The number of edges minus the number of vertices plus 2 of a graph. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001282The number of graphs with the same chromatic polynomial. St001596The number of two-by-two squares inside a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St000299The number of nonisomorphic vertex-induced subtrees. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001861The number of Bruhat lower covers of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000908The length of the shortest maximal antichain in a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000907The number of maximal antichains of minimal length in a poset. St001866The nesting alignments of a signed permutation. St000292The number of ascents of a binary word. St001862The number of crossings of a signed permutation. St001864The number of excedances of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001896The number of right descents of a signed permutations. St000259The diameter of a connected graph. St000291The number of descents of a binary word. St000390The number of runs of ones in a binary word. St001769The reflection length of a signed permutation. St001895The oddness of a signed permutation. St000068The number of minimal elements in a poset. St001725The harmonious chromatic number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. St000181The number of connected components of the Hasse diagram for the poset. St000456The monochromatic index of a connected graph. St000909The number of maximal chains of maximal size in a poset. St000327The number of cover relations in a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000070The number of antichains in a poset. St000665The number of rafts of a permutation. St001490The number of connected components of a skew partition. St000672The number of minimal elements in Bruhat order not less than the permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001905The number of preferred parking spots in a parking function less than the index of the car. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St000956The maximal displacement of a permutation. St000117The number of centered tunnels of a Dyck path. St000120The number of left tunnels of a Dyck path. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001768The number of reduced words of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001884The number of borders of a binary word. St001903The number of fixed points of a parking function. St001946The number of descents in a parking function. St001948The number of augmented double ascents of a permutation. St000237The number of small exceedances. St000630The length of the shortest palindromic decomposition of a binary word. St000664The number of right ropes of a permutation. St000758The length of the longest staircase fitting into an integer composition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000894The trace of an alternating sign matrix. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. 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 $eA$. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001462The number of factors of a standard tableaux under concatenation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001517The length of a longest pair of twins in a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001777The number of weak descents in an integer composition. St001856The number of edges in the reduced word graph of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000381The largest part of an integer composition. St000717The number of ordinal summands of a poset. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000808The number of up steps of the associated bargraph. St000911The number of maximal antichains of maximal size in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001298The number of repeated entries in the Lehmer code of a permutation. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001424The number of distinct squares in a binary word. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001566The length of the longest arithmetic progression in a permutation. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St000443The number of long tunnels of a Dyck path. St000519The largest length of a factor maximising the subword complexity. St000538The number of even inversions of a permutation. St000863The length of the first row of the shifted shape of a permutation. St000904The maximal number of repetitions of an integer composition. St000922The minimal number such that all substrings of this length are unique. St001180Number of indecomposable injective modules with projective dimension at most 1. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001416The length of a longest palindromic factor of a binary word. St000064The number of one-box pattern of a permutation. St000735The last entry on the main diagonal of a standard tableau. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral.
Sorry, this statistic was not found in the database
or
add this statistic to the database – it's very simple and we need your support!