Your data matches 703 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001526
(load all 38 compositions to match this statistic)
Mapping
St001526: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1 = 0 + 1
[1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0]
 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[1,1,1,1,0,0,0,0]
 => 4 = 3 + 1
Description
The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001405
(load all 81 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001405: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 0
[1,0,1,0]
 => [1,2] => 1
[1,1,0,0]
 => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => 2
[1,0,1,1,0,0]
 => [1,3,2] => 1
[1,1,0,0,1,0]
 => [2,1,3] => 1
[1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,1,0,0,0]
 => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 3
Description
The number of bonds in a permutation. For a permutation $\pi$, the pair $(\pi_i, \pi_{i+1})$ is a bond if $|\pi_i-\pi_{i+1}| = 1$.
Matching statistic: St001566
(load all 262 compositions to match this statistic)
Mapping
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001566: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => 1 = 0 + 1
[1,0,1,0]
 => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [1,3,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [2,3,1] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,2,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 3 = 2 + 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] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 4 = 3 + 1
Description
The length of the longest arithmetic progression in a permutation. For a permutation $\pi$ of length $n$, this is the biggest $k$ such that there exist $1 \leq i_1 < \dots < i_k \leq n$ with $$\pi(i_2) - \pi(i_1) = \pi(i_3) - \pi(i_2) = \dots = \pi(i_k) - \pi(i_{k-1}).$$
Matching statistic: St000628
(load all 8 compositions to match this statistic)
Mapping
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 11 => 0
[1,0,1,0]
 => 1010 => 1101 => 1
[1,1,0,0]
 => 1100 => 1101 => 1
[1,0,1,0,1,0]
 => 101010 => 110101 => 1
[1,0,1,1,0,0]
 => 101100 => 110101 => 1
[1,1,0,0,1,0]
 => 110010 => 110101 => 1
[1,1,0,1,0,0]
 => 110100 => 111001 => 2
[1,1,1,0,0,0]
 => 111000 => 111001 => 2
[1,0,1,0,1,0,1,0]
 => 10101010 => 11010101 => 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 11010101 => 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 11010101 => 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 11011001 => 2
[1,0,1,1,1,0,0,0]
 => 10111000 => 11011001 => 2
[1,1,0,0,1,0,1,0]
 => 11001010 => 11010101 => 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 11010101 => 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 11100101 => 2
[1,1,0,1,0,1,0,0]
 => 11010100 => 11101001 => 2
[1,1,0,1,1,0,0,0]
 => 11011000 => 11101001 => 2
[1,1,1,0,0,0,1,0]
 => 11100010 => 11100101 => 2
[1,1,1,0,0,1,0,0]
 => 11100100 => 11101001 => 2
[1,1,1,0,1,0,0,0]
 => 11101000 => 11110001 => 3
[1,1,1,1,0,0,0,0]
 => 11110000 => 11110001 => 3
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000836
(load all 160 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000836: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [2,1] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [3,2,1] => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,4,3,1] => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [3,4,2,1] => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [4,3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,4,1] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [2,4,3,5,1] => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,3,1] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,5,4,3,1] => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [3,2,4,5,1] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,2,5,4,1] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [3,4,2,5,1] => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,2,1] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [3,5,4,2,1] => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,3,2,5,1] => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [4,3,5,2,1] => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [4,5,3,2,1] => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [5,4,3,2,1] => 3
Description
The number of descents of distance 2 of a permutation. This is, $\operatorname{des}_2(\pi) = | \{ i : \pi(i) > \pi(i+2) \} |$.
Matching statistic: St000837
(load all 126 compositions to match this statistic)
Mapping
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000837: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,2] => 0
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [2,1,3] => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,2,3] => 1
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [2,3,1,4] => 1
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 1
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 2
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 2
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => 1
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => 1
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => 2
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => 2
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => 1
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => 1
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => 2
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => 2
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => 2
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => 2
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => 3
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 3
Description
The number of ascents of distance 2 of a permutation. This is, $\operatorname{asc}_2(\pi) = | \{ i : \pi(i) < \pi(i+2) \} |$.
Matching statistic: St001096
(load all 35 compositions to match this statistic)
Mapping
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001096: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 0
[1,0,1,0]
 => [1,2] => [1,2] => 1
[1,1,0,0]
 => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,2,3] => 2
[1,0,1,1,0,0]
 => [1,3,2] => [2,1,3] => 1
[1,1,0,0,1,0]
 => [2,1,3] => [1,3,2] => 1
[1,1,0,1,0,0]
 => [2,3,1] => [3,1,2] => 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [2,3,1,4] => 2
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,2,4,3] => 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [2,1,4,3] => 2
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [1,3,2,4] => 2
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [3,2,1,4] => 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [1,3,4,2] => 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [3,4,1,2] => 2
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [3,1,4,2] => 2
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [1,4,3,2] => 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [4,3,1,2] => 1
[1,1,1,0,1,0,0,0]
 => [3,4,2,1] => [4,1,3,2] => 2
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 3
Description
The size of the overlap set of a permutation. For a permutation $\pi\in\mathfrak S_n$ this is the number of indices $i < n$ such that the standardisation of $\pi_1\dots\pi_{n-i}$ equals the standardisation of $\pi_{i+1}\dots\pi_n$. In particular, for $n > 1$, the statistic is at least one, because the standardisations of $\pi_1$ and $\pi_n$ are both $1$. For example, for $\pi=2143$, the standardisations of $21$ and $43$ are equal, and so are the standardisations of $2$ and $3$. Thus, the statistic on $\pi$ is $2$.
Matching statistic: St001949
(load all 9 compositions to match this statistic)
Mapping
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001949: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => ([],1)
 => ([],1)
 => 0
[1,0,1,0]
 => ([(0,1)],2)
 => ([],2)
 => 1
[1,1,0,0]
 => ([],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 2
[1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => ([(1,2)],3)
 => 1
[1,1,0,1,0,0]
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 2
[1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 3
[1,0,1,0,1,1,0,0]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => ([(0,3),(1,3),(2,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => ([(2,3)],4)
 => 2
[1,1,0,0,1,1,0,0]
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2)],4)
 => 2
[1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => ([(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => ([(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,1,0,0,0,0]
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
Description
The rigidity index of a graph. A base of a permutation group is a set $B$ such that the pointwise stabilizer of $B$ is trivial. For example, a base of the symmetric group on $n$ letters must contain all but one letter. This statistic yields the minimal size of a base for the automorphism group of a graph.
Matching statistic: St000308
(load all 14 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [2,1] => 1 = 0 + 1
[1,0,1,0]
 => [3,1,2] => [1,3,2] => 2 = 1 + 1
[1,1,0,0]
 => [2,3,1] => [2,3,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,2,4,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [2,1,4,3] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,4,3,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [2,3,4,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,2,3,5,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [4,1,5,2,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,1,5,2,4] => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,5,2,4,3] => 3 = 2 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,4,1,5,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [2,1,3,5,4] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,5,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,3,5,2,4] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,2,5,4,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,3,5,1,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [2,3,1,5,4] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [2,5,1,4,3] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,3,5,4,2] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [2,3,4,5,1] => 4 = 3 + 1
Description
The height of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The statistic is given by the height of this tree. See also [[St000325]] for the width of this tree.
Matching statistic: St000808
(load all 5 compositions to match this statistic)
Mapping
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00071: Permutations —descent composition⟶ Integer compositions
St000808: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,1] => 1 = 0 + 1
[1,0,1,0]
 => [3,1,2] => [1,2] => 2 = 1 + 1
[1,1,0,0]
 => [2,3,1] => [2,1] => 2 = 1 + 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [1,3] => 3 = 2 + 1
[1,0,1,1,0,0]
 => [3,1,4,2] => [1,2,1] => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [2,2] => 2 = 1 + 1
[1,1,0,1,0,0]
 => [4,3,1,2] => [1,1,2] => 2 = 1 + 1
[1,1,1,0,0,0]
 => [2,3,4,1] => [3,1] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [1,4] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [1,3,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [1,3,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [2,3] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [2,2,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [1,1,3] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [1,1,3] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [1,1,2,1] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [3,2] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [2,1,2] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [1,2,2] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [4,1] => 4 = 3 + 1
Description
The number of up steps of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the number of up steps.
The following 693 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000956The maximal displacement of a permutation. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001315The dissociation number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000080The rank of the poset. St000141The maximum drop size of a permutation. St000245The number of ascents of a permutation. St000292The number of ascents of a binary word. St000306The bounce count of a Dyck path. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000392The length of the longest run of ones in a binary word. St000442The maximal area to the right of an up step of a Dyck path. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000624The normalized sum of the minimal distances to a greater element. St000662The staircase size of the code of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000864The number of circled entries of the shifted recording tableau of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000996The number of exclusive left-to-right maxima of a permutation. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001197The global dimension of $eAe$ for 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 $eA$. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. 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. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001742The difference of the maximal and the minimal degree in a graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000007The number of saliances of the permutation. St000013The height of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000075The orbit size of a standard tableau under promotion. St000166The depth minus 1 of an ordered tree. St000171The degree of the graph. St000172The Grundy number of a graph. St000203The number of external nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000271The chromatic index of a graph. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000443The number of long tunnels of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000619The number of cyclic descents of a permutation. St000626The minimal period of a binary word. St000636The hull number of a graph. St000638The number of up-down runs of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000831The number of indices that are either descents or recoils. St000904The maximal number of repetitions of an integer composition. St000918The 2-limited packing number of a graph. St000991The number of right-to-left minima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001058The breadth of the ordered tree. St001118The acyclic chromatic index of a graph. St001119The length of a shortest maximal path in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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: 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001439The number of even weak deficiencies and of odd weak exceedences. St001486The number of corners of the ribbon associated with an integer composition. St001530The depth of a Dyck path. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001806The upper middle entry of a permutation. St001962The proper pathwidth of a graph. St000050The depth or height of a binary tree. St000094The depth of an ordered tree. St000863The length of the first row of the shifted shape of a permutation. St000982The length of the longest constant subword. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001415The length of the longest palindromic prefix of a binary word. St001554The number of distinct nonempty subtrees of a binary tree. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000326The position of the first one in a binary word after appending a 1 at the end. St000538The number of even inversions of a permutation. St000886The number of permutations with the same antidiagonal sums. St000988The orbit size of a permutation under Foata's bijection. St001052The length of the exterior of a permutation. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000646The number of big ascents of a permutation. St000710The number of big deficiencies of a permutation. St000938The number of zeros of the symmetric group character corresponding to the partition. St000239The number of small weak excedances. St000382The first part of an integer composition. St000485The length of the longest cycle of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000876The number of factors in the Catalan decomposition of a binary word. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001081The number of minimal length factorizations of a permutation into star transpositions. St001461The number of topologically connected components of the chord diagram of a permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001570The minimal number of edges to add to make a graph Hamiltonian. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001733The number of weak left to right maxima of a Dyck path. St001807The lower middle entry of a permutation. St001884The number of borders of a binary word. St000216The absolute length of a permutation. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000290The major index of a binary word. St000295The length of the border of a binary word. St000354The number of recoils of a permutation. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000491The number of inversions of a set partition. St000539The number of odd inversions of a permutation. St000554The number of occurrences of the pattern {{1,2},{3}} in a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000565The major index of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000691The number of changes of a binary word. St000732The number of double deficiencies of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000989The number of final rises of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001388The number of non-attacking neighbors of a permutation. St001485The modular major index of a binary word. St001552The number of inversions between excedances and fixed points of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001811The Castelnuovo-Mumford regularity of a permutation. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001638The book thickness of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000654The first descent of a permutation. St001948The number of augmented double ascents of a permutation. St000489The number of cycles of a permutation of length at most 3. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000990The first ascent of a permutation. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000877The depth of the binary word interpreted as a path. St000932The number of occurrences of the pattern UDU in a Dyck path. 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 $eA$. St001432The order dimension of the partition. St000331The number of upper interactions of a Dyck path. St000393The number of strictly increasing runs in a binary word. St000759The smallest missing part in an integer partition. St000922The minimal number such that all substrings of this length are unique. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000120The number of left tunnels of a Dyck path. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000383The last part of an integer composition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000527The width of the poset. St000657The smallest part of an integer composition. St000741The Colin de Verdière graph invariant. St000758The length of the longest staircase fitting into an integer composition. St000767The number of runs in an integer composition. St000862The number of parts of the shifted shape of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. 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. 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. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001498The normalised height of a Nakayama algebra with magnitude 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001741The largest integer such that all patterns of this size are contained in the permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001955The number of natural descents for set-valued two row standard Young tableaux. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001569The maximal modular displacement of a permutation. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000454The largest eigenvalue of a graph if it is integral. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000260The radius of a connected graph. St000668The least common multiple of the parts of the partition. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001568The smallest positive integer that does not appear twice in the partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000402Half the size of the symmetry class of a permutation. St000444The length of the maximal rise of a Dyck path. St000568The hook number of a binary tree. St000630The length of the shortest palindromic decomposition of a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000993The multiplicity of the largest part of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001769The reflection length of a signed permutation. St001864The number of excedances of a signed permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001637The number of (upper) dissectors of a poset. St001330The hat guessing number of a graph. St001644The dimension of a graph. St000035The number of left outer peaks of a permutation. St000253The crossing number of a set partition. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000834The number of right outer peaks of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001571The Cartan determinant of the integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001863The number of weak excedances of a signed permutation. St000455The second largest eigenvalue of a graph if it is integral. St000032The number of elements smaller than the given Dyck path in the Tamari Order. St000038The product of the heights of the descending steps of a Dyck path. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000144The pyramid weight of the Dyck path. St000184The size of the centralizer of any permutation of given cycle type. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000228The size of a partition. St000254The nesting number of a set partition. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000352The Elizalde-Pak rank of a permutation. St000384The maximal part of the shifted composition of an integer partition. St000395The sum of the heights of the peaks of a Dyck path. St000418The number of Dyck paths that are weakly below a Dyck path. St000459The hook length of the base cell of a partition. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000519The largest length of a factor maximising the subword complexity. St000531The leading coefficient of the rook polynomial of an integer partition. St000532The total number of rook placements on a Ferrers board. St000631The number of distinct palindromic decompositions of a binary word. St000667The greatest common divisor of the parts of the partition. St000674The number of hills of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000784The maximum of the length and the largest part of the integer partition. St000885The number of critical steps in the Catalan decomposition of a binary word. St000933The number of multipartitions of sizes given by an integer partition. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001378The product of the cohook lengths of the integer partition. St001400The total number of Littlewood-Richardson tableaux of given shape. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001488The number of corners of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001527The cyclic permutation representation number of an integer partition. St001531Number of partial orders contained in the poset determined by the Dyck path. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001660The number of ways to place as many non-attacking rooks as possible on a skew Ferrers board. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001814The number of partitions interlacing the given partition. St001885The number of binary words with the same proper border set. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000891The number of distinct diagonal sums of a permutation matrix. St000647The number of big descents of a permutation. St000665The number of rafts of a permutation. St001060The distinguishing index of a graph. St001668The number of points of the poset minus the width of the poset. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000456The monochromatic index of a connected graph. St000871The number of very big ascents of a permutation. St001394The genus of a permutation. St000237The number of small exceedances. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000731The number of double exceedences of a permutation. St000762The sum of the positions of the weak records of an integer composition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. 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 $eA$. 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 $eA$. St001249Sum of the odd parts of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001623The number of doubly irreducible elements of a lattice. St000717The number of ordinal summands of a poset. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St000023The number of inner peaks of a permutation. St000089The absolute variation of a composition. St000091The descent variation of a composition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000241The number of cyclical small excedances. St000317The cycle descent number of a permutation. St000353The number of inner valleys of a permutation. St000461The rix statistic of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000562The number of internal points of a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000632The jump number of the poset. St000663The number of right floats of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000850The number of 1/2-balanced pairs in a poset. St000873The aix statistic of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001520The number of strict 3-descents. St001556The number of inversions of the third entry of a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001665The number of pure excedances of a permutation. St001728The number of invisible descents of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function. St000058The order of a permutation. St000307The number of rowmotion orbits of a poset. St000729The minimal arc length of a set partition. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000921The number of internal inversions of a binary word. St000942The number of critical left to right maxima of the parking functions. St001062The maximal size of a block of a set partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001517The length of a longest pair of twins in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001691The number of kings in a graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001937The size of the center of a parking function. St000297The number of leading ones in a binary word. St000906The length of the shortest maximal chain in a poset. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St001285The number of primes in the column sums of the two line notation of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001875The number of simple modules with projective dimension at most 1. St001372The length of a longest cyclic run of ones of a binary word. St000515The number of invariant set partitions when acting with a permutation of given cycle type. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000815The number of semistandard Young tableaux of partition weight of given shape. St001128The exponens consonantiae of a partition. St001564The value of the forgotten symmetric functions when all variables set to 1. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001621The number of atoms of a lattice. St001626The number of maximal proper sublattices of a lattice. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000014The number of parking functions supported by a Dyck path. St000015The number of peaks of a Dyck path. St000026The position of the first return of a Dyck path. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000347The inversion sum of a binary word. St000391The sum of the positions of the ones in a binary word. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000420The number of Dyck paths that are weakly above a Dyck path. St000439The position of the first down step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000529The number of permutations whose descent word is the given binary word. St000543The size of the conjugacy class of a binary word. St000655The length of the minimal rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000715The number of semistandard Young tableaux of given shape and entries at most 3. St000734The last entry in the first row of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St000770The major index of an integer partition when read from bottom to top. St000792The Grundy value for the game of ruler on a binary word. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000947The major index east count of a Dyck path. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001019Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001228The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St001254The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001267The length of the Lyndon factorization of the binary word. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001437The flex of a binary word. St001471The magnitude of a Dyck path. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001481The minimal height of a peak of a Dyck path. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001545The second Elser number of a connected graph. St001614The cyclic permutation representation number of a skew partition. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001658The total number of rook placements on a Ferrers board. St001721The degree of a binary word. St001732The number of peaks visible from the left. St001809The index of the step at the first peak of maximal height in a Dyck path. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001930The weak major index of a binary word. St000219The number of occurrences of the pattern 231 in a permutation. St000516The number of stretching pairs of a permutation. St000650The number of 3-rises of a permutation. St001130The number of two successive successions in a permutation. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001720The minimal length of a chain of small intervals in a lattice. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000781The number of proper colouring schemes of a Ferrers diagram. St000782The indicator function of whether a given perfect matching is an L & P matching. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. 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. St001645The pebbling number of a connected graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. 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. St001383The BG-rank of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000477The weight of a partition according to Alladi. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. St000997The even-odd crank of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000101The cocharge of a semistandard tableau. St001856The number of edges in the reduced word graph of a permutation. St000739The first entry in the last row of a semistandard tableau. St001401The number of distinct entries in a semistandard tableau. St001926Sparre Andersen's position of the maximum of a signed permutation. St000002The number of occurrences of the pattern 123 in a permutation. St000095The number of triangles of a graph. St000102The charge of a semistandard tableau. St000357The number of occurrences of the pattern 12-3. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001114The number of odd descents of a permutation. 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 $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. 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. 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. St001578The minimal number of edges to add or remove to make a graph a line graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001684The reduced word complexity of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001935The number of ascents in a parking function. St001964The interval resolution global dimension of a poset. St000116The major index of a semistandard tableau obtained by standardizing. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000736The last entry in the first row of a semistandard tableau. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000958The number of Bruhat factorizations of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001487The number of inner corners of a skew partition. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001596The number of two-by-two squares inside a skew partition. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001768The number of reduced words of a signed permutation. St001812The biclique partition number of a graph. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St000004The major index of a permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000210Minimum over maximum difference of elements in cycles. St000211The rank of the set partition. St000251The number of nonsingleton blocks of a set partition. St000316The number of non-left-to-right-maxima of a permutation. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000488The number of cycles of a permutation of length at most 2. St000493The los statistic of a set partition. St000499The rcb statistic of a set partition. St000504The cardinality of the first block of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000653The last descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000794The mak of a permutation. St000798The makl of a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001075The minimal size of a block of a set partition. St001117The game chromatic index of a graph. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001497The position of the largest weak excedence of a permutation. St001642The Prague dimension of a graph. St001649The length of a longest trail in a graph. St001722The number of minimal chains with small intervals between a binary word and the top element. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St000135The number of lucky cars of the parking function. St000521The number of distinct subtrees of an ordered tree. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001927Sparre Andersen's number of positives of a signed permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000735The last entry on the main diagonal of a standard tableau.