searching the database
Your data matches 204 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: St000209
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 2
[3,1,2] => 2
[3,2,1] => 2
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 2
[1,4,2,3] => 2
[1,4,3,2] => 2
[2,1,3,4] => 1
[2,1,4,3] => 1
[2,3,1,4] => 2
[2,3,4,1] => 3
[2,4,1,3] => 3
[2,4,3,1] => 3
[3,1,2,4] => 2
[3,1,4,2] => 3
[3,2,1,4] => 2
[3,2,4,1] => 3
[3,4,1,2] => 2
[3,4,2,1] => 3
[4,1,2,3] => 3
[4,1,3,2] => 3
[4,2,1,3] => 3
[4,2,3,1] => 3
[4,3,1,2] => 3
[4,3,2,1] => 3
Description
Maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the maximum of this value over all cycles in the permutation.
Matching statistic: St000141
(load all 41 compositions to match this statistic)
(load all 41 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ 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
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => 3
[2,4,1,3] => [3,4,1,2] => 2
[2,4,3,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => 3
[3,4,1,2] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => 3
[4,1,3,2] => [4,2,3,1] => 3
[4,2,1,3] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => 3
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: St000503
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00151: Permutations —to cycle type⟶ Set partitions
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000503: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => {{1},{2}}
=> 0
[2,1] => {{1,2}}
=> 1
[1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => {{1,2,3}}
=> 2
[3,1,2] => {{1,2,3}}
=> 2
[3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,1,4] => {{1,2,3},{4}}
=> 2
[2,3,4,1] => {{1,2,3,4}}
=> 3
[2,4,1,3] => {{1,2,3,4}}
=> 3
[2,4,3,1] => {{1,2,4},{3}}
=> 3
[3,1,2,4] => {{1,2,3},{4}}
=> 2
[3,1,4,2] => {{1,2,3,4}}
=> 3
[3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,4,1] => {{1,3,4},{2}}
=> 3
[3,4,1,2] => {{1,3},{2,4}}
=> 2
[3,4,2,1] => {{1,2,3,4}}
=> 3
[4,1,2,3] => {{1,2,3,4}}
=> 3
[4,1,3,2] => {{1,2,4},{3}}
=> 3
[4,2,1,3] => {{1,3,4},{2}}
=> 3
[4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,3,1,2] => {{1,2,3,4}}
=> 3
[4,3,2,1] => {{1,4},{2,3}}
=> 3
Description
The maximal difference between two elements in a common block.
Matching statistic: St000956
(load all 32 compositions to match this statistic)
(load all 32 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000956: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0
[2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => 2
[3,1,2] => [3,2,1] => 2
[3,2,1] => [3,2,1] => 2
[1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => 2
[1,4,2,3] => [1,4,3,2] => 2
[1,4,3,2] => [1,4,3,2] => 2
[2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [4,2,3,1] => 3
[2,4,1,3] => [3,4,1,2] => 2
[2,4,3,1] => [4,3,2,1] => 3
[3,1,2,4] => [3,2,1,4] => 2
[3,1,4,2] => [4,2,3,1] => 3
[3,2,1,4] => [3,2,1,4] => 2
[3,2,4,1] => [4,2,3,1] => 3
[3,4,1,2] => [4,3,2,1] => 3
[3,4,2,1] => [4,3,2,1] => 3
[4,1,2,3] => [4,2,3,1] => 3
[4,1,3,2] => [4,2,3,1] => 3
[4,2,1,3] => [4,3,2,1] => 3
[4,2,3,1] => [4,3,2,1] => 3
[4,3,1,2] => [4,3,2,1] => 3
[4,3,2,1] => [4,3,2,1] => 3
Description
The maximal displacement of a permutation.
This is $\max\{ |\pi(i)-i| \mid 1 \leq i \leq n\}$ for a permutation $\pi$ of $\{1,\ldots,n\}$.
This statistic without the absolute value is the maximal drop size [[St000141]].
Matching statistic: St000028
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000028: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 1
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [3,2,1] => [2,3,1] => 2
[3,1,2] => [3,2,1] => [2,3,1] => 2
[3,2,1] => [3,2,1] => [2,3,1] => 2
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,3,4,2] => [1,4,3,2] => [1,3,4,2] => 2
[1,4,2,3] => [1,4,3,2] => [1,3,4,2] => 2
[1,4,3,2] => [1,4,3,2] => [1,3,4,2] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 1
[2,3,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[2,3,4,1] => [4,2,3,1] => [2,3,4,1] => 3
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => 2
[2,4,3,1] => [4,3,2,1] => [3,2,4,1] => 3
[3,1,2,4] => [3,2,1,4] => [2,3,1,4] => 2
[3,1,4,2] => [4,2,3,1] => [2,3,4,1] => 3
[3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 2
[3,2,4,1] => [4,2,3,1] => [2,3,4,1] => 3
[3,4,1,2] => [4,3,2,1] => [3,2,4,1] => 3
[3,4,2,1] => [4,3,2,1] => [3,2,4,1] => 3
[4,1,2,3] => [4,2,3,1] => [2,3,4,1] => 3
[4,1,3,2] => [4,2,3,1] => [2,3,4,1] => 3
[4,2,1,3] => [4,3,2,1] => [3,2,4,1] => 3
[4,2,3,1] => [4,3,2,1] => [3,2,4,1] => 3
[4,3,1,2] => [4,3,2,1] => [3,2,4,1] => 3
[4,3,2,1] => [4,3,2,1] => [3,2,4,1] => 3
Description
The number of stack-sorts needed to sort a permutation.
A permutation is (West) $t$-stack sortable if it is sortable using $t$ stacks in series.
Let $W_t(n,k)$ be the number of permutations of size $n$
with $k$ descents which are $t$-stack sortable. Then the polynomials $W_{n,t}(x) = \sum_{k=0}^n W_t(n,k)x^k$
are symmetric and unimodal.
We have $W_{n,1}(x) = A_n(x)$, the Eulerian polynomials. One can show that $W_{n,1}(x)$ and $W_{n,2}(x)$ are real-rooted.
Precisely the permutations that avoid the pattern $231$ have statistic at most $1$, see [3]. These are counted by $\frac{1}{n+1}\binom{2n}{n}$ ([[OEIS:A000108]]). Precisely the permutations that avoid the pattern $2341$ and the barred pattern $3\bar 5241$ have statistic at most $2$, see [4]. These are counted by $\frac{2(3n)!}{(n+1)!(2n+1)!}$ ([[OEIS:A000139]]).
Matching statistic: St000171
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => ([],2)
=> 0
[2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3] => [1,2,3] => ([],3)
=> 0
[1,3,2] => [1,3,2] => ([(1,2)],3)
=> 1
[2,1,3] => [2,1,3] => ([(1,2)],3)
=> 1
[2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,2,3,4] => [1,2,3,4] => ([],4)
=> 0
[1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> 1
[2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,1,2,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,2,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,1,3] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,2,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
Description
The degree of the graph.
This is the maximal vertex degree of a graph.
Matching statistic: St000442
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000442: 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
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
=> 0
[2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,1,2] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 2
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[2,4,1,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 2
[2,4,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,1,4,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,1,3,2] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 3
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000651
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00088: Permutations —Kreweras complement⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 1
[1,2,3] => [1,2,3] => [2,3,1] => 1
[1,3,2] => [1,3,2] => [2,1,3] => 2
[2,1,3] => [2,1,3] => [3,2,1] => 0
[2,3,1] => [1,3,2] => [2,1,3] => 2
[3,1,2] => [3,1,2] => [3,1,2] => 1
[3,2,1] => [3,2,1] => [1,3,2] => 2
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 1
[1,2,4,3] => [1,2,4,3] => [2,3,1,4] => 3
[1,3,2,4] => [1,3,2,4] => [2,4,3,1] => 2
[1,3,4,2] => [1,2,4,3] => [2,3,1,4] => 3
[1,4,2,3] => [1,4,2,3] => [2,4,1,3] => 2
[1,4,3,2] => [1,4,3,2] => [2,1,4,3] => 3
[2,1,3,4] => [2,1,3,4] => [3,2,4,1] => 2
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 3
[2,3,1,4] => [1,3,2,4] => [2,4,3,1] => 2
[2,3,4,1] => [1,2,4,3] => [2,3,1,4] => 3
[2,4,1,3] => [2,4,1,3] => [4,2,1,3] => 2
[2,4,3,1] => [1,4,3,2] => [2,1,4,3] => 3
[3,1,2,4] => [3,1,2,4] => [3,4,2,1] => 1
[3,1,4,2] => [2,1,4,3] => [3,2,1,4] => 3
[3,2,1,4] => [3,2,1,4] => [4,3,2,1] => 0
[3,2,4,1] => [2,1,4,3] => [3,2,1,4] => 3
[3,4,1,2] => [2,4,1,3] => [4,2,1,3] => 2
[3,4,2,1] => [1,4,3,2] => [2,1,4,3] => 3
[4,1,2,3] => [4,1,2,3] => [3,4,1,2] => 1
[4,1,3,2] => [4,1,3,2] => [3,1,4,2] => 3
[4,2,1,3] => [4,2,1,3] => [4,3,1,2] => 1
[4,2,3,1] => [4,1,3,2] => [3,1,4,2] => 3
[4,3,1,2] => [4,3,1,2] => [4,1,3,2] => 2
[4,3,2,1] => [4,3,2,1] => [1,4,3,2] => 3
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000653
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00277: Permutations —catalanization⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000653: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => 0
[2,1] => [2,1] => [2,1] => 1
[1,2,3] => [1,2,3] => [1,2,3] => 0
[1,3,2] => [1,3,2] => [1,3,2] => 2
[2,1,3] => [2,1,3] => [2,1,3] => 1
[2,3,1] => [2,3,1] => [3,2,1] => 2
[3,1,2] => [2,3,1] => [3,2,1] => 2
[3,2,1] => [3,2,1] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 2
[1,3,4,2] => [1,3,4,2] => [1,4,3,2] => 3
[1,4,2,3] => [1,3,4,2] => [1,4,3,2] => 3
[1,4,3,2] => [1,4,3,2] => [1,4,2,3] => 2
[2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 3
[2,3,1,4] => [2,3,1,4] => [3,2,1,4] => 2
[2,3,4,1] => [2,3,4,1] => [4,2,3,1] => 3
[2,4,1,3] => [4,3,1,2] => [2,4,1,3] => 2
[2,4,3,1] => [2,4,3,1] => [4,2,1,3] => 2
[3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 2
[3,1,4,2] => [2,3,4,1] => [4,2,3,1] => 3
[3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 1
[3,2,4,1] => [3,2,4,1] => [4,3,2,1] => 3
[3,4,1,2] => [4,3,2,1] => [4,1,2,3] => 1
[3,4,2,1] => [3,4,2,1] => [4,1,3,2] => 3
[4,1,2,3] => [2,3,4,1] => [4,2,3,1] => 3
[4,1,3,2] => [2,4,3,1] => [4,2,1,3] => 2
[4,2,1,3] => [3,2,4,1] => [4,3,2,1] => 3
[4,2,3,1] => [3,4,2,1] => [4,1,3,2] => 3
[4,3,1,2] => [3,4,2,1] => [4,1,3,2] => 3
[4,3,2,1] => [4,3,2,1] => [4,1,2,3] => 1
Description
The last descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the largest index $0 \leq i < n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(0) = n+1$.
Matching statistic: St000730
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000730: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => {{1},{2}}
=> 0
[2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 0
[1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3] => [2,1,3] => {{1,2},{3}}
=> 1
[2,3,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,1,2] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,4,2,3] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[1,4,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 2
[2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[2,1,4,3] => [2,1,4,3] => {{1,2},{3,4}}
=> 1
[2,3,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[2,3,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[2,4,1,3] => [3,4,1,2] => {{1,3},{2,4}}
=> 2
[2,4,3,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[3,1,2,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,1,4,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[3,2,1,4] => [3,2,1,4] => {{1,3},{2},{4}}
=> 2
[3,2,4,1] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[3,4,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[3,4,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,1,2,3] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,1,3,2] => [4,2,3,1] => {{1,4},{2},{3}}
=> 3
[4,2,1,3] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,2,3,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,3,1,2] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
[4,3,2,1] => [4,3,2,1] => {{1,4},{2,3}}
=> 3
Description
The maximal arc length of a set partition.
The arcs of a set partition are those $i < j$ that are consecutive elements in the blocks. If there are no arcs, the maximal arc length is $0$.
The following 194 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001090The number of pop-stack-sorts needed to sort a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000013The height of a Dyck path. St000058The order of a permutation. St000444The length of the maximal rise of a Dyck path. St000485The length of the longest cycle of a permutation. St000839The largest opener of a set partition. St000019The cardinality of the support of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000214The number of adjacencies of a permutation. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000271The chromatic index of a graph. St000304The load of a permutation. St000306The bounce count of a Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000441The number of successions of a permutation. St000502The number of successions of a set partitions. St000662The staircase size of the code of a permutation. St000731The number of double exceedences of a permutation. St000809The reduced reflection length of the permutation. St000831The number of indices that are either descents or recoils. St000932The number of occurrences of the pattern UDU in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. 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. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. 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 $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001274The number of indecomposable injective modules with projective dimension equal to two. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001726The number of visible inversions of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000025The number of initial rises of a Dyck path. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000147The largest part of an integer partition. St000166The depth minus 1 of an ordered tree. St000240The number of indices that are not small excedances. St000308The height of the tree associated to a permutation. St000335The difference of lower and upper interactions. St000381The largest part of an integer composition. St000392The length of the longest run of ones in a binary word. St000628The balance of a binary word. St000668The least common multiple of the parts of the partition. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000708The product of the parts of an integer partition. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000738The first entry in the last row of a standard tableau. St000740The last entry of a permutation. St000808The number of up steps of the associated bargraph. St000844The size of the largest block in the direct sum decomposition of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000982The length of the longest constant subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001062The maximal size of a block of a set partition. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001110The 3-dynamic chromatic number of a graph. 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. 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:
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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001372The length of a longest cyclic run of ones of a binary word. St001497The position of the largest weak excedence of a permutation. St001530The depth of a Dyck path. St001555The order of a signed permutation. St001589The nesting number of a perfect matching. St001733The number of weak left to right maxima of a Dyck path. St000094The depth of an ordered tree. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. 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 $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. 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. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. 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. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000454The largest eigenvalue of a graph if it is integral. St001330The hat guessing number of a graph. St001118The acyclic chromatic index of a graph. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000741The Colin de Verdière graph invariant. St000681The Grundy value of Chomp on Ferrers diagrams. St001645The pebbling number of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001596The number of two-by-two squares inside a skew partition. St001060The distinguishing index of a graph. St000259The diameter of a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001877Number of indecomposable injective modules with projective dimension 2. St001498The normalised height of a Nakayama algebra with magnitude 1. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. 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 $eA$. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000264The girth of a graph, which is not a tree. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000762The sum of the positions of the weak records of an integer composition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000455The second largest eigenvalue of a graph if it is integral. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001568The smallest positive integer that does not appear twice in the partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000675The number of centered multitunnels of a Dyck path. St000707The product of the factorials of the parts. St000770The major index of an integer partition when read from bottom to top. St000937The number of positive values of the symmetric group character corresponding to the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000260The radius of a connected graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. 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. St001624The breadth of a lattice. St001626The number of maximal proper sublattices of a lattice. St000091The descent variation of a composition. St001926Sparre Andersen's position of the maximum of a signed permutation. St000173The segment statistic of a semistandard tableau. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000456The monochromatic index of a connected graph. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001778The largest greatest common divisor of an element and its image in a permutation. St001273The projective dimension of the first term in an injective coresolution of the regular module. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. 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. 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. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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 $eA$. St001875The number of simple modules with projective dimension at most 1. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000284The Plancherel distribution on integer partitions. 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. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. 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. 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. St001128The exponens consonantiae of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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!