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Your data matches 223 different statistics following compositions of up to 3 maps.
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Matching statistic: St000335
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
St000335: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 4
Description
The difference of lower and upper interactions.
An ''upper interaction'' in a Dyck path is the occurrence of a factor $0^k 1^k$ with $k \geq 1$ (see [[St000331]]), and a ''lower interaction'' is the occurrence of a factor $1^k 0^k$ with $k \geq 1$. In both cases, $1$ denotes an up-step $0$ denotes a a down-step.
Matching statistic: St001235
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 4
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001418
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001418: 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
St001418: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The stable Auslander algebra is by definition the stable endomorphism ring of the direct sum of all indecomposable modules.
Matching statistic: St001431
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001431: 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
St001431: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 4
Description
Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path.
The modified algebra B is obtained from the stable Auslander algebra kQ/I by deleting all relations which contain walks of length at least three (conjectural this step of deletion is not necessary as the stable higher Auslander algebras might be quadratic) and taking as B then the algebra kQ^(op)/J when J is the quadratic perp of the ideal I.
See http://www.findstat.org/DyckPaths/NakayamaAlgebras for the definition of Loewy length and Nakayama algebras associated to Dyck paths.
Matching statistic: St001652
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St001652: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4
Description
The length of a longest interval of consecutive numbers.
For a permutation $\pi=\pi_1,\dots,\pi_n$, this statistic returns the length of a longest subsequence $\pi_k,\dots,\pi_\ell$ such that $\pi_{i+1} = \pi_i + 1$ for $i\in\{k,\dots,\ell-1\}$.
Matching statistic: St001662
(load all 23 compositions to match this statistic)
(load all 23 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St001662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00069: Permutations —complement⟶ Permutations
St001662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [1,2] => 2
[1,0,1,0,1,0]
=> [1,2,3] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,3,1] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [2,1,3] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [4,2,1,3] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,1,2,3] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,4,1] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [3,2,1,4] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,4,1] => 3
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,3,1,4] => 2
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [1,3,2,4] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [1,2,3,4] => 4
Description
The length of the longest factor of consecutive numbers in a permutation.
Matching statistic: St000314
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St000314: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 1
[1,0,1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 1
[1,0,1,1,0,0]
=> [2,3,1] => [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => [2,3,1] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,2,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [4,2,3,1] => [3,1,4,2] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [4,1,3,2] => 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,4,1,2] => [2,4,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,4,3,1] => [4,3,1,2] => 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,4,2,1] => [4,2,1,3] => 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [3,1,2,4] => 2
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,4,2] => [1,4,2,3] => 2
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,3,4,1] => [4,1,2,3] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 4
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000374
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0]
=> [3,1,2] => [1,3,2] => [2,3,1] => 1
[1,1,0,0]
=> [2,3,1] => [2,3,1] => [3,1,2] => 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [2,3,4,1] => 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,3,4,2] => [2,4,1,3] => 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [4,2,1,3] => [3,1,4,2] => 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,4,3,2] => [3,4,2,1] => 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [4,1,2,3] => 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,2,3,5,4] => [2,3,4,5,1] => 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,2,4,5,3] => [2,3,5,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [3,1,4,5,2] => 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,5,2,4,3] => [3,2,5,4,1] => 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [2,5,1,3,4] => 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => [3,2,1,5,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,4,5,2,3] => 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,3,2,5,4] => [3,4,2,5,1] => 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,2,5,4,3] => [3,4,5,2,1] => 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,4,3,5,2] => [3,5,2,1,4] => 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [4,1,2,5,3] => 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [5,4,2,1,3] => [4,1,5,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => [4,5,2,3,1] => 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [5,1,2,3,4] => 4
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St000381
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [2] => [1,1] => 1
[1,1,0,0]
=> [1,0,1,0]
=> [1,1] => [2] => 2
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => [1,1,1] => 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => [1,2] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => [2,1] => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3] => [1,1,1] => 1
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => [3] => 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,3] => 3
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [2,1,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => [1,1,2] => 2
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [3,1] => 3
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => [2,1,1] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [4] => [1,1,1,1] => 1
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [4] => 4
Description
The largest part of an integer composition.
Matching statistic: St000444
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000444: 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
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,0,1,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
Description
The length of the maximal rise of a Dyck path.
The following 213 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000542The number of left-to-right-minima of a permutation. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000730The maximal arc length of a set partition. St000808The number of up steps of the associated bargraph. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001530The depth of a Dyck path. St001720The minimal length of a chain of small intervals in a lattice. St000214The number of adjacencies of a permutation. St000538The number of even inversions of a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000877The depth of the binary word interpreted as a path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001826The maximal number of leaves on a vertex of a graph. St000392The length of the longest run of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001330The hat guessing number of a graph. St000383The last part of an integer composition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St000439The position of the first down step of a Dyck path. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000454The largest eigenvalue of a graph if it is integral. St001651The Frankl number of a lattice. St001867The number of alignments of type EN of a signed permutation. St000352The Elizalde-Pak rank of a permutation. St000054The first entry of the permutation. St000441The number of successions of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000075The orbit size of a standard tableau under promotion. St000366The number of double descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000942The number of critical left to right maxima of the parking functions. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000056The decomposition (or block) number of a permutation. St000456The monochromatic index of a connected graph. St000492The rob statistic of a set partition. St000654The first descent of a permutation. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000991The number of right-to-left minima of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001461The number of topologically connected components of the chord diagram of a permutation. St001479The number of bridges of a graph. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001896The number of right descents of a signed permutations. St001937The size of the center of a parking function. St000234The number of global ascents of a permutation. St000839The largest opener of a set partition. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001545The second Elser number of a connected graph. St001712The number of natural descents of a standard Young tableau. St001728The number of invisible descents of a permutation. St001781The interlacing number of a set partition. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001904The length of the initial strictly increasing segment of a parking function. St000230Sum of the minimal elements of the blocks of a set partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001644The dimension of a graph. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St001060The distinguishing index of a graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000264The girth of a graph, which is not a tree. St000893The number of distinct diagonal sums of an alternating sign matrix. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000259The diameter of a connected graph. St000527The width of the poset. St000909The number of maximal chains of maximal size in a 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 $eA$. St001645The pebbling number of a connected graph. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000736The last entry in the first row of a semistandard tableau. St001569The maximal modular displacement of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001520The number of strict 3-descents. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001948The number of augmented double ascents of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000079The number of alternating sign matrices for a given Dyck path. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000174The flush statistic of a semistandard tableau. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000315The number of isolated vertices of a graph. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000958The number of Bruhat factorizations of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. 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. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001413Half the length of the longest even length palindromic prefix of a binary word. St001487The number of inner corners of a skew partition. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001823The Stasinski-Voll length of a signed permutation. 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. St001862The number of crossings of a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001935The number of ascents in a parking function. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000166The depth minus 1 of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000308The height of the tree associated to a permutation. St000365The number of double ascents of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000498The lcs statistic of a set partition. St000522The number of 1-protected nodes of a rooted tree. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000632The jump number of the poset. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000739The first entry in the last row of a semistandard tableau. St000836The number of descents of distance 2 of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001153The number of blocks with even minimum in a set partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001623The number of doubly irreducible elements of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001684The reduced word complexity of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001928The number of non-overlapping descents in a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000007The number of saliances of the permutation. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
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