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Your data matches 81 different statistics following compositions of up to 3 maps.
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Matching statistic: St001314
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Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Matching statistic: St000356
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000356: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [2,1] => 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [2,3,1] => 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [3,1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [2,3,4,1] => 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [4,3,1,5,2] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 0
Description
The number of occurrences of the pattern 13-2.
See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $13\!\!-\!\!2$.
Matching statistic: St000424
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000424: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St000424: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => [1] => 0
[1,1] => [1,0,1,0]
=> [2,1] => [2,1] => 0
[2] => [1,1,0,0]
=> [1,2] => [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => 1
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 0
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,4,3] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,3,2,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,4,2,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [2,4,1,3] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0
Description
The number of occurrences of the pattern 132 or of the pattern 231 in a permutation.
Matching statistic: St001723
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2] => ([],2)
=> ([],2)
=> ([],1)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[3] => ([],3)
=> ([],3)
=> ([],1)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[4] => ([],4)
=> ([],4)
=> ([],1)
=> 0
Description
The differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
The differential of a graph is the maximal differential of a set of vertices.
Matching statistic: St001724
Values
[1] => ([],1)
=> ([],1)
=> ([],1)
=> 0
[1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 0
[2] => ([],2)
=> ([],2)
=> ([],1)
=> 0
[1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ([(1,2)],3)
=> 0
[2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 0
[3] => ([],3)
=> ([],3)
=> ([],1)
=> 0
[1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> 1
[1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[1,3] => ([(2,3)],4)
=> ([(2,3)],4)
=> ([(1,2)],3)
=> 0
[2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,2] => ([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> 0
[3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 0
[4] => ([],4)
=> ([],4)
=> ([],1)
=> 0
Description
The 2-packing differential of a graph.
The external neighbourhood (or boundary) of a set of vertices $S\subseteq V(G)$ is the set of vertices not in $S$ which are adjacent to a vertex in $S$.
The differential of a set of vertices $S\subseteq V(G)$ is the difference of the size of the external neighbourhood of $S$ and the size of $S$.
A set $S\subseteq V(G)$ is $2$-packing if the closed neighbourhoods of any two vertices in $S$ have empty intersection.
The $2$-packing differential of a graph is the maximal differential of any $2$-packing set of vertices.
Matching statistic: St001282
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001282: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001282: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => ([],1)
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,1] => [2] => ([],2)
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2] => [1,1] => ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1] => [3] => ([],3)
=> ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1] => [1,2] => ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,1] => [4] => ([],4)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,1] => [1,3] => ([(2,3)],4)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
Description
The number of graphs with the same chromatic polynomial.
Matching statistic: St000369
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 0
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000376
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00124: Dyck paths —Adin-Bagno-Roichman transformation⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1,0]
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
Description
The bounce deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St000435
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000435: Permutations ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000435: Permutations ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> [1] => ? = 0
[1,1] => [1,0,1,0]
=> [1,0,1,0]
=> [2,1] => 0
[2] => [1,1,0,0]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1,3] => 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,3,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [3,1,2] => 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
Description
The number of occurrences of the pattern 213 or of the pattern 231 in a permutation.
Matching statistic: St000555
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> {{1}}
=> ? = 0
[1,1] => [1,0,1,0]
=> [1,1,0,0]
=> {{1,2}}
=> 0
[2] => [1,1,0,0]
=> [1,0,1,0]
=> {{1},{2}}
=> 0
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
[3] => [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 0
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 0
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 0
Description
The number of occurrences of the pattern {{1,3},{2}} in a set partition.
The following 71 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000961The shifted major index of a permutation. St001845The number of join irreducibles minus the rank of a lattice. St001651The Frankl number of a lattice. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001470The cyclic holeyness of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001344The neighbouring number of a permutation. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000264The girth of a graph, which is not a tree. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001570The minimal number of edges to add to make a graph Hamiltonian. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. St001890The maximum magnitude of the Möbius function of a poset. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St000259The diameter of a connected graph. St000260The radius of a connected graph. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St001645The pebbling number of a connected graph. 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 $eA$. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000102The charge of a semistandard tableau. St000478Another weight of a partition according to Alladi. 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. St000934The 2-degree of an integer partition. St001556The number of inversions of the third entry of a permutation. St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. 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. St001118The acyclic chromatic index of a graph. 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 $K[x]/(x^n)$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001569The maximal modular displacement of a permutation. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001568The smallest positive integer that does not appear twice in the partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition.
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