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Your data matches 185 different statistics following compositions of up to 3 maps.
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Matching statistic: St000269
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Mp00011: Binary trees —to graph⟶ Graphs
St000269: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000269: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 4
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 8
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
Description
The number of acyclic orientations of a graph.
Matching statistic: St000270
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 4
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 8
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
Description
The number of forests contained in a graph.
That is, for a graph $G = (V,E)$ with vertices $V$ and edges $E$, the number of subsets $E' \subseteq E$ for which the subgraph $(V,E')$ is acyclic.
If $T_G(x,y)$ is the Tutte polynomial [2] of $G$, then the number of forests contained in $G$ is given by $T_G(2,1)$.
Matching statistic: St000972
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St000972: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000972: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 4
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 8
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
Description
The composition number of a graph.
This is the number of set partitions of the vertex set of the graph, such that the subgraph induced by each block is connected.
Matching statistic: St001474
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00011: Binary trees —to graph⟶ Graphs
St001474: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001474: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> ([],1)
=> 1
[[.,.],.]
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> ([(0,2),(1,2)],3)
=> 4
[[[.,.],.],.]
=> ([(0,2),(1,2)],3)
=> 4
[[.,[[.,.],.]],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(2,3)],4)
=> 8
[[[[.,.],.],.],.]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[.,[[[.,.],.],.]],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[.,.],.],[.,.]],.]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 16
[[[[[.,.],.],.],.],.]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 32
[[[[[[.,.],.],.],.],.],.]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
Description
The evaluation of the Tutte polynomial of the graph at (x,y) equal to (2,-1).
Matching statistic: St000040
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(load all 10 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000040: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [2,1] => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [2,3,1] => 4
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3,1,2] => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [2,4,1,3] => 8
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 8
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 8
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => 16
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => 16
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => 16
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 16
[[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => 32
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => 32
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => 32
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => 32
[[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [3,1,7,2,4,5,6] => 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,1,2,7,3,5,6] => 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [5,1,2,3,7,4,6] => 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [6,1,2,3,4,7,5] => 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => 64
Description
The number of regions of the inversion arrangement of a permutation.
The inversion arrangement $\mathcal{A}_w$ consists of the hyperplanes $x_i-x_j=0$ such that $(i,j)$ is an inversion of $w$.
Postnikov [4] conjectured that the number of regions in $\mathcal{A}_w$ equals the number of permutations in the interval $[id,w]$ in the strong Bruhat order if and only if $w$ avoids $4231$, $35142$, $42513$, $351624$. This conjecture was proved by Hultman-Linusson-Shareshian-Sjöstrand [1].
Oh-Postnikov-Yoo [3] showed that the number of regions of $\mathcal{A}_w$ is $|\chi_{G_w}(-1)|$ where $\chi_{G_w}$ is the chromatic polynomial of the inversion graph $G_w$. This is the graph with vertices ${1,2,\ldots,n}$ and edges $(i,j)$ for $i\lneq j$ $w_i\gneq w_j$.
For a permutation $w=w_1\cdots w_n$, Lewis-Morales [2] and Hultman (see appendix in [2]) showed that this number equals the number of placements of $n$ non-attacking rooks on the south-west Rothe diagram of $w$.
Matching statistic: St001645
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00011: Binary trees —to graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00011: Binary trees —to graph⟶ Graphs
St001645: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [.,.]
=> ([],1)
=> 1
[[.,.],.]
=> [.,[.,.]]
=> ([(0,1)],2)
=> 2
[[.,[.,.]],.]
=> [.,[[.,.],.]]
=> ([(0,2),(1,2)],3)
=> 4
[[[.,.],.],.]
=> [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 4
[[.,[[.,.],.]],.]
=> [.,[[[.,.],.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[.,.],[.,.]],.]
=> [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[[[.,.],.],.],.]
=> [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 8
[[.,[[[.,.],.],.]],.]
=> [.,[[[[.,.],.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[.,.],[[.,.],.]],.]
=> [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[[.,.],.],[.,.]],.]
=> [[[.,.],.],[[.,.],.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[[[[.,.],.],.],.],.]
=> [[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> [.,[[[[[.,.],.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> [[.,.],[[[[.,.],.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> [[[.,.],.],[[[.,.],.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> [[[[.,.],.],.],[[.,.],.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[[[[[.,.],.],.],.],.],.]
=> [[[[[.,.],.],.],.],[.,.]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [.,[[[[[[.,.],.],.],.],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [[.,.],[[[[[.,.],.],.],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [[[.,.],.],[[[[.,.],.],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [[[[.,.],.],.],[[[.,.],.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [[[[[.,.],.],.],.],[[.,.],.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [[[[[[.,.],.],.],.],.],[.,.]]
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 64
Description
The pebbling number of a connected graph.
Matching statistic: St001800
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001800: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
St001800: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> 4
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 8
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 8
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 8
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 16
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 16
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 16
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 32
[[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 64
Description
The number of 3-Catalan paths having this Dyck path as first and last coordinate projections.
A 3-Catalan path is a lattice path from $(0,0,0)$ to $(n,n,n)$ consisting of steps $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$ such that for each point $(x,y,z)$ on the path we have $x \geq y \geq z$.
Its first and last coordinate projections, denoted by $D_{xy}$ and $D_{yz}$, are the Dyck paths obtained by projecting the Catalan path onto the $x,y$-plane and the $y,z$-plane, respectively.
For a given Dyck path $D$ this is the number of Catalan paths $C$ such that $D_{xy}(C) = D_{yz}(C) = D$.
If $D$ is of semilength $n$, $r_i(D)$ denotes the number of downsteps between the $i$-th and $(i+1)$-st upstep, and $s_i(D)$ denotes the number of upsteps between the $i$-th and $(i+1)$-st downstep, then this number is given by $\prod\limits_{i=1}^{n-1} \binom{r_i(D) + s_i(D)}{r_i(D)}$.
Matching statistic: St000038
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St000038: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> [1,0]
=> 1
[[.,.],.]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 4
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 8
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 8
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 8
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 16
[[[.,.],[[.,.],.]],.]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 16
[[[[.,.],.],[.,.]],.]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 16
[[[[[.,.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 32
[[[[[[.,.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 64
Description
The product of the heights of the descending steps of a Dyck path.
A Dyck path with 2n letters defines a partition inside an [n] x [n] board. This statistic counts the number of placements of n non-attacking rooks on the board.
By the Gessel-Viennot theory of orthogonal polynomials this corresponds to the 0-moment of the Hermite polynomials.
Summing the values of the statistic over all Dyck paths of fixed size n the number of perfect matchings (2n+1)!! is obtained: up steps are openers, down steps closers and the rooks determine a pairing of openers and closers.
Matching statistic: St000207
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => [1]
=> 1
[[.,.],.]
=> [1,2] => [1,2] => [2]
=> 2
[[.,[.,.]],.]
=> [2,1,3] => [1,2,3] => [3]
=> 4
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [3]
=> 4
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,2,3,4] => [4]
=> 8
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,3,4] => [4]
=> 8
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [4]
=> 8
[[.,[[[.,.],.],.]],.]
=> [2,3,4,1,5] => [1,2,3,4,5] => [5]
=> 16
[[[.,.],[[.,.],.]],.]
=> [1,3,4,2,5] => [1,2,3,4,5] => [5]
=> 16
[[[[.,.],.],[.,.]],.]
=> [1,2,4,3,5] => [1,2,3,4,5] => [5]
=> 16
[[[[[.,.],.],.],.],.]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5]
=> 16
[[.,[[[[.,.],.],.],.]],.]
=> [2,3,4,5,1,6] => [1,2,3,4,5,6] => [6]
=> 32
[[[.,.],[[[.,.],.],.]],.]
=> [1,3,4,5,2,6] => [1,2,3,4,5,6] => [6]
=> 32
[[[[.,.],.],[[.,.],.]],.]
=> [1,2,4,5,3,6] => [1,2,3,4,5,6] => [6]
=> 32
[[[[[.,.],.],.],[.,.]],.]
=> [1,2,3,5,4,6] => [1,2,3,4,5,6] => [6]
=> 32
[[[[[[.,.],.],.],.],.],.]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [6]
=> 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [2,3,4,5,6,1,7] => [1,2,3,4,5,6,7] => [7]
=> 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,3,4,5,6,2,7] => [1,2,3,4,5,6,7] => [7]
=> 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,2,4,5,6,3,7] => [1,2,3,4,5,6,7] => [7]
=> 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,2,3,5,6,4,7] => [1,2,3,4,5,6,7] => [7]
=> 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,2,3,4,6,5,7] => [1,2,3,4,5,6,7] => [7]
=> 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7]
=> 64
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000289
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000289: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 1 => 1
[[.,.],.]
=> [1,1,0,0]
=> [2] => 10 => 2
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [3] => 100 => 4
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [3] => 100 => 4
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [4] => 1000 => 8
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [4] => 1000 => 8
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1000 => 8
[[.,[[[.,.],.],.]],.]
=> [1,1,0,1,1,1,0,0,0,0]
=> [5] => 10000 => 16
[[[.,.],[[.,.],.]],.]
=> [1,1,1,0,0,1,1,0,0,0]
=> [5] => 10000 => 16
[[[[.,.],.],[.,.]],.]
=> [1,1,1,1,0,0,0,1,0,0]
=> [5] => 10000 => 16
[[[[[.,.],.],.],.],.]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => 10000 => 16
[[.,[[[[.,.],.],.],.]],.]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [6] => 100000 => 32
[[[.,.],[[[.,.],.],.]],.]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [6] => 100000 => 32
[[[[.,.],.],[[.,.],.]],.]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [6] => 100000 => 32
[[[[[.,.],.],.],[.,.]],.]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [6] => 100000 => 32
[[[[[[.,.],.],.],.],.],.]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => 100000 => 32
[[.,[[[[[.,.],.],.],.],.]],.]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [7] => 1000000 => 64
[[[.,.],[[[[.,.],.],.],.]],.]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> [7] => 1000000 => 64
[[[[.,.],.],[[[.,.],.],.]],.]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [7] => 1000000 => 64
[[[[[.,.],.],.],[[.,.],.]],.]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> [7] => 1000000 => 64
[[[[[[.,.],.],.],.],[.,.]],.]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [7] => 1000000 => 64
[[[[[[[.,.],.],.],.],.],.],.]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => 1000000 => 64
Description
The decimal representation of a binary word.
The following 175 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001365The number of lattice paths of the same length weakly above the path given by a binary word. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000412The number of binary trees with the same underlying unordered tree. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000418The number of Dyck paths that are weakly below a Dyck path. St000827The decimal representation of a binary word with a leading 1. St000421The number of Dyck paths that are weakly below a Dyck path, except for the path itself. St000694The number of affine bounded permutations that project to a given permutation. St000511The number of invariant subsets when acting with a permutation of given cycle type. St000343The number of spanning subgraphs of a graph. St000109The number of elements less than or equal to the given element in Bruhat order. St000948The chromatic discriminant of a graph. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001815The number of order preserving surjections from a poset to a total order. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St001531Number of partial orders contained in the poset determined by the Dyck path. St001959The product of the heights of the peaks of a Dyck path. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000413The number of ordered trees with the same underlying unordered tree. St001616The number of neutral elements in a lattice. St000525The number of posets with the same zeta polynomial. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St000918The 2-limited packing number of a graph. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001286The annihilation number of a graph. St001304The number of maximally independent sets of vertices of a graph. St001512The minimum rank of a graph. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St000060The greater neighbor of the maximum. St000272The treewidth of a graph. St000340The number of non-final maximal constant sub-paths of length greater than one. St000388The number of orbits of vertices of a graph under automorphisms. St000443The number of long tunnels of a Dyck path. St000482The (zero)-forcing number of a graph. St000526The number of posets with combinatorially isomorphic order polytopes. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St000539The number of odd inversions of a permutation. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000570The Edelman-Greene number of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000796The stat' of a permutation. St000886The number of permutations with the same antidiagonal sums. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001726The number of visible inversions of a permutation. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001754The number of tolerances of a finite lattice. St001760The number of prefix or suffix reversals needed to sort a permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St000024The number of double up and double down steps of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000136The dinv of a parking function. St000141The maximum drop size of a permutation. St000144The pyramid weight of the Dyck path. St000154The sum of the descent bottoms of a permutation. St000171The degree of the graph. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000204The number of internal nodes of a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000222The number of alignments in the permutation. St000271The chromatic index of a graph. St000307The number of rowmotion orbits of a poset. St000316The number of non-left-to-right-maxima of a permutation. St000331The number of upper interactions of a Dyck path. St000332The positive inversions of an alternating sign matrix. St000355The number of occurrences of the pattern 21-3. St000357The number of occurrences of the pattern 12-3. St000359The number of occurrences of the pattern 23-1. St000363The number of minimal vertex covers of a graph. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000376The bounce deficit of a Dyck path. St000491The number of inversions of a set partition. St000496The rcs statistic of a set partition. St000538The number of even inversions of a permutation. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000677The standardized bi-alternating inversion number of a permutation. St000680The Grundy value for Hackendot on posets. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000743The number of entries in a standard Young tableau such that the next integer is a neighbour. St000747A variant of the major index of a set partition. St000797The stat`` of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001104The number of descents of the invariant in a tensor power of the adjoint representation of the rank two general linear group. St001119The length of a shortest maximal path in a graph. 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. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001209The pmaj statistic of a parking function. St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. 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. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over 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. St001315The dissociation number of a graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001343The dimension of the reduced incidence algebra of a poset. St001350Half of the Albertson index of a graph. St001388The number of non-attacking neighbors of a permutation. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001480The number of simple summands of the module J^2/J^3. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001535The number of cyclic alignments of a permutation. St001565The number of arithmetic progressions of length 2 in a permutation. St001580The acyclic chromatic number of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001584The area statistic between a Dyck path and its bounce path. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001727The number of invisible inversions of a permutation. St001731The factorization defect of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001843The Z-index of a set partition. St001874Lusztig's a-function for the symmetric group. St001883The mutual visibility number of a graph. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. 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. St001875The number of simple modules with projective dimension at most 1. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000045The number of linear extensions of a binary tree. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000422The energy of a graph, if it is integral.
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