- FindStat

Your data matches 582 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000849
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000849: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0

Description

The number of 1/3-balanced pairs in a poset. A pair of elements

$x,y$ of a poset is

$\alpha$ -balanced if the proportion of linear extensions where

$x$ comes before

$y$ is between

$\alpha$ and

$1-\alpha$ . Kislitsyn [1] conjectured that every poset which is not a chain has a

$1/3$ -balanced pair. Brightwell, Felsner and Trotter [2] show that at least a

$(1-\sqrt 5)/10$ -balanced pair exists in posets which are not chains. Olson and Sagan [3] show that posets corresponding to skew Ferrers diagrams have a

$1/3$ -balanced pair.

Matching statistic: St001314
(load all 7 compositions to match this statistic)

Mapping

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [1,1,0,0]
 => 0
[[.,.],.]
 => [1,0,1,0]
 => 0
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => 0
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => 0
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => 0
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => 0
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => 0
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => 2
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => 2
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => 2
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => 0
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => 0
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => 0
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => 0
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => 0
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => 0
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => 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: St001397
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St001397: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 0

Description

Number of pairs of incomparable elements in a finite poset. For a finite poset

$(P,\leq)$ , this is the number of unordered pairs

$\{x,y\} \in \binom{P}{2}$ with

$x \not\leq y$ and

$y \not\leq x$ .

Matching statistic: St000100
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1

Description

The number of linear extensions of a poset.

Matching statistic: St001268
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St001268: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1

Description

The size of the largest ordinal summand in the poset. The ordinal sum of two posets

$P$ and

$Q$ is the poset having elements

$(p,0)$ and

$(q,1)$ for

$p\in P$ and

$q\in Q$ , and relations

$(a,0) < (b,0)$ if

$a < b$ in

$P$ ,

$(a,1) < (b,1)$ if

$a < b$ in

$Q$ , and

$(a,0) < (b,1)$ . This statistic is the maximal cardinality of a summand in the longest ordinal decomposition of a poset.

Matching statistic: St001779
(load all 3 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
St001779: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],.]
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => 3 = 2 + 1
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => 2 = 1 + 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1

Description

The order of promotion on the set of linear extensions of a poset.

Matching statistic: St000081

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000081: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0

Description

The number of edges of a graph.

Matching statistic: St000145
(load all 2 compositions to match this statistic)

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00307: Posets —promotion cycle type⟶ Integer partitions
St000145: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => [1]
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => [1]
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => [2]
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => [1]
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => [3]
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => [2]
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => [1]
 => 0

Description

The Dyson rank of a partition. This rank is defined as the largest part minus the number of parts. It was introduced by Dyson [1] in connection to Ramanujan's partition congruences

$p(5n+4) \equiv 0 \pmod 5$ and

$p(7n+6) \equiv 0 \pmod 7.$

Matching statistic: St000171

Mapping

Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000171: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => 0
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => 0
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => 1
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => 0
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => 0

Description

The degree of the graph. This is the maximal vertex degree of a graph.

Matching statistic: St000220
(load all 23 compositions to match this statistic)

Mapping

Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[.,[.,.]]
 => [2,1] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [1,3,2] => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,2,3] => 0
[[.,[.,.]],.]
 => [2,1,3] => [1,2,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,4,2,3] => 2
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,3,2,4] => 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [1,2,4,3] => 2
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,3,4,2] => 2
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,2,3,4] => 0
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,2,4,3] => 2
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,2,3,4] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [1,2,3,4] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,3,2,4] => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,2,3,4] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,3,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,2,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0

Description

The number of occurrences of the pattern 132 in a permutation.

The following 572 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000271The chromatic index of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000355The number of occurrences of the pattern 21-3. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001341The number of edges in the center of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001388The number of non-attacking neighbors of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001479The number of bridges of a graph. St001512The minimum rank of a graph. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001649The length of a longest trail in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001869The maximum cut size of a graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000047The number of standard immaculate tableaux of a given shape. St000078The number of alternating sign matrices whose left key is the permutation. St000086The number of subgraphs. St000088The row sums of the character table of the symmetric group. St000147The largest part of an integer partition. St000184The size of the centralizer of any permutation of given cycle type. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000228The size of a partition. St000255The number of reduced Kogan faces with the permutation as type. St000299The number of nonisomorphic vertex-induced subtrees. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000384The maximal part of the shifted composition of an integer partition. St000388The number of orbits of vertices of a graph under automorphisms. St000453The number of distinct Laplacian eigenvalues of a graph. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000468The Hosoya index of a graph. St000531The leading coefficient of the rook polynomial of an integer partition. St000667The greatest common divisor of the parts of the partition. St000784The maximum of the length and the largest part of the integer partition. St000812The sum of the entries in the column specified by the partition of the change of basis matrix from complete homogeneous symmetric functions to monomial symmetric functions. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000935The number of ordered refinements of an integer partition. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001093The detour number of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001352The number of internal nodes in the modular decomposition of a graph. St001360The number of covering relations in Young's lattice below a partition. St001367The smallest number which does not occur as degree of a vertex in a graph. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001612The number of coloured multisets of cycles such that the multiplicities of colours are given by a partition. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001674The number of vertices of the largest induced star graph in the graph. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001725The harmonious chromatic number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000063The number of linear extensions of a certain poset defined for an integer partition. St000108The number of partitions contained in the given partition. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000532The total number of rook placements on a Ferrers board. St000886The number of permutations with the same antidiagonal sums. St001400The total number of Littlewood-Richardson tableaux of given shape. St001814The number of partitions interlacing the given partition. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000057The Shynar inversion number of a standard tableau. St000091The descent variation of a composition. St000095The number of triangles of a graph. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000217The number of occurrences of the pattern 312 in a permutation. St000218The number of occurrences of the pattern 213 in a permutation. St000222The number of alignments in the permutation. St000225Difference between largest and smallest parts in a partition. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000293The number of inversions of a binary word. St000295The length of the border of a binary word. St000359The number of occurrences of the pattern 23-1. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000377The dinv defect of an integer partition. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000462The major index minus the number of excedences of a permutation. St000463The number of admissible inversions of a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000539The number of odd inversions of a permutation. St000572The dimension exponent of a set partition. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000747A variant of the major index of a set partition. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000769The major index of a composition regarded as a word. St000833The comajor index of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000868The aid statistic in the sense of Shareshian-Wachs. St000940The number of characters of the symmetric group whose value on the partition is zero. St001082The number of boxed occurrences of 123 in a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001176The size of a partition minus its first part. St001308The number of induced paths on three vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001377The major index minus the number of inversions of a permutation. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001535The number of cyclic alignments of a permutation. St001569The maximal modular displacement of a permutation. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001584The area statistic between a Dyck path and its bounce path. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001705The number of occurrences of the pattern 2413 in a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001727The number of invisible inversions of a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001867The number of alignments of type EN of a signed permutation. St000010The length of the partition. St000045The number of linear extensions of a binary tree. St000071The number of maximal chains in a poset. St000148The number of odd parts of a partition. St000160The multiplicity of the smallest part of a partition. St000240The number of indices that are not small excedances. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000346The number of coarsenings of a partition. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000475The number of parts equal to 1 in a partition. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000548The number of different non-empty partial sums of an integer partition. St000626The minimal period of a binary word. St000638The number of up-down runs of a permutation. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000708The product of the parts of an integer partition. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000820The number of compositions obtained by rotating the composition. St000847The number of standard Young tableaux whose descent set is the binary word. St000867The sum of the hook lengths in the first row of an integer partition. St000909The number of maximal chains of maximal size in a poset. St000933The number of multipartitions of sizes given by an integer partition. St000983The length of the longest alternating subword. St000988The orbit size of a permutation under Foata's bijection. St001127The sum of the squares of the parts of a partition. St001128The exponens consonantiae of a partition. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001282The number of graphs with the same chromatic polynomial. St001313The number of Dyck paths above the lattice path given by a binary word. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001415The length of the longest palindromic prefix of a binary word. St001740The number of graphs with the same symmetric edge polytope as the given graph. 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. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001933The largest multiplicity of a part in an integer partition. St000064The number of one-box pattern of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000219The number of occurrences of the pattern 231 in a permutation. St000454The largest eigenvalue of a graph if it is integral. St000632The jump number of the poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000850The number of 1/2-balanced pairs in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001399The distinguishing number of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001964The interval resolution global dimension of a poset. St001846The number of elements which do not have a complement in the lattice. St000701The protection number of a binary tree. St000741The Colin de Verdière graph invariant. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000422The energy of a graph, if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001438The number of missing boxes of a skew partition. St001541The Gini index of an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St001330The hat guessing number of a graph. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000523The number of 2-protected nodes of a rooted tree. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000974The length of the trunk of an ordered tree. St001175The size of a partition minus the hook length of the base cell. St001586The number of odd parts smaller than the largest even part in an integer partition. St001651The Frankl number of a lattice. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000700The protection number of an ordered tree. St001119The length of a shortest maximal path in a graph. St000674The number of hills of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000944The 3-degree of an integer partition. St001139The number of occurrences of hills of size 2 in a Dyck path. St001172The number of 1-rises at odd height of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000975The length of the boundary minus the length of the trunk of an ordered tree. St000655The length of the minimal rise of a Dyck path. St001394The genus 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. St000352The Elizalde-Pak rank of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000534The number of 2-rises of a permutation. 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. St000929The constant term of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000214The number of adjacencies of a permutation. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St001732The number of peaks visible from the left. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000011The number of touch points (or returns) of a Dyck path. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001625The Möbius invariant of a lattice. St000374The number of exclusive right-to-left minima of a permutation. St000386The number of factors DDU in a Dyck path. St000409The number of pitchforks in a binary tree. St000441The number of successions of a permutation. St000665The number of rafts of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000567The sum of the products of all pairs of parts. St000681The Grundy value of Chomp on Ferrers diagrams. St000687The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001857The number of edges in the reduced word graph of a signed permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001481The minimal height of a peak of a Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001949The rigidity index of a graph. St000482The (zero)-forcing number of a graph. St000906The length of the shortest maximal chain in a poset. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001205The number of non-simple indecomposable projective-injective modules of the algebra

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001723The differential of a graph. St000686The finitistic dominant dimension of a Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000317The cycle descent number of a permutation. St000323The minimal crossing number of a graph. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000699The toughness times the least common multiple of 1,. St000709The number of occurrences of 14-2-3 or 14-3-2. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000779The tier of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St000951The dimension of

$Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has

$Ext^i(D(A),A)=0$ for

$i>0$ . St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra

$A/I$ when

$I$ is the tilting module corresponding to the permutation in the Auslander algebra of

$K[x]/(x^n)$ . St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001961The sum of the greatest common divisors of all pairs of parts. St000025The number of initial rises of a Dyck path. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000056The decomposition (or block) number of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000306The bounce count of a Dyck path. St000486The number of cycles of length at least 3 of a permutation. St000544The cop number of a graph. St000617The number of global maxima of a Dyck path. St000627The exponent of a binary word. St000703The number of deficiencies of a permutation. St000710The number of big deficiencies of a permutation. St000732The number of double deficiencies of a permutation. St000776The maximal multiplicity of an eigenvalue in a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path)

$[c_0,c_1,...,c_{n−1}]$ by adding

$c_0$ to

$c_{n−1}$ . St000990The first ascent of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck 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. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001188The number of simple modules

$S$ with grade

$\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra

$A$ 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. St001191Number of simple modules

$S$ with

$Ext_A^i(S,A)=0$ for all

$i=0,1,...,g-1$ in the corresponding Nakayama algebra

$A$ with global dimension

$g$ . St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001272The number of graphs with the same degree sequence. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001340The cardinality of a minimal non-edge isolating set of a graph. St001344The neighbouring number of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001405The number of bonds in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001501The 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. St001510The number of self-evacuating linear extensions of a finite poset. St001518The number of graphs with the same ordinary spectrum as the given graph. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001665The number of pure excedances of a permutation. St001694The number of maximal dissociation sets in a graph. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001839The number of excedances of a set partition. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000258The burning number of a graph. St000273The domination number of a graph. St000309The number of vertices with even degree. St000335The difference of lower and upper interactions. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000955Number of times one has

$Ext^i(D(A),A)>0$ for

$i>0$ for the corresponding LNakayama algebra. St001027Number of simple modules with projective dimension equal to injective dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra

$eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001206The maximal dimension of an indecomposable projective

$eAe$ -module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module

$eA$ . St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001623The number of doubly irreducible elements of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001741The largest integer such that all patterns of this size are contained in the permutation. St001792The arboricity of a graph. St001829The common independence number of a graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000015The number of peaks of a Dyck path. St000314The number of left-to-right-maxima of a permutation. St000831The number of indices that are either descents or recoils. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000981The length of the longest zigzag subpath. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001875The number of simple modules with projective dimension at most 1. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001180Number of indecomposable injective modules with projective dimension at most 1. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000879The number of long braid edges in the graph of braid moves of a permutation. St001271The competition number of a graph. St001435The number of missing boxes in the first row. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000893The number of distinct diagonal sums of an alternating sign matrix. St001624The breadth of a lattice. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001301The first Betti number of the order complex associated with the poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001596The number of two-by-two squares inside a skew partition. St000680The Grundy value for Hackendot on posets. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001597The Frobenius rank of a skew partition. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000717The number of ordinal summands of a poset. St001545The second Elser number of a connected graph. St001717The largest size of an interval in a poset. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000479The Ramsey number of a graph.