- FindStat

Your data matches 236 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000233
(load all 106 compositions to match this statistic)

Mapping

Mp00091: Set partitions —rotate increasing⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1},{2,3}}
 => 0
{{1,3},{2}}
 => {{1,2},{3}}
 => 0
{{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1},{2,3,4}}
 => 0
{{1,2,4},{3}}
 => {{1,2,3},{4}}
 => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => 1
{{1,2},{3},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1,3,4},{2}}
 => {{1,2,4},{3}}
 => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => 0
{{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
{{1},{2,3},{4}}
 => {{1},{2},{3,4}}
 => 0
{{1,4},{2},{3}}
 => {{1,2},{3},{4}}
 => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0

Description

The number of nestings of a set partition. This is given by the number of

$i < i' < j' < j$ such that

$i,j$ are two consecutive entries on one block, and

$i',j'$ are consecutive entries in another block.

Matching statistic: St000407
(load all 185 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St000407: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => 0
{{1,2}}
 => [2,1] => 0
{{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => [2,1,3] => 0
{{1,3},{2}}
 => [3,2,1] => 0
{{1},{2,3}}
 => [1,3,2] => 0
{{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 0

Description

The number of occurrences of the pattern 2143 in a permutation. A permutation

$\pi$ avoids this pattern if and only if it is ''vexillary'' as introduced in [1].

Matching statistic: St001208
(load all 185 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => 1 = 0 + 1
{{1,2}}
 => [2,1] => 1 = 0 + 1
{{1},{2}}
 => [1,2] => 1 = 0 + 1
{{1,2,3}}
 => [2,3,1] => 1 = 0 + 1
{{1,2},{3}}
 => [2,1,3] => 1 = 0 + 1
{{1,3},{2}}
 => [3,2,1] => 1 = 0 + 1
{{1},{2,3}}
 => [1,3,2] => 1 = 0 + 1
{{1},{2},{3}}
 => [1,2,3] => 1 = 0 + 1
{{1,2,3,4}}
 => [2,3,4,1] => 1 = 0 + 1
{{1,2,3},{4}}
 => [2,3,1,4] => 1 = 0 + 1
{{1,2,4},{3}}
 => [2,4,3,1] => 1 = 0 + 1
{{1,2},{3,4}}
 => [2,1,4,3] => 2 = 1 + 1
{{1,2},{3},{4}}
 => [2,1,3,4] => 1 = 0 + 1
{{1,3,4},{2}}
 => [3,2,4,1] => 1 = 0 + 1
{{1,3},{2,4}}
 => [3,4,1,2] => 1 = 0 + 1
{{1,3},{2},{4}}
 => [3,2,1,4] => 1 = 0 + 1
{{1,4},{2,3}}
 => [4,3,2,1] => 1 = 0 + 1
{{1},{2,3,4}}
 => [1,3,4,2] => 1 = 0 + 1
{{1},{2,3},{4}}
 => [1,3,2,4] => 1 = 0 + 1
{{1,4},{2},{3}}
 => [4,2,3,1] => 1 = 0 + 1
{{1},{2,4},{3}}
 => [1,4,3,2] => 1 = 0 + 1
{{1},{2},{3,4}}
 => [1,2,4,3] => 1 = 0 + 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 1 = 0 + 1

Description

The number of connected components of the quiver of

$A/T$ when

$T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra

$A$ of

$K[x]/(x^n)$ .

Matching statistic: St000232
(load all 110 compositions to match this statistic)

Mapping

Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00219: Set partitions —inverse Yip⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => {{1}}
 => {{1}}
 => 0
{{1,2}}
 => {{1,2}}
 => {{1,2}}
 => 0
{{1},{2}}
 => {{1},{2}}
 => {{1},{2}}
 => 0
{{1,2,3}}
 => {{1,2,3}}
 => {{1,2,3}}
 => 0
{{1,2},{3}}
 => {{1,3},{2}}
 => {{1},{2,3}}
 => 0
{{1,3},{2}}
 => {{1},{2,3}}
 => {{1,3},{2}}
 => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => {{1,2},{3}}
 => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => {{1},{2},{3}}
 => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => {{1,2,3,4}}
 => 0
{{1,2,3},{4}}
 => {{1,2,4},{3}}
 => {{1,2},{3,4}}
 => 0
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => {{1},{2,3,4}}
 => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => {{1,3},{2,4}}
 => 1
{{1,2},{3},{4}}
 => {{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => 0
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => {{1,3,4},{2}}
 => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => {{1,4},{2,3}}
 => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => {{1},{2,4},{3}}
 => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => {{1,2,4},{3}}
 => 0
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => {{1,2,3},{4}}
 => 0
{{1},{2,3},{4}}
 => {{1,2},{3},{4}}
 => {{1,2},{3},{4}}
 => 0
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => {{1,4},{2},{3}}
 => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => {{1},{2,3},{4}}
 => 0
{{1},{2},{3,4}}
 => {{1},{2,3},{4}}
 => {{1,3},{2},{4}}
 => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => 0

Description

The number of crossings of a set partition. This is given by the number of

$i < i' < j < j'$ such that

$i,j$ are two consecutive entries on one block, and

$i',j'$ are consecutive entries in another block.

Matching statistic: St000375
(load all 31 compositions to match this statistic)

Mapping

Mp00176: Set partitions —rotate decreasing⟶ Set partitions
Mp00080: Set partitions —to permutation⟶ Permutations
St000375: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => {{1}}
 => [1] => 0
{{1,2}}
 => {{1,2}}
 => [2,1] => 0
{{1},{2}}
 => {{1},{2}}
 => [1,2] => 0
{{1,2,3}}
 => {{1,2,3}}
 => [2,3,1] => 0
{{1,2},{3}}
 => {{1,3},{2}}
 => [3,2,1] => 0
{{1,3},{2}}
 => {{1},{2,3}}
 => [1,3,2] => 0
{{1},{2,3}}
 => {{1,2},{3}}
 => [2,1,3] => 0
{{1},{2},{3}}
 => {{1},{2},{3}}
 => [1,2,3] => 0
{{1,2,3,4}}
 => {{1,2,3,4}}
 => [2,3,4,1] => 0
{{1,2,3},{4}}
 => {{1,2,4},{3}}
 => [2,4,3,1] => 0
{{1,2,4},{3}}
 => {{1,3,4},{2}}
 => [3,2,4,1] => 0
{{1,2},{3,4}}
 => {{1,4},{2,3}}
 => [4,3,2,1] => 1
{{1,2},{3},{4}}
 => {{1,4},{2},{3}}
 => [4,2,3,1] => 0
{{1,3,4},{2}}
 => {{1},{2,3,4}}
 => [1,3,4,2] => 0
{{1,3},{2,4}}
 => {{1,3},{2,4}}
 => [3,4,1,2] => 0
{{1,3},{2},{4}}
 => {{1},{2,4},{3}}
 => [1,4,3,2] => 0
{{1,4},{2,3}}
 => {{1,2},{3,4}}
 => [2,1,4,3] => 0
{{1},{2,3,4}}
 => {{1,2,3},{4}}
 => [2,3,1,4] => 0
{{1},{2,3},{4}}
 => {{1,2},{3},{4}}
 => [2,1,3,4] => 0
{{1,4},{2},{3}}
 => {{1},{2},{3,4}}
 => [1,2,4,3] => 0
{{1},{2,4},{3}}
 => {{1,3},{2},{4}}
 => [3,2,1,4] => 0
{{1},{2},{3,4}}
 => {{1},{2,3},{4}}
 => [1,3,2,4] => 0
{{1},{2},{3},{4}}
 => {{1},{2},{3},{4}}
 => [1,2,3,4] => 0
{{1},{2},{3},{4},{5}}
 => {{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => 0

Description

The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . Given a permutation

$\pi = [\pi_1,\ldots,\pi_n]$ , this statistic counts the number of position

$j$ such that

$\pi_j < j$ and there exist indices

$i,k$ with

$i < j < k$ and

$\pi_i > \pi_j > \pi_k$ . See also [[St000213]] and [[St000119]].

Matching statistic: St000404
(load all 24 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000404: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 0
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,3,4,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,4,1,2] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,4,1] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,4,2,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,1,2] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [2,4,1,3] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,2,3,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,4,3,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [3,2,1,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [3,1,4,2] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [3,1,4,5,2] => 0

Description

The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. A permutation avoids these two pattern if and only if it is an ''input-restricted deques'', see [1].

Matching statistic: St000406
(load all 57 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St000406: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1] => 0
{{1,2}}
 => [2,1] => [2,1] => 0
{{1},{2}}
 => [1,2] => [1,2] => 0
{{1,2,3}}
 => [2,3,1] => [1,3,2] => 0
{{1,2},{3}}
 => [2,1,3] => [3,1,2] => 0
{{1,3},{2}}
 => [3,2,1] => [1,2,3] => 0
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => 0
{{1},{2},{3}}
 => [1,2,3] => [3,2,1] => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,3,4,2] => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [3,4,1,2] => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1,2,4] => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,4,1] => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [3,4,2,1] => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [4,1,3,2] => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,4,2,3] => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [4,3,1,2] => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [2,4,1,3] => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [4,2,1,3] => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [4,3,2,1] => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,2,3,4] => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [2,4,3,1] => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [3,2,1,4] => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [3,1,4,2] => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [3,1,4,5,2] => 0

Description

The number of occurrences of the pattern 3241 in a permutation.

Matching statistic: St000687
(load all 21 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000687: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1,0]
 => 0
{{1,2}}
 => [2,1] => [1,1,0,0]
 => 0
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => 0
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => 0
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => 0
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => 0
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => 0
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 0

Description

The dimension of

$Hom(I,P)$ for the LNakayama algebra of a Dyck path. In this expression,

$I$ is the direct sum of all injective non-projective indecomposable modules and

$P$ is the direct sum of all projective non-injective indecomposable modules. This statistic was discussed in [Theorem 5.7, 1].

Matching statistic: St001175
(load all 18 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => [1]
 => 0
{{1,2}}
 => [2,1] => [2]
 => 0
{{1},{2}}
 => [1,2] => [1,1]
 => 0
{{1,2,3}}
 => [2,3,1] => [2,1]
 => 0
{{1,2},{3}}
 => [2,1,3] => [2,1]
 => 0
{{1,3},{2}}
 => [3,2,1] => [3]
 => 0
{{1},{2,3}}
 => [1,3,2] => [2,1]
 => 0
{{1},{2},{3}}
 => [1,2,3] => [1,1,1]
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,1]
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => [2,1,1]
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => [3,1]
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => [2,2]
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,1]
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => [3,1]
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => [2,1,1]
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,1]
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => [4]
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => [2,1,1]
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,1,1]
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => [3,1]
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,1]
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,1,1]
 => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,1,1,1]
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,1,1,1,1]
 => 0

Description

The size of a partition minus the hook length of the base cell. This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.

Matching statistic: St001301
(load all 15 compositions to match this statistic)

Mapping

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001301: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

{{1}}
 => [1] => ([],1)
 => 0
{{1,2}}
 => [2,1] => ([],2)
 => 0
{{1},{2}}
 => [1,2] => ([(0,1)],2)
 => 0
{{1,2,3}}
 => [2,3,1] => ([(1,2)],3)
 => 0
{{1,2},{3}}
 => [2,1,3] => ([(0,2),(1,2)],3)
 => 0
{{1,3},{2}}
 => [3,2,1] => ([],3)
 => 0
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2)],3)
 => 0
{{1},{2},{3}}
 => [1,2,3] => ([(0,2),(2,1)],3)
 => 0
{{1,2,3,4}}
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 0
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 0
{{1,2,4},{3}}
 => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 0
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 1
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 0
{{1,3,4},{2}}
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 0
{{1,3},{2,4}}
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 0
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 0
{{1,4},{2,3}}
 => [4,3,2,1] => ([],4)
 => 0
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 0
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
{{1,4},{2},{3}}
 => [4,2,3,1] => ([(2,3)],4)
 => 0
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 0
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 0
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 0
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 0

Description

The first Betti number of the order complex associated with the poset. The order complex of a poset is the simplicial complex whose faces are the chains of the poset. This statistic is the rank of the first homology group of the order complex.

The following 226 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001705The number of occurrences of the pattern 2413 in a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000183The side length of the Durfee square of an integer partition. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000057The Shynar inversion number of a standard tableau. St000119The number of occurrences of the pattern 321 in a permutation. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000142The number of even parts of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000223The number of nestings in the permutation. St000256The number of parts from which one can substract 2 and still get an integer partition. St000355The number of occurrences of the pattern 21-3. St000356The number of occurrences of the pattern 13-2. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000360The number of occurrences of the pattern 32-1. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000660The number of rises of length at least 3 of a Dyck path. St000664The number of right ropes of a permutation. St000731The number of double exceedences of a permutation. St000768The number of peaks in an integer composition. St000871The number of very big ascents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001092The number of distinct even parts of 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. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001394The genus of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001537The number of cyclic crossings of a permutation. St001596The number of two-by-two squares inside a skew partition. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001867The number of alignments of type EN of a signed permutation. St001871The number of triconnected components of a graph. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000775The multiplicity of the largest eigenvalue in a graph. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000913The number of ways to refine the partition into singletons. St000920The logarithmic height of a Dyck path. St000935The number of ordered refinements of an integer partition. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001282The number of graphs with the same chromatic polynomial. St001393The induced matching number of a graph. St001481The minimal height of a peak of a Dyck path. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001597The Frobenius rank of a skew partition. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001261The Castelnuovo-Mumford regularity of a graph. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000516The number of stretching pairs of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000570The Edelman-Greene number of a permutation. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000750The number of occurrences of the pattern 4213 in a permutation. St000562The number of internal points of a set partition. St000650The number of 3-rises of a permutation. St000661The number of rises of length 3 of a Dyck path. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000800The number of occurrences of the vincular pattern |231 in 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. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000872The number of very big descents of a permutation. St000921The number of internal inversions of a binary word. St000931The number of occurrences of the pattern UUU in a Dyck path. St000963The 2-shifted major index of a permutation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001552The number of inversions between excedances and fixed points of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St001162The minimum jump of a permutation. St000842The breadth of a permutation. St001651The Frankl number of a lattice. St000068The number of minimal elements in a poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000219The number of occurrences of the pattern 231 in a permutation. St001845The number of join irreducibles minus the rank of a lattice. St000914The sum of the values of the Möbius function of a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the 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. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001571The Cartan determinant of the integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001198The number of simple modules in the algebra

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

$A$ with minimal faithful projective-injective module

$eA$ . St001206The maximal dimension of an indecomposable projective

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

$eA$ . St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000478Another weight of a partition according to Alladi. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. 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. St000944The 3-degree of an integer partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St000618The number of self-evacuating tableaux of given shape. St001432The order dimension of the partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001780The order of promotion on the set of standard tableaux of given shape. 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. St001924The number of cells in an integer partition whose arm and leg length coincide. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000264The girth of a graph, which is not a tree. St001964The interval resolution global dimension of a poset. St000181The number of connected components of the Hasse diagram for the poset. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St001890The maximum magnitude of the Möbius function of a poset. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001570The minimal number of edges to add to make a graph Hamiltonian. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St000455The second largest eigenvalue of a graph if it is integral. St001857The number of edges in the reduced word graph of a signed permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000084The number of subtrees. St000328The maximum number of child nodes in a tree. St000907The number of maximal antichains of minimal length in a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000782The indicator function of whether a given perfect matching is an L & P matching. St000102The charge of a semistandard tableau.