- FindStat

Your data matches 704 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000955
(load all 58 compositions to match this statistic)

Mapping

St000955: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => 1
[1,0,1,0]
 => 1
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 1
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => 2
[1,1,0,0,1,1,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => 1

Description

Number of times one has

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

$i>0$ for the corresponding LNakayama algebra.

Matching statistic: St000021
(load all 11 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of descents of a permutation. This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].

Matching statistic: St000035
(load all 30 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of left outer peaks of a permutation. A left outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ . In other words, it is a peak in the word

$[0,w_1,..., w_n]$ . This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.

Matching statistic: St000092
(load all 119 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000092: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of outer peaks of a permutation. An outer peak in a permutation

$w = [w_1,..., w_n]$ is either a position

$i$ such that

$w_{i-1} < w_i > w_{i+1}$ or

$1$ if

$w_1 > w_2$ or

$n$ if

$w_{n} > w_{n-1}$ . In other words, it is a peak in the word

$[0,w_1,..., w_n,0]$ .

Matching statistic: St000099
(load all 140 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000099: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of valleys of a permutation, including the boundary. The number of valleys excluding the boundary is [[St000353]].

Matching statistic: St000243
(load all 54 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
St000243: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclic valleys and cyclic peaks of a permutation. This is given by the number of indices

$i$ such that

$\pi_{i-1} > \pi_i < \pi_{i+1}$ with indices considered cyclically. Equivalently, this is the number of indices

$i$ such that

$\pi_{i-1} < \pi_i > \pi_{i+1}$ with indices considered cyclically.

Matching statistic: St000862
(load all 127 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000862: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts of the shifted shape of a permutation. The diagram of a strict partition

$\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of

$n$ is a tableau with

$\ell$ rows, the

$i$ -th row being indented by

$i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair

$(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in

$Q$ may be circled. This statistic records the number of parts of the shifted shape.

Matching statistic: St000920
(load all 7 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1,0]
 => [1,0]
 => 1
[1,0,1,0]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[1,1,0,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[1,1,1,0,0,0]
 => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[1,0,1,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 1
[1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2

Description

The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Matching statistic: St001359
(load all 42 compositions to match this statistic)

Mapping

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00309: Permutations —inverse toric promotion⟶ Permutations
St001359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. In other words, this is

$2^k$ where

$k$ is the number of cycles of length at least three ([[St000486]]) in its cycle decomposition. The generating function for the number of equivalence classes,

$f(n)$ , is

$\sum_{n\geq 0} f(n)\frac{x^n}{n!} = \frac{e(\frac{x}{2} + \frac{x^2}{4})}{\sqrt{1-x}}.$

Matching statistic: St001569
(load all 65 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St001569: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximal modular displacement of a permutation. This is

$\max_{1\leq i \leq n} \left(\min(\pi(i)-i\pmod n, i-\pi(i)\pmod n)\right)$ for a permutation

$\pi$ of

$\{1,\dots,n\}$ .

The following 694 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St001928The number of non-overlapping descents in a permutation. St000023The number of inner peaks of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000159The number of distinct parts of the integer partition. St000183The side length of the Durfee square of an integer partition. St000201The number of leaf nodes in a binary tree. St000245The number of ascents of a permutation. St000288The number of ones in a binary word. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000354The number of recoils of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000396The register function (or Horton-Strahler number) of a binary tree. St000451The length of the longest pattern of the form k 1 2. St000522The number of 1-protected nodes of a rooted tree. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000568The hook number of a binary tree. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000662The staircase size of the code of a permutation. St000703The number of deficiencies of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000758The length of the longest staircase fitting into an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000889The number of alternating sign matrices with the same antidiagonal sums. St000897The number of different multiplicities of parts of an integer partition. St000905The number of different multiplicities of parts of an integer composition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001096The size of the overlap set of a permutation. St001151The number of blocks with odd minimum. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001286The annihilation number of a graph. St001405The number of bonds in a permutation. St001427The number of descents of a signed permutation. St001432The order dimension of the partition. St001461The number of topologically connected components of the chord diagram of a permutation. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001597The Frobenius rank of a skew partition. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001716The 1-improper chromatic number of a graph. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001741The largest integer such that all patterns of this size are contained in the permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between

$e_i J$ and

$e_j J$ (the radical of the indecomposable projective modules). St001884The number of borders of a binary word. St000052The number of valleys of a Dyck path not on the x-axis. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000150The floored half-sum of the multiplicities of a partition. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000213The number of weak exceedances (also weak excedences) of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000317The cycle descent number of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000345The number of refinements of a partition. St000353The number of inner valleys of a permutation. St000356The number of occurrences of the pattern 13-2. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length

$3$ . St000386The number of factors DDU in a Dyck path. 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. St000481The number of upper covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000647The number of big descents of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000663The number of right floats of a permutation. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000702The number of weak deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000761The number of ascents in an integer composition. St000779The tier of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000872The number of very big descents of a permutation. St000884The number of isolated descents of a permutation. St000893The number of distinct diagonal sums of an alternating sign matrix. St000935The number of ordered refinements of an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. 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. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path 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. 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. St001090The number of pop-stack-sorts needed to sort a permutation. St001115The number of even descents of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001263The index of the maximal parabolic seaweed algebra associated with the composition. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001335The cardinality of a minimal cycle-isolating set of a graph. St001389The number of partitions of the same length below the given integer partition. St001394The genus of a permutation. St001423The number of distinct cubes in a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001469The holeyness of a permutation. St001470The cyclic holeyness of a permutation. St001471The magnitude of a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St001513The number of nested exceedences of a permutation. St001566The length of the longest arithmetic progression in a permutation. St001665The number of pure excedances of a permutation. St001673The degree of asymmetry of an integer composition. 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. St001712The number of natural descents of a standard Young tableau. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001731The factorization defect of a permutation. St001737The number of descents of type 2 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001777The number of weak descents in an integer composition. St001811The Castelnuovo-Mumford regularity of a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001874Lusztig's a-function for the symmetric group. St001960The number of descents of a permutation minus one if its first entry is not one. St000624The normalized sum of the minimal distances to a greater element. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001424The number of distinct squares in a binary word. St000058The order of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000886The number of permutations with the same antidiagonal sums. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001487The number of inner corners of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001642The Prague dimension of a graph. St000291The number of descents of a binary word. St000295The length of the border of a binary word. St000352The Elizalde-Pak rank of a permutation. St000539The number of odd inversions of a permutation. St000628The balance of a binary word. St000646The number of big ascents of a permutation. St000651The maximal size of a rise in a permutation. St000670The reversal length of a permutation. St000710The number of big deficiencies of a permutation. St000837The number of ascents of distance 2 of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. 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. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001280The number of parts of an integer partition that are at least two. St001298The number of repeated entries in the Lehmer code of a permutation. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001435The number of missing boxes in the first row. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001524The degree of symmetry of a binary word. St001557The number of inversions of the second entry of a permutation. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001760The number of prefix or suffix reversals needed to sort a permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1. St001820The size of the image of the pop stack sorting operator. St000485The length of the longest cycle of a permutation. St000489The number of cycles of a permutation of length at most 3. St000216The absolute length of a permutation. St000640The rank of the largest boolean interval in a poset. St000677The standardized bi-alternating inversion number of a permutation. St000809The reduced reflection length of the permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001195The global dimension of the algebra

$A/AfA$ of the corresponding Nakayama algebra

$A$ with minimal left faithful projective-injective module

$Af$ . St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000397The Strahler number of a rooted tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001644The dimension of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000444The length of the maximal rise of a Dyck path. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000933The number of multipartitions of sizes given by an integer partition. St001896The number of right descents of a signed permutations. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000259The diameter of a connected graph. St000409The number of pitchforks in a binary tree. 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

$eA$ . St001864The number of excedances of a signed permutation. St000563The number of overlapping pairs of blocks of a set partition. St000661The number of rises of length 3 of a Dyck path. St000731The number of double exceedences of a permutation. St001866The nesting alignments of a signed permutation. St000260The radius of a connected graph. St000759The smallest missing part in an integer partition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001570The minimal number of edges to add to make a graph Hamiltonian. St001885The number of binary words with the same proper border set. St000233The number of nestings of a set partition. St001846The number of elements which do not have a complement in the lattice. St001862The number of crossings of a signed permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000124The cardinality of the preimage of the Simion-Schmidt map. St000237The number of small exceedances. St000454The largest eigenvalue of a graph if it is integral. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents 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. St000455The second largest eigenvalue of a graph if it is integral. St000534The number of 2-rises of a permutation. St000562The number of internal points of a set partition. St000648The number of 2-excedences of a permutation. St000842The breadth of a permutation. St001868The number of alignments of type NE of a signed permutation. St001964The interval resolution global dimension of a poset. St000068The number of minimal elements in a poset. St000214The number of adjacencies of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000223The number of nestings in the permutation. St000232The number of crossings of a set partition. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000441The number of successions of a permutation. St001330The hat guessing number of a graph. St001396Number of triples of incomparable elements in a finite poset. St001438The number of missing boxes of a skew partition. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001946The number of descents in a parking function. St000091The descent variation of a composition. St000236The number of cyclical small weak excedances. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000365The number of double ascents of a permutation. St000650The number of 3-rises of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St000078The number of alternating sign matrices whose left key is the permutation. St000255The number of reduced Kogan faces with the permutation as type. St000456The monochromatic index of a connected graph. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000785The number of distinct colouring schemes of a graph. St000789The number of crossing-similar perfect matchings of a perfect matching. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000988The orbit size of a permutation under Foata's bijection. St001061The number of indices that are both descents and recoils of a permutation. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001208The number of connected components of the quiver of

$A/T$ when

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

$A$ of

$K[x]/(x^n)$ . St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. 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. St001272The number of graphs with the same degree sequence. St001344The neighbouring number of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001413Half the length of the longest even length palindromic prefix of a binary word. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001768The number of reduced words of a signed permutation. St001769The reflection length of a signed permutation. St001863The number of weak excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000090The variation of a composition. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000217The number of occurrences of the pattern 312 in a permutation. St000247The number of singleton blocks of a set partition. St000338The number of pixed points of a permutation. St000357The number of occurrences of the pattern 12-3. St000358The number of occurrences of the pattern 31-2. St000367The number of simsun double descents of a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000461The rix statistic of a permutation. St000516The number of stretching pairs of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. 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. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000649The number of 3-excedences of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000873The aix statistic of a permutation. St000962The 3-shifted major index of a permutation. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001114The number of odd descents of a permutation. St001126Number of simple module that are 1-regular 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. St001171The vector space dimension of

$Ext_A^1(I_o,A)$ when

$I_o$ is the tilting module corresponding to the permutation

$o$ in the Auslander algebra

$A$ of

$K[x]/(x^n)$ . St001200The number of simple modules in

$eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001316The domatic number of a graph. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001520The number of strict 3-descents. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. 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. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001732The number of peaks visible from the left. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001781The interlacing number of a set partition. St001822The number of alignments of a signed permutation. St001823The Stasinski-Voll length of a signed permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001871The number of triconnected components of a graph. St001903The number of fixed points of a parking function. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. 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. St001948The number of augmented double ascents of a permutation. St000917The open packing number of a graph. St000735The last entry on the main diagonal of a standard tableau. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001488The number of corners of a skew partition. St001624The breadth of a lattice. 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. St000015The number of peaks of a Dyck path. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000418The number of Dyck paths that are weakly below a Dyck path. 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. St000674The number of hills of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000908The length of the shortest maximal antichain in a poset. St000911The number of maximal antichains of maximal size in a poset. St000914The sum of the values of the Möbius function of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000953The largest degree of an irreducible factor of the Coxeter polynomial of the Dyck path over the rational numbers. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001118The acyclic chromatic index of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001201The grade of the simple module

$S_0$ in the special CNakayama algebra corresponding to the 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: St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001523The degree of symmetry of a Dyck path. St001530The depth of a Dyck path. St001531Number of partial orders contained in the poset determined by the Dyck path. St001545The second Elser number of a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001733The number of weak left to right maxima of a Dyck path. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001881The number of factors of a lattice as a Cartesian product of lattices. St001959The product of the heights of the peaks of a Dyck path. St000219The number of occurrences of the pattern 231 in a permutation. 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. St000298The order dimension or Dushnik-Miller dimension of a poset. St000422The energy of a graph, if it is integral. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000907The number of maximal antichains of minimal length in a poset. 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. St001616The number of neutral elements in a lattice. St001631The number of simple modules

$S$ with

$dim Ext^1(S,A)=1$ in the incidence algebra

$A$ of the poset. St001720The minimal length of a chain of small intervals in a lattice. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001618The cardinality of the Frattini sublattice of a lattice. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000706The product of the factorials of the multiplicities of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000782The indicator function of whether a given perfect matching is an L & P matching. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000993The multiplicity of the largest part of an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St000879The number of long braid edges in the graph of braid moves of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001260The permanent of an alternating sign matrix. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000894The trace of an alternating sign matrix. St000071The number of maximal chains in a poset. St000100The number of linear extensions of a poset. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000527The width of the poset. St000681The Grundy value of Chomp on Ferrers diagrams. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000909The number of maximal chains of maximal size in a poset. St000997The even-odd crank of an integer partition. St001128The exponens consonantiae of a partition. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000102The charge of a semistandard tableau. St001556The number of inversions of the third entry of a permutation. St000056The decomposition (or block) number of a permutation. St000096The number of spanning trees of a graph. St000154The sum of the descent bottoms of a permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000254The nesting number of a set partition. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000284The Plancherel distribution on integer partitions. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000307The number of rowmotion orbits of a poset. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000450The number of edges minus the number of vertices plus 2 of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000694The number of affine bounded permutations that project to a given permutation. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000729The minimal arc length of a set partition. St000739The first entry in the last row of a semistandard tableau. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000883The number of longest increasing subsequences of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. 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. St000991The number of right-to-left minima of a permutation. St001050The number of terminal closers of a set partition. St001052The length of the exterior of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001192The maximal dimension of

$Ext_A^2(S,A)$ for a simple module

$S$ over the corresponding Nakayama algebra

$A$ . St001204Call 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. St001256Number of simple reflexive modules that are 2-stable reflexive. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001410The minimal entry of a semistandard tableau. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001481The minimal height of a peak of a Dyck path. St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001778The largest greatest common divisor of an element and its image in a permutation. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001806The upper middle entry of a permutation. St001828The Euler characteristic of a graph. St001889The size of the connectivity set of a signed permutation. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000022The number of fixed points of a permutation. St000075The orbit size of a standard tableau under promotion. St000084The number of subtrees. St000089The absolute variation of a composition. St000095The number of triangles of a graph. St000101The cocharge of a semistandard tableau. St000105The number of blocks in the set partition. St000117The number of centered tunnels of a Dyck path. St000133The "bounce" of a permutation. St000134The size of the orbit of an alternating sign matrix under gyration. St000153The number of adjacent cycles of a permutation. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000221The number of strong fixed points of a permutation. St000234The number of global ascents of a permutation. St000251The number of nonsingleton blocks of a set partition. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by

$4$ . St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000328The maximum number of child nodes in a tree. St000355The number of occurrences of the pattern 21-3. St000360The number of occurrences of the pattern 32-1. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000401The size of the symmetry class of a permutation. St000402Half the size of the symmetry class of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000417The size of the automorphism group of the ordered tree. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000462The major index minus the number of excedences of a permutation. St000487The length of the shortest cycle of a permutation. St000496The rcs statistic of a set partition. St000504The cardinality of the first block of a set partition. St000542The number of left-to-right-minima of a permutation. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. 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. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000623The number of occurrences of the pattern 52341 in a permutation. St000632The jump number of the poset. St000679The pruning number of an ordered tree. St000750The number of occurrences of the pattern 4213 in a permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. 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. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000822The Hadwiger number of the graph. St000823The number of unsplittable factors of the set partition. St000836The number of descents of distance 2 of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000943The number of spots the most unlucky car had to go further in a parking function. St000989The number of final rises of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001058The breadth of the ordered tree. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001062The maximal size of a block of a set partition. St001075The minimal size of a block of a set partition. St001082The number of boxed occurrences of 123 in a permutation. St001185The number of indecomposable injective modules of grade at least 2 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001371The length of the longest Yamanouchi prefix of a binary word. St001381The fertility of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001429The number of negative entries in a signed permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001517The length of a longest pair of twins in a permutation. St001536The number of cyclic misalignments of a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001555The order of a signed permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001734The lettericity of a graph. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001783The number of odd automorphisms of a graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001851The number of Hecke atoms of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001867The number of alignments of type EN of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001893The flag descent of a signed permutation. St001895The oddness of a signed permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St000495The number of inversions of distance at most 2 of a permutation. St000638The number of up-down runs of a permutation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.