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Your data matches 419 different statistics following compositions of up to 3 maps.
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Matching statistic: St001515
(load all 71 compositions to match this statistic)

Mapping

St001515: 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]
 => 2
[1,0,1,0,1,0]
 => 2
[1,0,1,1,0,0]
 => 2
[1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => 2
[1,1,1,0,0,0]
 => 3
[1,0,1,0,1,0,1,0]
 => 3
[1,0,1,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => 3
[1,0,1,1,1,0,0,0]
 => 3
[1,1,0,0,1,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => 3
[1,1,0,1,0,0,1,0]
 => 3
[1,1,0,1,0,1,0,0]
 => 3
[1,1,0,1,1,0,0,0]
 => 3
[1,1,1,0,0,0,1,0]
 => 3
[1,1,1,0,0,1,0,0]
 => 3
[1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,0,0,0,0]
 => 4

Description

The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).

Matching statistic: St000672
(load all 40 compositions to match this statistic)

Mapping

Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000672: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of minimal elements in Bruhat order not less than the permutation. The minimal elements in question are biGrassmannian, that is

$1\dots r\ \ a+1\dots b\ \ r+1\dots a\ \ b+1\dots$ for some

$(r,a,b)$ . This is also the size of Fulton's essential set of the reverse permutation, according to [ex.4.7, 2].

Matching statistic: St000325
(load all 20 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000325: 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] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 2
[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] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 4

Description

The width of the tree associated to a permutation. A permutation can be mapped to a rooted tree with vertices

$\{0,1,2,\ldots,n\}$ and root

$0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1]. The width of the tree is given by the number of leaves of this tree. Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]]. See also [[St000308]] for the height of this tree.

Matching statistic: St000470
(load all 20 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000470: 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] => 2
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 2
[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] => 2
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 2
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 3
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 3
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 3
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 3
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 3
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 3
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 3
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 2
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 3
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 3
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 3
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 3
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => 3
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 4

Description

The number of runs in a permutation. A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence. This is the same as the number of descents plus 1.

Matching statistic: St001773
(load all 9 compositions to match this statistic)

Mapping

Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001773: Signed 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]
 => [2,1] => [2,1] => 1
[1,1,0,0]
 => [1,2] => [1,2] => 2
[1,0,1,0,1,0]
 => [2,3,1] => [2,3,1] => 2
[1,0,1,1,0,0]
 => [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
 => [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
 => [3,1,2] => [3,1,2] => 2
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => 3
[1,0,1,0,1,0,1,0]
 => [2,3,4,1] => [2,3,4,1] => 3
[1,0,1,0,1,1,0,0]
 => [2,3,1,4] => [2,3,1,4] => 3
[1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
 => [2,4,1,3] => [2,4,1,3] => 3
[1,0,1,1,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => 3
[1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [1,3,4,2] => 3
[1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [1,3,2,4] => 3
[1,1,0,1,0,0,1,0]
 => [3,1,4,2] => [3,1,4,2] => 3
[1,1,0,1,0,1,0,0]
 => [3,4,1,2] => [3,4,1,2] => 3
[1,1,0,1,1,0,0,0]
 => [3,1,2,4] => [3,1,2,4] => 3
[1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [1,2,4,3] => 3
[1,1,1,0,0,1,0,0]
 => [1,4,2,3] => [1,4,2,3] => 3
[1,1,1,0,1,0,0,0]
 => [4,1,2,3] => [4,1,2,3] => 3
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => 4

Description

The number of minimal elements in Bruhat order not less than the signed permutation. The minimal elements in question are biGrassmannian, that is both the element and its inverse have at most one descent. This is the size of the essential set of the signed permutation, see [1].

Matching statistic: St000021
(load all 20 compositions to match this statistic)

Mapping

Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[1,0]
 => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
 => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
 => [2,1] => [2,1] => 1 = 2 - 1
[1,0,1,0,1,0]
 => [1,2,3] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
 => [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,0,1,0]
 => [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,1,0,0]
 => [2,3,1] => [2,3,1] => 1 = 2 - 1
[1,1,1,0,0,0]
 => [3,2,1] => [3,2,1] => 2 = 3 - 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => [1,4,3,2] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
 => [1,4,3,2] => [1,4,3,2] => 2 = 3 - 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => [2,1,4,3] => 2 = 3 - 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => [2,4,1,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
 => [2,4,3,1] => [2,4,3,1] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
 => [3,2,1,4] => [3,2,1,4] => 2 = 3 - 1
[1,1,1,0,0,1,0,0]
 => [3,2,4,1] => [3,2,4,1] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
 => [4,2,3,1] => [4,2,3,1] => 2 = 3 - 1
[1,1,1,1,0,0,0,0]
 => [4,3,2,1] => [4,3,2,1] => 3 = 4 - 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: St000863
(load all 4 compositions to match this statistic)

Mapping

Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St000863: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The length of the first row 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 length of the first row of

$P$ and

$Q$ .

Matching statistic: St001211
(load all 32 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.

Matching statistic: St001489
(load all 29 compositions to match this statistic)

Mapping

Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The maximum of the number of descents and the number of inverse descents. This is, the maximum of [[St000021]] and [[St000354]].

Matching statistic: St001492
(load all 48 compositions to match this statistic)

Mapping

Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.

The following 409 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000007The number of saliances of the permutation. St000015The number of peaks of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000149The number of cells of the partition whose leg is zero and arm is odd. St000213The number of weak exceedances (also weak excedences) of a permutation. St000245The number of ascents of a permutation. St000308The height of the tree associated to a permutation. St000354The number of recoils of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000393The number of strictly increasing runs in a binary word. St000542The number of left-to-right-minima of a permutation. St000553The number of blocks of a graph. St000619The number of cyclic descents of a permutation. St000686The finitistic dominant dimension of a Dyck path. St000702The number of weak deficiencies of a permutation. St000744The length of the path to the largest entry in a standard Young tableau. St000862The number of parts of the shifted shape of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima of a permutation. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. 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: St001286The annihilation number of a graph. St001298The number of repeated entries in the Lehmer code of a permutation. St001389The number of partitions of the same length below the given integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001497The position of the largest weak excedence of a permutation. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000141The maximum drop size of a permutation. St000144The pyramid weight of the Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000157The number of descents of a standard tableau. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001176The size of a partition minus its first part. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of

$projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001197The global dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

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

$eAe$ in the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001321The number of vertices of the largest induced subforest of a graph. St001427The number of descents of a signed permutation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001622The number of join-irreducible elements of a lattice. St001668The number of points of the poset minus the width of the poset. St001760The number of prefix or suffix reversals needed to sort a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001812The biclique partition number of a graph. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000288The number of ones in a binary word. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001372The length of a longest cyclic run of ones of a binary word. 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

$eA$ . St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001626The number of maximal proper sublattices of a lattice. St001083The number of boxed occurrences of 132 in a permutation. St001875The number of simple modules with projective dimension at most 1. St001863The number of weak excedances of a signed permutation. 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. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St000699The toughness times the least common multiple of 1,. 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. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001488The number of corners of a skew partition. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation. St000834The number of right outer peaks of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000871The number of very big ascents of a permutation. St001394The genus of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000035The number of left outer peaks of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St000356The number of occurrences of the pattern 13-2. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001928The number of non-overlapping descents in a permutation. St001470The cyclic holeyness of a permutation. St001566The length of the longest arithmetic progression in a permutation. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000251The number of nonsingleton blocks of a set partition. St000253The crossing number of a set partition. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000387The matching number of a graph. St000488The number of cycles of a permutation of length at most 2. St000539The number of odd inversions of a permutation. St000806The semiperimeter of the associated bargraph. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001060The distinguishing index of a graph. St001114The number of odd descents of a permutation. St001151The number of blocks with odd minimum. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001405The number of bonds in a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001864The number of 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. St001962The proper pathwidth of a graph. St000023The number of inner peaks of a permutation. St000039The number of crossings of a permutation. St000236The number of cyclical small weak excedances. St000252The number of nodes of degree 3 of a binary tree. St000353The number of inner valleys of a permutation. St000390The number of runs of ones in a binary word. St000486The number of cycles of length at least 3 of a permutation. St000562The number of internal points of a set partition. St000624The normalized sum of the minimal distances to a greater element. St000710The number of big deficiencies of a permutation. St000711The number of big exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000779The tier of a permutation. St000824The sum of the number of descents and the number of recoils of a permutation. St000872The number of very big descents of a permutation. St001471The magnitude of a Dyck path. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001728The number of invisible descents of a permutation. St001806The upper middle entry of a permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000735The last entry on the main diagonal of a standard tableau. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000741The Colin de Verdière graph invariant. St000969We 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

$c_{n-1}$ . St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001644The dimension of a graph. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001638The book thickness of a graph. St000307The number of rowmotion orbits of a poset. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. 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

$eA$ . St001632The number of indecomposable injective modules

$I$ with

$dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001623The number of doubly irreducible elements of a lattice. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000736The last entry in the first row of a semistandard tableau. 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

$K[x]/(x^n)$ . St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001569The maximal modular displacement of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000075The orbit size of a standard tableau under promotion. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000177The number of free tiles in the pattern. St000178Number of free entries. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001520The number of strict 3-descents. St001811The Castelnuovo-Mumford regularity of a permutation. St001948The number of augmented double ascents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000079The number of alternating sign matrices for a given Dyck path. St000174The flush statistic of a semistandard tableau. St000239The number of small weak excedances. St000315The number of isolated vertices of a graph. St000317The cycle descent number of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000366The number of double descents of a permutation. St000502The number of successions of a set partitions. St000570The Edelman-Greene number of a permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000652The maximal difference between successive positions of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000730The maximal arc length of a set partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000988The orbit size of a permutation under Foata's bijection. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St001115The number of even descents of a permutation. St001162The minimum jump of a permutation. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. 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. St001191Number of simple modules

$S$ with

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

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

$A$ with global dimension

$g$ . St001194The injective dimension of

$A/AfA$ in the corresponding Nakayama algebra

$A$ when

$Af$ is the minimal faithful projective-injective left

$A$ -module St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra 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. 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. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001413Half the length of the longest even length palindromic prefix of a binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001469The holeyness of a permutation. St001487The number of inner corners of a skew partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the 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. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001823The Stasinski-Voll length of a signed permutation. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001862The number of crossings of a signed permutation. 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). St001889The size of the connectivity set of a signed permutation. St001896The number of right descents of a signed permutations. St001904The length of the initial strictly increasing segment of a parking function. St001937The size of the center of a parking function. St000089The absolute variation of a composition. St000090The variation of a composition. St000091The descent variation of a composition. St000166The depth minus 1 of an ordered tree. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000217The number of occurrences of the pattern 312 in a permutation. St000222The number of alignments in the permutation. St000232The number of crossings of a set partition. 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. St000254The nesting number of a set partition. St000287The number of connected components of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000335The difference of lower and upper interactions. St000338The number of pixed points of a permutation. St000348The non-inversion sum of a binary word. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St000407The number of occurrences of the pattern 2143 in 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. St000450The number of edges minus the number of vertices plus 2 of a graph. St000451The length of the longest pattern of the form k 1 2. St000489The number of cycles of a permutation of length at most 3. St000491The number of inversions of a set partition. St000492The rob statistic of a set partition. St000497The lcb statistic of a set partition. St000498The lcs statistic of a set partition. St000516The number of stretching pairs of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000534The number of 2-rises of a permutation. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. 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. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 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. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000654The first descent of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000739The first entry in the last row of a semistandard tableau. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000836The number of descents of distance 2 of a permutation. St000842The breadth of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000954Number of times the corresponding LNakayama algebra has

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

$i>0$ . St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences 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. St001153The number of blocks with even minimum in a set partition. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension 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

$K[x]/(x^n)$ . St001174The Gorenstein dimension of the algebra

$A/I$ when

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

$K[x]/(x^n)$ . St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. 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. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St001377The major index minus the number of inversions of a permutation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001530The depth of a Dyck path. St001535The number of cyclic alignments of a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001570The minimal number of edges to add to make a graph Hamiltonian. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001637The number of (upper) dissectors of a poset. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001684The reduced word complexity of 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. St001705The number of occurrences of the pattern 2413 in a permutation. St001712The number of natural descents of a standard Young tableau. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St001778The largest greatest common divisor of an element and its image in a permutation. St001822The number of alignments of a signed permutation. St001841The number of inversions of a set partition. St001856The number of edges in the reduced word graph of a permutation. St001857The number of edges in the reduced word graph of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001911A descent variant minus the number of inversions. St001926Sparre Andersen's position of the maximum of a signed permutation. St000054The first entry of the permutation. St000094The depth of an ordered tree. St000383The last part of an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000519The largest length of a factor maximising the subword complexity. St000521The number of distinct subtrees of an ordered tree. St000839The largest opener of a set partition. St000922The minimal number such that all substrings of this length are unique. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St000230Sum of the minimal elements of the blocks of a set partition. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one.