- FindStat

Your data matches 125 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000022
(load all 34 compositions to match this statistic)

Mapping

St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of fixed points of a permutation.

Matching statistic: St000215
(load all 6 compositions to match this statistic)

Mapping

St000215: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of adjacencies of a permutation, zero appended. An adjacency is a descent of the form $(e+1,e)$ in the word corresponding to the permutation in one-line notation. This statistic, $\operatorname{adj_0}$, counts adjacencies in the word with a zero appended. $(\operatorname{adj_0}, \operatorname{des})$ and $(\operatorname{fix}, \operatorname{exc})$ are equidistributed, see [1].

Matching statistic: St000241
(load all 12 compositions to match this statistic)

Mapping

St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of cyclical small excedances. A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.

Matching statistic: St000338
(load all 27 compositions to match this statistic)

Mapping

St000338: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of pixed points of a permutation. For a permutation $\sigma = p \tau_{1} \tau_{2} \cdots \tau_{k}$ in its hook factorization, [1] defines $$\textrm{pix} \, \sigma = \textrm{length} (p)$$.

Matching statistic: St000445
(load all 13 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000445: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of rises of length 1 of a Dyck path.

Matching statistic: St000475
(load all 5 compositions to match this statistic)

Mapping

Mp00108: Permutations —cycle type⟶ Integer partitions
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of parts equal to 1 in a partition.

Matching statistic: St001816
(load all 7 compositions to match this statistic)

Mapping

Mp00070: Permutations —Robinson-Schensted recording tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Eigenvalues of the top-to-random operator acting on a simple module. These eigenvalues are given in [1] and [3]. The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module. This statistic bears different names, such as the type in [2] or eig in [3]. Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].

Matching statistic: St000221
(load all 19 compositions to match this statistic)

Mapping

Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of strong fixed points of a permutation. $i$ is called a strong fixed point of $\pi$ if 1. $j < i$ implies $\pi_j < \pi_i$, and 2. $j > i$ implies $\pi_j > \pi_i$ This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3]. The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].

Matching statistic: St001126
(load all 2 compositions to match this statistic)

Mapping

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001126: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

Number of simple module that are 1-regular in the corresponding Nakayama algebra.

Matching statistic: St000117
(load all 2 compositions to match this statistic)

Mapping

Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

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

Description

The number of centered tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.

The following 115 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000315The number of isolated vertices of a graph. St000895The number of ones on the main diagonal of an alternating sign matrix. St000932The number of occurrences of the pattern UDU in a Dyck path. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St000461The rix statistic of a permutation. St000873The aix statistic of a permutation. St000247The number of singleton blocks of a set partition. St000248The number of anti-singletons of a set partition. St000674The number of hills of a Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000137The Grundy value of an integer partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001587Half of the largest even part of an integer partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000929The constant term of the character polynomial of an integer partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000984The number of boxes below precisely one peak. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001571The Cartan determinant of the integer partition. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000681The Grundy value of Chomp on Ferrers diagrams. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000237The number of small exceedances. St000894The trace of an alternating sign matrix. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001383The BG-rank of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001933The largest multiplicity of a part in an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000940The number of characters of the symmetric group whose value on the partition is zero. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001651The Frankl number of a lattice. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000456The monochromatic index of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000938The number of zeros of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000707The product of the factorials of the parts. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St001128The exponens consonantiae of a partition. St000239The number of small weak excedances. 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. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001903The number of fixed points of a parking function. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.