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Your data matches 358 different statistics following compositions of up to 3 maps.
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Matching statistic: St001937
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001937: Parking functions ⟶ 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
[1,1,1] => 3
[1,1,2] => 3
[1,2,1] => 3
[2,1,1] => 2
[1,1,3] => 3
[1,3,1] => 2
[3,1,1] => 2
[1,2,2] => 3
[2,1,2] => 2
[2,2,1] => 1
[1,2,3] => 3
[1,3,2] => 2
[2,1,3] => 1
[2,3,1] => 1
[3,1,2] => 2
[3,2,1] => 1
[1,1,1,1] => 4
[1,1,1,2] => 4
[1,1,2,1] => 4
[1,2,1,1] => 4
[2,1,1,1] => 3
[1,1,1,3] => 4
[1,1,3,1] => 4
[1,3,1,1] => 3
[3,1,1,1] => 3
[1,1,1,4] => 4
[1,1,4,1] => 3
[1,4,1,1] => 3
[4,1,1,1] => 3
[1,1,2,2] => 4
[1,2,1,2] => 4
[1,2,2,1] => 4
[2,1,1,2] => 3
[2,1,2,1] => 3
[2,2,1,1] => 2
[1,1,2,3] => 4
[1,1,3,2] => 4
[1,2,1,3] => 4
[1,2,3,1] => 4
[1,3,1,2] => 3
[1,3,2,1] => 3
[2,1,1,3] => 3
[2,1,3,1] => 2
[2,3,1,1] => 2
[3,1,1,2] => 3
[3,1,2,1] => 3
Description
The size of the center of a parking function.
The center of a parking function $p_1,\dots,p_n$ is the longest subsequence $a_1,\dots,a_k$ such that $a_i\leq i$.
Matching statistic: St000011
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,1,0,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,0,1,0]
=> 2
[1,1,1] => [1,1,1,0,0,0]
=> 1
[1,1,2] => [1,1,0,1,0,0]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 1
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000135
Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
St000135: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000135: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [1,1] => 1
[1,2] => [1,2] => 2
[2,1] => [1,2] => 2
[1,1,1] => [1,1,1] => 1
[1,1,2] => [1,1,2] => 1
[1,2,1] => [1,1,2] => 1
[2,1,1] => [1,1,2] => 1
[1,1,3] => [1,1,3] => 2
[1,3,1] => [1,1,3] => 2
[3,1,1] => [1,1,3] => 2
[1,2,2] => [1,2,2] => 2
[2,1,2] => [1,2,2] => 2
[2,2,1] => [1,2,2] => 2
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,2,3] => 3
[2,1,3] => [1,2,3] => 3
[2,3,1] => [1,2,3] => 3
[3,1,2] => [1,2,3] => 3
[3,2,1] => [1,2,3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,1,2] => [1,1,1,2] => 1
[1,1,2,1] => [1,1,1,2] => 1
[1,2,1,1] => [1,1,1,2] => 1
[2,1,1,1] => [1,1,1,2] => 1
[1,1,1,3] => [1,1,1,3] => 1
[1,1,3,1] => [1,1,1,3] => 1
[1,3,1,1] => [1,1,1,3] => 1
[3,1,1,1] => [1,1,1,3] => 1
[1,1,1,4] => [1,1,1,4] => 2
[1,1,4,1] => [1,1,1,4] => 2
[1,4,1,1] => [1,1,1,4] => 2
[4,1,1,1] => [1,1,1,4] => 2
[1,1,2,2] => [1,1,2,2] => 1
[1,2,1,2] => [1,1,2,2] => 1
[1,2,2,1] => [1,1,2,2] => 1
[2,1,1,2] => [1,1,2,2] => 1
[2,1,2,1] => [1,1,2,2] => 1
[2,2,1,1] => [1,1,2,2] => 1
[1,1,2,3] => [1,1,2,3] => 1
[1,1,3,2] => [1,1,2,3] => 1
[1,2,1,3] => [1,1,2,3] => 1
[1,2,3,1] => [1,1,2,3] => 1
[1,3,1,2] => [1,1,2,3] => 1
[1,3,2,1] => [1,1,2,3] => 1
[2,1,1,3] => [1,1,2,3] => 1
[2,1,3,1] => [1,1,2,3] => 1
[2,3,1,1] => [1,1,2,3] => 1
[3,1,1,2] => [1,1,2,3] => 1
[3,1,2,1] => [1,1,2,3] => 1
Description
The number of lucky cars of the parking function.
A lucky car is a car that was able to park in its prefered spot.
The generating function,
$$
q\prod_{i=1}^{n-1} (i + (n-i+1)q)
$$
was established in [1].
Matching statistic: St001202
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
St001202: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001202: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,1,0,0]
=> 1
[1,2] => [1,0,1,0]
=> 2
[2,1] => [1,0,1,0]
=> 2
[1,1,1] => [1,1,1,0,0,0]
=> 1
[1,1,2] => [1,1,0,1,0,0]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> 2
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> 2
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> 2
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> 2
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> 2
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> 2
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> 2
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> 2
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> 2
Description
Call 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.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$−1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra to which we can associate a Dyck path as the top boundary of the Auslander-Reiten quiver of the LNakayama algebra.
The statistic gives half the dominant dimension of hte first indecomposable projective module in the special CNakayama algebra.
Matching statistic: St001903
Mp00052: Parking functions —to non-decreasing parking function⟶ Parking functions
St001903: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001903: Parking functions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [1,1] => 1
[1,2] => [1,2] => 2
[2,1] => [1,2] => 2
[1,1,1] => [1,1,1] => 1
[1,1,2] => [1,1,2] => 1
[1,2,1] => [1,1,2] => 1
[2,1,1] => [1,1,2] => 1
[1,1,3] => [1,1,3] => 2
[1,3,1] => [1,1,3] => 2
[3,1,1] => [1,1,3] => 2
[1,2,2] => [1,2,2] => 2
[2,1,2] => [1,2,2] => 2
[2,2,1] => [1,2,2] => 2
[1,2,3] => [1,2,3] => 3
[1,3,2] => [1,2,3] => 3
[2,1,3] => [1,2,3] => 3
[2,3,1] => [1,2,3] => 3
[3,1,2] => [1,2,3] => 3
[3,2,1] => [1,2,3] => 3
[1,1,1,1] => [1,1,1,1] => 1
[1,1,1,2] => [1,1,1,2] => 1
[1,1,2,1] => [1,1,1,2] => 1
[1,2,1,1] => [1,1,1,2] => 1
[2,1,1,1] => [1,1,1,2] => 1
[1,1,1,3] => [1,1,1,3] => 1
[1,1,3,1] => [1,1,1,3] => 1
[1,3,1,1] => [1,1,1,3] => 1
[3,1,1,1] => [1,1,1,3] => 1
[1,1,1,4] => [1,1,1,4] => 2
[1,1,4,1] => [1,1,1,4] => 2
[1,4,1,1] => [1,1,1,4] => 2
[4,1,1,1] => [1,1,1,4] => 2
[1,1,2,2] => [1,1,2,2] => 1
[1,2,1,2] => [1,1,2,2] => 1
[1,2,2,1] => [1,1,2,2] => 1
[2,1,1,2] => [1,1,2,2] => 1
[2,1,2,1] => [1,1,2,2] => 1
[2,2,1,1] => [1,1,2,2] => 1
[1,1,2,3] => [1,1,2,3] => 1
[1,1,3,2] => [1,1,2,3] => 1
[1,2,1,3] => [1,1,2,3] => 1
[1,2,3,1] => [1,1,2,3] => 1
[1,3,1,2] => [1,1,2,3] => 1
[1,3,2,1] => [1,1,2,3] => 1
[2,1,1,3] => [1,1,2,3] => 1
[2,1,3,1] => [1,1,2,3] => 1
[2,3,1,1] => [1,1,2,3] => 1
[3,1,1,2] => [1,1,2,3] => 1
[3,1,2,1] => [1,1,2,3] => 1
Description
The number of fixed points of a parking function.
If $(a_1,\dots,a_n)$ is a parking function, a fixed point is an index $i$ such that $a_i = i$.
It can be shown [1] that the generating function for parking functions with respect to this statistic is
$$
\frac{1}{(n+1)^2} \left((q+n)^{n+1} - (q-1)^{n+1}\right).
$$
Matching statistic: St000007
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,1] => [1,1,0,0]
=> [1,2] => 1
[1,2] => [1,0,1,0]
=> [2,1] => 2
[2,1] => [1,0,1,0]
=> [2,1] => 2
[1,1,1] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,1,2] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,1,3] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [3,1,2] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [2,3,1] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [3,2,1] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
Matching statistic: St000010
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> 1
[1,1] => [2] => [2]
=> 1
[1,2] => [1,1] => [1,1]
=> 2
[2,1] => [1,1] => [1,1]
=> 2
[1,1,1] => [3] => [3]
=> 1
[1,1,2] => [3] => [3]
=> 1
[1,2,1] => [3] => [3]
=> 1
[2,1,1] => [3] => [3]
=> 1
[1,1,3] => [2,1] => [2,1]
=> 2
[1,3,1] => [2,1] => [2,1]
=> 2
[3,1,1] => [2,1] => [2,1]
=> 2
[1,2,2] => [1,2] => [2,1]
=> 2
[2,1,2] => [1,2] => [2,1]
=> 2
[2,2,1] => [1,2] => [2,1]
=> 2
[1,2,3] => [1,1,1] => [1,1,1]
=> 3
[1,3,2] => [1,1,1] => [1,1,1]
=> 3
[2,1,3] => [1,1,1] => [1,1,1]
=> 3
[2,3,1] => [1,1,1] => [1,1,1]
=> 3
[3,1,2] => [1,1,1] => [1,1,1]
=> 3
[3,2,1] => [1,1,1] => [1,1,1]
=> 3
[1,1,1,1] => [4] => [4]
=> 1
[1,1,1,2] => [4] => [4]
=> 1
[1,1,2,1] => [4] => [4]
=> 1
[1,2,1,1] => [4] => [4]
=> 1
[2,1,1,1] => [4] => [4]
=> 1
[1,1,1,3] => [4] => [4]
=> 1
[1,1,3,1] => [4] => [4]
=> 1
[1,3,1,1] => [4] => [4]
=> 1
[3,1,1,1] => [4] => [4]
=> 1
[1,1,1,4] => [3,1] => [3,1]
=> 2
[1,1,4,1] => [3,1] => [3,1]
=> 2
[1,4,1,1] => [3,1] => [3,1]
=> 2
[4,1,1,1] => [3,1] => [3,1]
=> 2
[1,1,2,2] => [4] => [4]
=> 1
[1,2,1,2] => [4] => [4]
=> 1
[1,2,2,1] => [4] => [4]
=> 1
[2,1,1,2] => [4] => [4]
=> 1
[2,1,2,1] => [4] => [4]
=> 1
[2,2,1,1] => [4] => [4]
=> 1
[1,1,2,3] => [4] => [4]
=> 1
[1,1,3,2] => [4] => [4]
=> 1
[1,2,1,3] => [4] => [4]
=> 1
[1,2,3,1] => [4] => [4]
=> 1
[1,3,1,2] => [4] => [4]
=> 1
[1,3,2,1] => [4] => [4]
=> 1
[2,1,1,3] => [4] => [4]
=> 1
[2,1,3,1] => [4] => [4]
=> 1
[2,3,1,1] => [4] => [4]
=> 1
[3,1,1,2] => [4] => [4]
=> 1
[3,1,2,1] => [4] => [4]
=> 1
Description
The length of the partition.
Matching statistic: St000015
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00057: Parking functions —to touch composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000015: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 1
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,2] => [3] => [1,1,1,0,0,0]
=> 1
[1,2,1] => [3] => [1,1,1,0,0,0]
=> 1
[2,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,3,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[3,1,1] => [2,1] => [1,1,0,0,1,0]
=> 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,1,2] => [1,2] => [1,0,1,1,0,0]
=> 2
[2,2,1] => [1,2] => [1,0,1,1,0,0]
=> 2
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[3,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,4] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,4,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,4,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[4,1,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
=> 2
[1,1,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,2,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,2,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,3,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,3,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,3,2,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,1,3] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,1,3,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[2,3,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,1,2] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[3,1,2,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
Description
The number of peaks of a Dyck path.
Matching statistic: St000025
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,0]
=> 1
[1,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[2,1] => [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,1] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,2] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,1] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,3,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,1,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,3,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,1,2] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[3,2,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000056
(load all 40 compositions to match this statistic)
(load all 40 compositions to match this statistic)
Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000056: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 1
[1,1] => [1,1,0,0]
=> [2,1] => 1
[1,2] => [1,0,1,0]
=> [1,2] => 2
[2,1] => [1,0,1,0]
=> [1,2] => 2
[1,1,1] => [1,1,1,0,0,0]
=> [3,2,1] => 1
[1,1,2] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,2,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[2,1,1] => [1,1,0,1,0,0]
=> [2,3,1] => 1
[1,1,3] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,3,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[3,1,1] => [1,1,0,0,1,0]
=> [2,1,3] => 2
[1,2,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[2,2,1] => [1,0,1,1,0,0]
=> [1,3,2] => 2
[1,2,3] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,3,2] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1,3] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,3,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,1,2] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[3,2,1] => [1,0,1,0,1,0]
=> [1,2,3] => 3
[1,1,1,1] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
[1,1,1,2] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,1,2,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,2,1,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[2,1,1,1] => [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 1
[1,1,1,3] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,3,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,3,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[3,1,1,1] => [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 1
[1,1,1,4] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,4,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,4,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[4,1,1,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 2
[1,1,2,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,2,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,2,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,1,1,2] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,1,2,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[2,2,1,1] => [1,1,0,1,1,0,0,0]
=> [2,4,3,1] => 1
[1,1,2,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,1,3,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,2,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,2,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,3,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[1,3,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,1,1,3] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,1,3,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[2,3,1,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[3,1,1,2] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
[3,1,2,1] => [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 1
Description
The decomposition (or block) number of a permutation.
For $\pi \in \mathcal{S}_n$, this is given by
$$\#\big\{ 1 \leq k \leq n : \{\pi_1,\ldots,\pi_k\} = \{1,\ldots,k\} \big\}.$$
This is also known as the number of connected components [1] or the number of blocks [2] of the permutation, considering it as a direct sum.
This is one plus [[St000234]].
The following 348 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000084The number of subtrees. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000759The smallest missing part in an integer partition. St000822The Hadwiger number of the graph. St000843The decomposition number of a perfect matching. St000991The number of right-to-left minima of a permutation. St001029The size of the core of a graph. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. 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:
St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001461The number of topologically connected components of the chord diagram of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001494The Alon-Tarsi number of a 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. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St000053The number of valleys of the Dyck path. St000234The number of global ascents of a permutation. St000237The number of small exceedances. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000362The size of a minimal vertex cover of a graph. St000439The position of the first down step of a Dyck path. St000536The pathwidth of a graph. St000546The number of global descents of a permutation. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001812The biclique partition number of a graph. St000013The height of a Dyck path. St000031The number of cycles in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000062The length of the longest increasing subsequence of the permutation. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000068The number of minimal elements in a poset. St000069The number of maximal elements of a poset. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000105The number of blocks in the set partition. St000144The pyramid weight of the Dyck path. St000147The largest part of an integer partition. St000153The number of adjacent cycles of a permutation. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000213The number of weak exceedances (also weak excedences) of a permutation. St000228The size of a partition. St000239The number of small weak excedances. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000325The width of the tree associated to a permutation. St000328The maximum number of child nodes in a tree. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000363The number of minimal vertex covers of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000378The diagonal inversion number of an integer partition. St000382The first part of an integer composition. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000395The sum of the heights of the peaks of a Dyck path. St000443The number of long tunnels of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000544The cop number of a graph. St000636The hull number of a graph. St000654The first descent of a permutation. 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. St000702The number of weak deficiencies of a permutation. St000703The number of deficiencies of a permutation. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000740The last entry of a permutation. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000908The length of the shortest maximal antichain in a poset. St000916The packing number of a graph. St000924The number of topologically connected components of a perfect matching. St000926The clique-coclique number of a graph. St000971The smallest closer of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001018Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001020Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path. St001050The number of terminal closers of a set partition. St001058The breadth of the ordered tree. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001342The number of vertices in the center of a graph. St001363The Euler characteristic of a graph according to Knill. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001372The length of a longest cyclic run of ones of a binary word. 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. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001530The depth of a Dyck path. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001809The index of the step at the first peak of maximal height in a Dyck path. St001829The common independence number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number 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. St000012The area of a Dyck path. St000018The number of inversions of a permutation. St000019The cardinality of the support of a permutation. St000021The number of descents of a permutation. St000024The number of double up and double down steps of a Dyck path. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000052The number of valleys of a Dyck path not on the x-axis. St000133The "bounce" of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000159The number of distinct parts of the integer partition. St000160The multiplicity of the smallest part of a partition. St000171The degree of the graph. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000214The number of adjacencies of a permutation. St000224The sorting index of a permutation. St000245The number of ascents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000292The number of ascents of a binary word. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000310The minimal degree of a vertex of a graph. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000340The number of non-final maximal constant sub-paths of length greater than one. St000358The number of occurrences of the pattern 31-2. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000648The number of 2-excedences of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000675The number of centered multitunnels of a Dyck path. St000732The number of double deficiencies of a permutation. St000738The first entry in the last row of a standard tableau. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000883The number of longest increasing subsequences of a permutation. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St000989The number of final rises of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001185The number of indecomposable injective modules of grade at least 2 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. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. 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. 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. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001234The number of indecomposable three dimensional modules with projective dimension one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001270The bandwidth of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001489The maximum of the number of descents and the number of inverse descents. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. 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. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001644The dimension of a graph. St001650The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001712The number of natural descents of a standard Young tableau. St001726The number of visible inversions of a permutation. St001727The number of invisible inversions of a permutation. St001733The number of weak left to right maxima of a Dyck path. St001777The number of weak descents in an integer composition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000967The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra. St001218Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1. St000061The number of nodes on the left branch of a binary tree. St000717The number of ordinal summands of a poset. St000326The position of the first one in a binary word after appending a 1 at the end. St000444The length of the maximal rise of a Dyck path. St000504The cardinality of the first block of a set partition. St000668The least common multiple of the parts of the partition. St000898The number of maximal entries in the last diagonal of the monotone triangle. St000906The length of the shortest maximal chain in a poset. St000914The sum of the values of the Möbius function of a poset. St000925The number of topologically connected components of a set partition. St000990The first ascent of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000502The number of successions of a set partitions. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000809The reduced reflection length of the permutation. St000874The position of the last double rise in a Dyck path. St000957The number of Bruhat lower covers of a permutation. St000984The number of boxes below precisely one peak. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). 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. 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 $eA$. St000939The number of characters of the symmetric group whose value on the partition is positive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001587Half of the largest even part of an integer partition. 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. St000260The radius of a connected graph. St000993The multiplicity of the largest part 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. St001118The acyclic chromatic index of a graph. St001060The distinguishing index of a graph. St001875The number of simple modules with projective dimension at most 1. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the 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. St001568The smallest positive integer that does not appear twice in the partition. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions 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. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000264The girth of a graph, which is not a tree. 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. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. 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. St000422The energy of a graph, if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000456The monochromatic index of a connected graph. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000307The number of rowmotion orbits of a poset. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. 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. St001665The number of pure excedances of a permutation. St001737The number of descents of type 2 in a permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001738The minimal order of a graph which is not an induced subgraph of the given graph.
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