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Matching statistic: St000054
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 1
[1,2,4,3] => [1,2,3] => 1
[1,3,2,4] => [1,3,2] => 1
[1,3,4,2] => [1,3,2] => 1
[1,4,2,3] => [1,2,3] => 1
[1,4,3,2] => [1,3,2] => 1
[2,1,3,4] => [2,1,3] => 2
[2,1,4,3] => [2,1,3] => 2
[2,3,1,4] => [2,3,1] => 2
[2,3,4,1] => [2,3,1] => 2
[2,4,1,3] => [2,1,3] => 2
[2,4,3,1] => [2,3,1] => 2
[3,1,2,4] => [3,1,2] => 3
[3,1,4,2] => [3,1,2] => 3
[3,2,1,4] => [3,2,1] => 3
[3,2,4,1] => [3,2,1] => 3
[3,4,1,2] => [3,1,2] => 3
[3,4,2,1] => [3,2,1] => 3
[4,1,2,3] => [1,2,3] => 1
[4,1,3,2] => [1,3,2] => 1
[4,2,1,3] => [2,1,3] => 2
[4,2,3,1] => [2,3,1] => 2
[4,3,1,2] => [3,1,2] => 3
[4,3,2,1] => [3,2,1] => 3
[1,2,3,4,5] => [1,2,3,4] => 1
[1,2,3,5,4] => [1,2,3,4] => 1
[1,2,4,3,5] => [1,2,4,3] => 1
[1,2,4,5,3] => [1,2,4,3] => 1
[1,2,5,3,4] => [1,2,3,4] => 1
[1,2,5,4,3] => [1,2,4,3] => 1
[1,3,2,4,5] => [1,3,2,4] => 1
[1,3,2,5,4] => [1,3,2,4] => 1
[1,3,4,2,5] => [1,3,4,2] => 1
[1,3,4,5,2] => [1,3,4,2] => 1
[1,3,5,2,4] => [1,3,2,4] => 1
[1,3,5,4,2] => [1,3,4,2] => 1
[1,4,2,3,5] => [1,4,2,3] => 1
[1,4,2,5,3] => [1,4,2,3] => 1
[1,4,3,2,5] => [1,4,3,2] => 1
[1,4,3,5,2] => [1,4,3,2] => 1
[1,4,5,2,3] => [1,4,2,3] => 1
[1,4,5,3,2] => [1,4,3,2] => 1
Description
The first entry of the permutation.
This can be described as 1 plus the number of occurrences of the vincular pattern ([2,1], {(0,0),(0,1),(0,2)}), i.e., the first column is shaded, see [1].
This statistic is related to the number of deficiencies [[St000703]] as follows: consider the arc diagram of a permutation $\pi$ of $n$, together with its rotations, obtained by conjugating with the long cycle $(1,\dots,n)$. Drawing the labels $1$ to $n$ in this order on a circle, and the arcs $(i, \pi(i))$ as straight lines, the rotation of $\pi$ is obtained by replacing each number $i$ by $(i\bmod n) +1$. Then, $\pi(1)-1$ is the number of rotations of $\pi$ where the arc $(1, \pi(1))$ is a deficiency. In particular, if $O(\pi)$ is the orbit of rotations of $\pi$, then the number of deficiencies of $\pi$ equals
$$
\frac{1}{|O(\pi)|}\sum_{\sigma\in O(\pi)} (\sigma(1)-1).
$$
Matching statistic: St000740
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 2
[1,3,2] => [1,2] => 2
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 2
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 3
[1,2,4,3] => [1,2,3] => 3
[1,3,2,4] => [1,3,2] => 2
[1,3,4,2] => [1,3,2] => 2
[1,4,2,3] => [1,2,3] => 3
[1,4,3,2] => [1,3,2] => 2
[2,1,3,4] => [2,1,3] => 3
[2,1,4,3] => [2,1,3] => 3
[2,3,1,4] => [2,3,1] => 1
[2,3,4,1] => [2,3,1] => 1
[2,4,1,3] => [2,1,3] => 3
[2,4,3,1] => [2,3,1] => 1
[3,1,2,4] => [3,1,2] => 2
[3,1,4,2] => [3,1,2] => 2
[3,2,1,4] => [3,2,1] => 1
[3,2,4,1] => [3,2,1] => 1
[3,4,1,2] => [3,1,2] => 2
[3,4,2,1] => [3,2,1] => 1
[4,1,2,3] => [1,2,3] => 3
[4,1,3,2] => [1,3,2] => 2
[4,2,1,3] => [2,1,3] => 3
[4,2,3,1] => [2,3,1] => 1
[4,3,1,2] => [3,1,2] => 2
[4,3,2,1] => [3,2,1] => 1
[1,2,3,4,5] => [1,2,3,4] => 4
[1,2,3,5,4] => [1,2,3,4] => 4
[1,2,4,3,5] => [1,2,4,3] => 3
[1,2,4,5,3] => [1,2,4,3] => 3
[1,2,5,3,4] => [1,2,3,4] => 4
[1,2,5,4,3] => [1,2,4,3] => 3
[1,3,2,4,5] => [1,3,2,4] => 4
[1,3,2,5,4] => [1,3,2,4] => 4
[1,3,4,2,5] => [1,3,4,2] => 2
[1,3,4,5,2] => [1,3,4,2] => 2
[1,3,5,2,4] => [1,3,2,4] => 4
[1,3,5,4,2] => [1,3,4,2] => 2
[1,4,2,3,5] => [1,4,2,3] => 3
[1,4,2,5,3] => [1,4,2,3] => 3
[1,4,3,2,5] => [1,4,3,2] => 2
[1,4,3,5,2] => [1,4,3,2] => 2
[1,4,5,2,3] => [1,4,2,3] => 3
[1,4,5,3,2] => [1,4,3,2] => 2
Description
The last entry of a permutation.
This statistic is undefined for the empty permutation.
Matching statistic: St001806
Mp00252: Permutations —restriction⟶ Permutations
St001806: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001806: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 2
[1,3,2] => [1,2] => 2
[2,1,3] => [2,1] => 1
[2,3,1] => [2,1] => 1
[3,1,2] => [1,2] => 2
[3,2,1] => [2,1] => 1
[1,2,3,4] => [1,2,3] => 2
[1,2,4,3] => [1,2,3] => 2
[1,3,2,4] => [1,3,2] => 3
[1,3,4,2] => [1,3,2] => 3
[1,4,2,3] => [1,2,3] => 2
[1,4,3,2] => [1,3,2] => 3
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 3
[2,3,4,1] => [2,3,1] => 3
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 3
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => 2
[4,1,3,2] => [1,3,2] => 3
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 3
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => 3
[1,2,3,5,4] => [1,2,3,4] => 3
[1,2,4,3,5] => [1,2,4,3] => 4
[1,2,4,5,3] => [1,2,4,3] => 4
[1,2,5,3,4] => [1,2,3,4] => 3
[1,2,5,4,3] => [1,2,4,3] => 4
[1,3,2,4,5] => [1,3,2,4] => 2
[1,3,2,5,4] => [1,3,2,4] => 2
[1,3,4,2,5] => [1,3,4,2] => 4
[1,3,4,5,2] => [1,3,4,2] => 4
[1,3,5,2,4] => [1,3,2,4] => 2
[1,3,5,4,2] => [1,3,4,2] => 4
[1,4,2,3,5] => [1,4,2,3] => 2
[1,4,2,5,3] => [1,4,2,3] => 2
[1,4,3,2,5] => [1,4,3,2] => 3
[1,4,3,5,2] => [1,4,3,2] => 3
[1,4,5,2,3] => [1,4,2,3] => 2
[1,4,5,3,2] => [1,4,3,2] => 3
Description
The upper middle entry of a permutation.
This is the entry $\sigma(\frac{n+1}{2})$ when $n$ is odd, and $\sigma(\frac{n}{2}+1)$ when $n$ is even, where $n$ is the size of the permutation $\sigma$.
Matching statistic: St001807
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001807: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => 1
[2,1] => [1] => 1
[1,2,3] => [1,2] => 1
[1,3,2] => [1,2] => 1
[2,1,3] => [2,1] => 2
[2,3,1] => [2,1] => 2
[3,1,2] => [1,2] => 1
[3,2,1] => [2,1] => 2
[1,2,3,4] => [1,2,3] => 2
[1,2,4,3] => [1,2,3] => 2
[1,3,2,4] => [1,3,2] => 3
[1,3,4,2] => [1,3,2] => 3
[1,4,2,3] => [1,2,3] => 2
[1,4,3,2] => [1,3,2] => 3
[2,1,3,4] => [2,1,3] => 1
[2,1,4,3] => [2,1,3] => 1
[2,3,1,4] => [2,3,1] => 3
[2,3,4,1] => [2,3,1] => 3
[2,4,1,3] => [2,1,3] => 1
[2,4,3,1] => [2,3,1] => 3
[3,1,2,4] => [3,1,2] => 1
[3,1,4,2] => [3,1,2] => 1
[3,2,1,4] => [3,2,1] => 2
[3,2,4,1] => [3,2,1] => 2
[3,4,1,2] => [3,1,2] => 1
[3,4,2,1] => [3,2,1] => 2
[4,1,2,3] => [1,2,3] => 2
[4,1,3,2] => [1,3,2] => 3
[4,2,1,3] => [2,1,3] => 1
[4,2,3,1] => [2,3,1] => 3
[4,3,1,2] => [3,1,2] => 1
[4,3,2,1] => [3,2,1] => 2
[1,2,3,4,5] => [1,2,3,4] => 2
[1,2,3,5,4] => [1,2,3,4] => 2
[1,2,4,3,5] => [1,2,4,3] => 2
[1,2,4,5,3] => [1,2,4,3] => 2
[1,2,5,3,4] => [1,2,3,4] => 2
[1,2,5,4,3] => [1,2,4,3] => 2
[1,3,2,4,5] => [1,3,2,4] => 3
[1,3,2,5,4] => [1,3,2,4] => 3
[1,3,4,2,5] => [1,3,4,2] => 3
[1,3,4,5,2] => [1,3,4,2] => 3
[1,3,5,2,4] => [1,3,2,4] => 3
[1,3,5,4,2] => [1,3,4,2] => 3
[1,4,2,3,5] => [1,4,2,3] => 4
[1,4,2,5,3] => [1,4,2,3] => 4
[1,4,3,2,5] => [1,4,3,2] => 4
[1,4,3,5,2] => [1,4,3,2] => 4
[1,4,5,2,3] => [1,4,2,3] => 4
[1,4,5,3,2] => [1,4,3,2] => 4
Description
The lower middle entry of a permutation.
This is the entry $\sigma(\frac{n+1}{2})$ when $n$ is odd, and $\sigma(\frac{n}{2})$ when $n$ is even, where $n$ is the size of the permutation $\sigma$.
Matching statistic: St000025
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> 1
[2,1] => [1] => [1,0]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 2
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,4,5,3,2] => [1,4,3,2] => [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: St000839
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => {{1}}
=> 1
[2,1] => [1] => {{1}}
=> 1
[1,2,3] => [1,2] => {{1},{2}}
=> 2
[1,3,2] => [1,2] => {{1},{2}}
=> 2
[2,1,3] => [2,1] => {{1,2}}
=> 1
[2,3,1] => [2,1] => {{1,2}}
=> 1
[3,1,2] => [1,2] => {{1},{2}}
=> 2
[3,2,1] => [2,1] => {{1,2}}
=> 1
[1,2,3,4] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,2,4,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,3,2,4] => [1,3,2] => {{1},{2,3}}
=> 2
[1,3,4,2] => [1,3,2] => {{1},{2,3}}
=> 2
[1,4,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[1,4,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[2,1,3,4] => [2,1,3] => {{1,2},{3}}
=> 3
[2,1,4,3] => [2,1,3] => {{1,2},{3}}
=> 3
[2,3,1,4] => [2,3,1] => {{1,2,3}}
=> 1
[2,3,4,1] => [2,3,1] => {{1,2,3}}
=> 1
[2,4,1,3] => [2,1,3] => {{1,2},{3}}
=> 3
[2,4,3,1] => [2,3,1] => {{1,2,3}}
=> 1
[3,1,2,4] => [3,1,2] => {{1,2,3}}
=> 1
[3,1,4,2] => [3,1,2] => {{1,2,3}}
=> 1
[3,2,1,4] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,4,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,4,1,2] => [3,1,2] => {{1,2,3}}
=> 1
[3,4,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[4,1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3
[4,1,3,2] => [1,3,2] => {{1},{2,3}}
=> 2
[4,2,1,3] => [2,1,3] => {{1,2},{3}}
=> 3
[4,2,3,1] => [2,3,1] => {{1,2,3}}
=> 1
[4,3,1,2] => [3,1,2] => {{1,2,3}}
=> 1
[4,3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4,5] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,3,5,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,4,3,5] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,2,4,5,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,2,5,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4
[1,2,5,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3
[1,3,2,4,5] => [1,3,2,4] => {{1},{2,3},{4}}
=> 4
[1,3,2,5,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 4
[1,3,4,2,5] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,4,5,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,3,5,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 4
[1,3,5,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2
[1,4,2,3,5] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,2,5,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,3,2,5] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,4,3,5,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
[1,4,5,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 2
[1,4,5,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 3
Description
The largest opener of a set partition.
An opener (or left hand endpoint) of a set partition is a number that is minimal in its block. For this statistic, singletons are considered as openers.
Matching statistic: St000971
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00151: Permutations —to cycle type⟶ Set partitions
St000971: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => {{1}}
=> 1
[2,1] => [1] => {{1}}
=> 1
[1,2,3] => [1,2] => {{1},{2}}
=> 1
[1,3,2] => [1,2] => {{1},{2}}
=> 1
[2,1,3] => [2,1] => {{1,2}}
=> 2
[2,3,1] => [2,1] => {{1,2}}
=> 2
[3,1,2] => [1,2] => {{1},{2}}
=> 1
[3,2,1] => [2,1] => {{1,2}}
=> 2
[1,2,3,4] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,2,4,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,3,2,4] => [1,3,2] => {{1},{2,3}}
=> 1
[1,3,4,2] => [1,3,2] => {{1},{2,3}}
=> 1
[1,4,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[1,4,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[2,1,3,4] => [2,1,3] => {{1,2},{3}}
=> 2
[2,1,4,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,3,1,4] => [2,3,1] => {{1,2,3}}
=> 3
[2,3,4,1] => [2,3,1] => {{1,2,3}}
=> 3
[2,4,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[2,4,3,1] => [2,3,1] => {{1,2,3}}
=> 3
[3,1,2,4] => [3,1,2] => {{1,2,3}}
=> 3
[3,1,4,2] => [3,1,2] => {{1,2,3}}
=> 3
[3,2,1,4] => [3,2,1] => {{1,3},{2}}
=> 2
[3,2,4,1] => [3,2,1] => {{1,3},{2}}
=> 2
[3,4,1,2] => [3,1,2] => {{1,2,3}}
=> 3
[3,4,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[4,1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 1
[4,1,3,2] => [1,3,2] => {{1},{2,3}}
=> 1
[4,2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2
[4,2,3,1] => [2,3,1] => {{1,2,3}}
=> 3
[4,3,1,2] => [3,1,2] => {{1,2,3}}
=> 3
[4,3,2,1] => [3,2,1] => {{1,3},{2}}
=> 2
[1,2,3,4,5] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,3,5,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,4,3,5] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,2,4,5,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,2,5,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 1
[1,2,5,4,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 1
[1,3,2,4,5] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,2,5,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,4,2,5] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,3,4,5,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,3,5,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 1
[1,3,5,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 1
[1,4,2,3,5] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,4,2,5,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,4,3,2,5] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,3,5,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
[1,4,5,2,3] => [1,4,2,3] => {{1},{2,3,4}}
=> 1
[1,4,5,3,2] => [1,4,3,2] => {{1},{2,4},{3}}
=> 1
Description
The smallest closer of a set partition.
A closer (or right hand endpoint) of a set partition is a number that is maximal in its block. For this statistic, singletons are considered as closers.
In other words, this is the smallest among the maximal elements of the blocks.
Matching statistic: St001184
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> 1
[2,1] => [1] => [1,0]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 1
[1,3,2] => [1,2] => [1,0,1,0]
=> 1
[2,1,3] => [2,1] => [1,1,0,0]
=> 2
[2,3,1] => [2,1] => [1,1,0,0]
=> 2
[3,1,2] => [1,2] => [1,0,1,0]
=> 1
[3,2,1] => [2,1] => [1,1,0,0]
=> 2
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 3
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 3
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 3
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001291
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1,0]
=> 1
[2,1] => [1] => [1,0]
=> 1
[1,2,3] => [1,2] => [1,0,1,0]
=> 2
[1,3,2] => [1,2] => [1,0,1,0]
=> 2
[2,1,3] => [2,1] => [1,1,0,0]
=> 1
[2,3,1] => [2,1] => [1,1,0,0]
=> 1
[3,1,2] => [1,2] => [1,0,1,0]
=> 2
[3,2,1] => [2,1] => [1,1,0,0]
=> 1
[1,2,3,4] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,2,4,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,3,2,4] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,3,4,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,4,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[1,4,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[2,1,3,4] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[2,1,4,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[2,3,1,4] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,3,4,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[2,4,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[2,4,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[3,1,2,4] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,1,4,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,2,1,4] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,2,4,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[3,4,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[3,4,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[4,1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 3
[4,1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 2
[4,2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 3
[4,2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 2
[4,3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[4,3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,2,3,4,5] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,3,5,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,4,3,5] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,4,5,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,2,5,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
[1,2,5,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 3
[1,3,2,4,5] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,2,5,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,4,2,5] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,3,4,5,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,3,5,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
[1,3,5,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 3
[1,4,2,3,5] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,2,5,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,2,5] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,3,5,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,5,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 2
[1,4,5,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 2
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000051
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [.,.]
=> 0 = 1 - 1
[2,1] => [1] => [.,.]
=> 0 = 1 - 1
[1,2,3] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[1,3,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[2,1,3] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[2,3,1] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[3,1,2] => [1,2] => [.,[.,.]]
=> 0 = 1 - 1
[3,2,1] => [2,1] => [[.,.],.]
=> 1 = 2 - 1
[1,2,3,4] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,2,4,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,3,2,4] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,3,4,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[1,4,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[1,4,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[2,1,3,4] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,1,4,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,3,1,4] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[2,3,4,1] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[2,4,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[2,4,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[3,1,2,4] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,1,4,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,2,1,4] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[3,2,4,1] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[3,4,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[3,4,2,1] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[4,1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> 0 = 1 - 1
[4,1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> 0 = 1 - 1
[4,2,1,3] => [2,1,3] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,2,3,1] => [2,3,1] => [[.,[.,.]],.]
=> 2 = 3 - 1
[4,3,1,2] => [3,1,2] => [[.,.],[.,.]]
=> 1 = 2 - 1
[4,3,2,1] => [3,2,1] => [[[.,.],.],.]
=> 2 = 3 - 1
[1,2,3,4,5] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,3,5,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,4,3,5] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,4,5,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,2,5,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> 0 = 1 - 1
[1,2,5,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> 0 = 1 - 1
[1,3,2,4,5] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,3,2,5,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,3,4,2,5] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,4,5,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,3,5,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,3,5,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> 0 = 1 - 1
[1,4,2,3,5] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,2,5,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,3,2,5] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,3,5,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
[1,4,5,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> 0 = 1 - 1
[1,4,5,3,2] => [1,4,3,2] => [.,[[[.,.],.],.]]
=> 0 = 1 - 1
Description
The size of the left subtree of a binary tree.
The following 105 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000133The "bounce" of a permutation. St000439The position of the first down step of a Dyck path. 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. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000026The position of the first return of a Dyck path. St000066The column of the unique '1' in the first row of the alternating sign matrix. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000991The number of right-to-left minima of a permutation. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001497The position of the largest weak excedence of a permutation. St000010The length of the partition. St000141The maximum drop size of a permutation. St000147The largest part of an integer partition. St000316The number of non-left-to-right-maxima of a permutation. St000738The first entry in the last row of a standard tableau. St001161The major index north count of a Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000504The cardinality of the first block of a set partition. St000693The modular (standard) major index of a standard tableau. St000061The number of nodes on the left branch of a binary tree. St000654The first descent of a permutation. St000678The number of up steps after the last double rise of a Dyck path. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. 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. St000653The last descent of a permutation. St000989The number of final rises of a permutation. St001480The number of simple summands of the module J^2/J^3. St001933The largest multiplicity of a part in an integer partition. St000993The multiplicity of the largest part of an integer partition. St000288The number of ones in a binary word. St000734The last entry in the first row of a standard tableau. St000733The row containing the largest entry of a standard tableau. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000259The diameter of a connected graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001568The smallest positive integer that does not appear twice in the partition. St000260The radius of a connected graph. St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St000840The number of closers smaller than the largest opener in a perfect matching. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. 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. St001434The number of negative sum pairs of a signed permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. 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. St000681The Grundy value of Chomp on Ferrers diagrams. St001875The number of simple modules with projective dimension at most 1. St000422The energy of a graph, if it is integral. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000741The Colin de Verdière graph invariant. St001557The number of inversions of the second entry of a permutation. St000060The greater neighbor of the maximum. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001927Sparre Andersen's number of positives of a signed permutation. St001964The interval resolution global dimension of a poset. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the 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. St001556The number of inversions of the third entry of a permutation. St000392The length of the longest run of ones in a binary word. St000982The length of the longest constant subword. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001822The number of alignments of a signed permutation. St001372The length of a longest cyclic run of ones of a binary word. St001926Sparre Andersen's position of the maximum of a signed permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001623The number of doubly irreducible elements of a lattice. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree.
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