- FindStat

Your data matches 51 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000734

Mapping

St000734: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 1
[[1,2]]
 => 2
[[1],[2]]
 => 1
[[1,2,3]]
 => 3
[[1,3],[2]]
 => 3
[[1,2],[3]]
 => 2
[[1],[2],[3]]
 => 1
[[1,2,3,4]]
 => 4
[[1,3,4],[2]]
 => 4
[[1,2,4],[3]]
 => 4
[[1,2,3],[4]]
 => 3
[[1,3],[2,4]]
 => 3
[[1,2],[3,4]]
 => 2
[[1,4],[2],[3]]
 => 4
[[1,3],[2],[4]]
 => 3
[[1,2],[3],[4]]
 => 2
[[1],[2],[3],[4]]
 => 1
[[1,2,3,4,5]]
 => 5
[[1,3,4,5],[2]]
 => 5
[[1,2,4,5],[3]]
 => 5
[[1,2,3,5],[4]]
 => 5
[[1,2,3,4],[5]]
 => 4
[[1,3,5],[2,4]]
 => 5
[[1,2,5],[3,4]]
 => 5
[[1,3,4],[2,5]]
 => 4
[[1,2,4],[3,5]]
 => 4
[[1,2,3],[4,5]]
 => 3
[[1,4,5],[2],[3]]
 => 5
[[1,3,5],[2],[4]]
 => 5
[[1,2,5],[3],[4]]
 => 5
[[1,3,4],[2],[5]]
 => 4
[[1,2,4],[3],[5]]
 => 4
[[1,2,3],[4],[5]]
 => 3
[[1,4],[2,5],[3]]
 => 4
[[1,3],[2,5],[4]]
 => 3
[[1,2],[3,5],[4]]
 => 2
[[1,3],[2,4],[5]]
 => 3
[[1,2],[3,4],[5]]
 => 2
[[1,5],[2],[3],[4]]
 => 5
[[1,4],[2],[3],[5]]
 => 4
[[1,3],[2],[4],[5]]
 => 3
[[1,2],[3],[4],[5]]
 => 2
[[1],[2],[3],[4],[5]]
 => 1
[[1,2,3,4,5,6]]
 => 6
[[1,3,4,5,6],[2]]
 => 6
[[1,2,4,5,6],[3]]
 => 6
[[1,2,3,5,6],[4]]
 => 6
[[1,2,3,4,6],[5]]
 => 6
[[1,2,3,4,5],[6]]
 => 5
[[1,3,5,6],[2,4]]
 => 6

Description

The last entry in the first row of a standard tableau.

Matching statistic: St000738
(load all 2 compositions to match this statistic)

Mapping

St000738: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => 1
[[1,2]]
 => 1
[[1],[2]]
 => 2
[[1,2,3]]
 => 1
[[1,3],[2]]
 => 2
[[1,2],[3]]
 => 3
[[1],[2],[3]]
 => 3
[[1,2,3,4]]
 => 1
[[1,3,4],[2]]
 => 2
[[1,2,4],[3]]
 => 3
[[1,2,3],[4]]
 => 4
[[1,3],[2,4]]
 => 2
[[1,2],[3,4]]
 => 3
[[1,4],[2],[3]]
 => 3
[[1,3],[2],[4]]
 => 4
[[1,2],[3],[4]]
 => 4
[[1],[2],[3],[4]]
 => 4
[[1,2,3,4,5]]
 => 1
[[1,3,4,5],[2]]
 => 2
[[1,2,4,5],[3]]
 => 3
[[1,2,3,5],[4]]
 => 4
[[1,2,3,4],[5]]
 => 5
[[1,3,5],[2,4]]
 => 2
[[1,2,5],[3,4]]
 => 3
[[1,3,4],[2,5]]
 => 2
[[1,2,4],[3,5]]
 => 3
[[1,2,3],[4,5]]
 => 4
[[1,4,5],[2],[3]]
 => 3
[[1,3,5],[2],[4]]
 => 4
[[1,2,5],[3],[4]]
 => 4
[[1,3,4],[2],[5]]
 => 5
[[1,2,4],[3],[5]]
 => 5
[[1,2,3],[4],[5]]
 => 5
[[1,4],[2,5],[3]]
 => 3
[[1,3],[2,5],[4]]
 => 4
[[1,2],[3,5],[4]]
 => 4
[[1,3],[2,4],[5]]
 => 5
[[1,2],[3,4],[5]]
 => 5
[[1,5],[2],[3],[4]]
 => 4
[[1,4],[2],[3],[5]]
 => 5
[[1,3],[2],[4],[5]]
 => 5
[[1,2],[3],[4],[5]]
 => 5
[[1],[2],[3],[4],[5]]
 => 5
[[1,2,3,4,5,6]]
 => 1
[[1,3,4,5,6],[2]]
 => 2
[[1,2,4,5,6],[3]]
 => 3
[[1,2,3,5,6],[4]]
 => 4
[[1,2,3,4,6],[5]]
 => 5
[[1,2,3,4,5],[6]]
 => 6
[[1,3,5,6],[2,4]]
 => 2

Description

The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].

Matching statistic: St000054
(load all 18 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000054: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => 1
[[1,2]]
 => [1,2] => 1
[[1],[2]]
 => [2,1] => 2
[[1,2,3]]
 => [1,2,3] => 1
[[1,3],[2]]
 => [2,1,3] => 2
[[1,2],[3]]
 => [3,1,2] => 3
[[1],[2],[3]]
 => [3,2,1] => 3
[[1,2,3,4]]
 => [1,2,3,4] => 1
[[1,3,4],[2]]
 => [2,1,3,4] => 2
[[1,2,4],[3]]
 => [3,1,2,4] => 3
[[1,2,3],[4]]
 => [4,1,2,3] => 4
[[1,3],[2,4]]
 => [2,4,1,3] => 2
[[1,2],[3,4]]
 => [3,4,1,2] => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => 4
[[1,2],[3],[4]]
 => [4,3,1,2] => 4
[[1],[2],[3],[4]]
 => [4,3,2,1] => 4
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 5
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 4
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 5
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 5
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 5
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 5
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 5
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 5
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 5
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 5
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 6
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => 2

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: St000505
(load all 3 compositions to match this statistic)

Mapping

Mp00284: Standard tableaux —rows⟶ Set partitions
St000505: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => {{1}}
 => 1
[[1,2]]
 => {{1,2}}
 => 2
[[1],[2]]
 => {{1},{2}}
 => 1
[[1,2,3]]
 => {{1,2,3}}
 => 3
[[1,3],[2]]
 => {{1,3},{2}}
 => 3
[[1,2],[3]]
 => {{1,2},{3}}
 => 2
[[1],[2],[3]]
 => {{1},{2},{3}}
 => 1
[[1,2,3,4]]
 => {{1,2,3,4}}
 => 4
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 4
[[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 4
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 3
[[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 3
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 2
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 4
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 3
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 2
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 1
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 5
[[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => 5
[[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 5
[[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 5
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 4
[[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => 5
[[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 5
[[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => 4
[[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 4
[[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 3
[[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 5
[[1,3,5],[2],[4]]
 => {{1,3,5},{2},{4}}
 => 5
[[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 5
[[1,3,4],[2],[5]]
 => {{1,3,4},{2},{5}}
 => 4
[[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 4
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 3
[[1,4],[2,5],[3]]
 => {{1,4},{2,5},{3}}
 => 4
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 3
[[1,2],[3,5],[4]]
 => {{1,2},{3,5},{4}}
 => 2
[[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => 3
[[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 2
[[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 5
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => 4
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => 3
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 2
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 1
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 6
[[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => 6
[[1,2,4,5,6],[3]]
 => {{1,2,4,5,6},{3}}
 => 6
[[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => 6
[[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => 6
[[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 5
[[1,3,5,6],[2,4]]
 => {{1,3,5,6},{2,4}}
 => 6

Description

The biggest entry in the block containing the 1.

Matching statistic: St000740
(load all 47 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000740: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => 1
[[1,2]]
 => [1,2] => 2
[[1],[2]]
 => [2,1] => 1
[[1,2,3]]
 => [1,2,3] => 3
[[1,3],[2]]
 => [2,1,3] => 3
[[1,2],[3]]
 => [3,1,2] => 2
[[1],[2],[3]]
 => [3,2,1] => 1
[[1,2,3,4]]
 => [1,2,3,4] => 4
[[1,3,4],[2]]
 => [2,1,3,4] => 4
[[1,2,4],[3]]
 => [3,1,2,4] => 4
[[1,2,3],[4]]
 => [4,1,2,3] => 3
[[1,3],[2,4]]
 => [2,4,1,3] => 3
[[1,2],[3,4]]
 => [3,4,1,2] => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => 4
[[1,3],[2],[4]]
 => [4,2,1,3] => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => 5
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => 5
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => 5
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => 5
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => 4
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => 5
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => 5
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => 4
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => 4
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => 3
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => 5
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => 5
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => 4
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => 4
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => 3
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => 4
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => 3
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => 5
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => 4
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => 6
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => 6
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => 6
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => 6
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => 6
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => 5
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => 6

Description

The last entry of a permutation. This statistic is undefined for the empty permutation.

Matching statistic: St000839
(load all 43 compositions to match this statistic)

Mapping

Mp00284: Standard tableaux —rows⟶ Set partitions
St000839: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => {{1}}
 => 1
[[1,2]]
 => {{1,2}}
 => 1
[[1],[2]]
 => {{1},{2}}
 => 2
[[1,2,3]]
 => {{1,2,3}}
 => 1
[[1,3],[2]]
 => {{1,3},{2}}
 => 2
[[1,2],[3]]
 => {{1,2},{3}}
 => 3
[[1],[2],[3]]
 => {{1},{2},{3}}
 => 3
[[1,2,3,4]]
 => {{1,2,3,4}}
 => 1
[[1,3,4],[2]]
 => {{1,3,4},{2}}
 => 2
[[1,2,4],[3]]
 => {{1,2,4},{3}}
 => 3
[[1,2,3],[4]]
 => {{1,2,3},{4}}
 => 4
[[1,3],[2,4]]
 => {{1,3},{2,4}}
 => 2
[[1,2],[3,4]]
 => {{1,2},{3,4}}
 => 3
[[1,4],[2],[3]]
 => {{1,4},{2},{3}}
 => 3
[[1,3],[2],[4]]
 => {{1,3},{2},{4}}
 => 4
[[1,2],[3],[4]]
 => {{1,2},{3},{4}}
 => 4
[[1],[2],[3],[4]]
 => {{1},{2},{3},{4}}
 => 4
[[1,2,3,4,5]]
 => {{1,2,3,4,5}}
 => 1
[[1,3,4,5],[2]]
 => {{1,3,4,5},{2}}
 => 2
[[1,2,4,5],[3]]
 => {{1,2,4,5},{3}}
 => 3
[[1,2,3,5],[4]]
 => {{1,2,3,5},{4}}
 => 4
[[1,2,3,4],[5]]
 => {{1,2,3,4},{5}}
 => 5
[[1,3,5],[2,4]]
 => {{1,3,5},{2,4}}
 => 2
[[1,2,5],[3,4]]
 => {{1,2,5},{3,4}}
 => 3
[[1,3,4],[2,5]]
 => {{1,3,4},{2,5}}
 => 2
[[1,2,4],[3,5]]
 => {{1,2,4},{3,5}}
 => 3
[[1,2,3],[4,5]]
 => {{1,2,3},{4,5}}
 => 4
[[1,4,5],[2],[3]]
 => {{1,4,5},{2},{3}}
 => 3
[[1,3,5],[2],[4]]
 => {{1,3,5},{2},{4}}
 => 4
[[1,2,5],[3],[4]]
 => {{1,2,5},{3},{4}}
 => 4
[[1,3,4],[2],[5]]
 => {{1,3,4},{2},{5}}
 => 5
[[1,2,4],[3],[5]]
 => {{1,2,4},{3},{5}}
 => 5
[[1,2,3],[4],[5]]
 => {{1,2,3},{4},{5}}
 => 5
[[1,4],[2,5],[3]]
 => {{1,4},{2,5},{3}}
 => 3
[[1,3],[2,5],[4]]
 => {{1,3},{2,5},{4}}
 => 4
[[1,2],[3,5],[4]]
 => {{1,2},{3,5},{4}}
 => 4
[[1,3],[2,4],[5]]
 => {{1,3},{2,4},{5}}
 => 5
[[1,2],[3,4],[5]]
 => {{1,2},{3,4},{5}}
 => 5
[[1,5],[2],[3],[4]]
 => {{1,5},{2},{3},{4}}
 => 4
[[1,4],[2],[3],[5]]
 => {{1,4},{2},{3},{5}}
 => 5
[[1,3],[2],[4],[5]]
 => {{1,3},{2},{4},{5}}
 => 5
[[1,2],[3],[4],[5]]
 => {{1,2},{3},{4},{5}}
 => 5
[[1],[2],[3],[4],[5]]
 => {{1},{2},{3},{4},{5}}
 => 5
[[1,2,3,4,5,6]]
 => {{1,2,3,4,5,6}}
 => 1
[[1,3,4,5,6],[2]]
 => {{1,3,4,5,6},{2}}
 => 2
[[1,2,4,5,6],[3]]
 => {{1,2,4,5,6},{3}}
 => 3
[[1,2,3,5,6],[4]]
 => {{1,2,3,5,6},{4}}
 => 4
[[1,2,3,4,6],[5]]
 => {{1,2,3,4,6},{5}}
 => 5
[[1,2,3,4,5],[6]]
 => {{1,2,3,4,5},{6}}
 => 6
[[1,3,5,6],[2,4]]
 => {{1,3,5,6},{2,4}}
 => 2

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: St000025
(load all 4 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ 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%

Values

[[1]]
 => [1] => [1,0]
 => 1
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 2
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 3
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 3
[[1,2,3,4]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 4
[[1,3],[2,4]]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 4
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 4
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 4
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 4
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 6
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 2

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: St000051
(load all 4 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00072: Permutations —binary search tree: left to right⟶ Binary trees
St000051: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [.,.]
 => 0 = 1 - 1
[[1,2]]
 => [1,2] => [.,[.,.]]
 => 0 = 1 - 1
[[1],[2]]
 => [2,1] => [[.,.],.]
 => 1 = 2 - 1
[[1,2,3]]
 => [1,2,3] => [.,[.,[.,.]]]
 => 0 = 1 - 1
[[1,3],[2]]
 => [2,1,3] => [[.,.],[.,.]]
 => 1 = 2 - 1
[[1,2],[3]]
 => [3,1,2] => [[.,[.,.]],.]
 => 2 = 3 - 1
[[1],[2],[3]]
 => [3,2,1] => [[[.,.],.],.]
 => 2 = 3 - 1
[[1,2,3,4]]
 => [1,2,3,4] => [.,[.,[.,[.,.]]]]
 => 0 = 1 - 1
[[1,3,4],[2]]
 => [2,1,3,4] => [[.,.],[.,[.,.]]]
 => 1 = 2 - 1
[[1,2,4],[3]]
 => [3,1,2,4] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[1,2,3],[4]]
 => [4,1,2,3] => [[.,[.,[.,.]]],.]
 => 3 = 4 - 1
[[1,3],[2,4]]
 => [2,4,1,3] => [[.,.],[[.,.],.]]
 => 1 = 2 - 1
[[1,2],[3,4]]
 => [3,4,1,2] => [[.,[.,.]],[.,.]]
 => 2 = 3 - 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [[[.,.],.],[.,.]]
 => 2 = 3 - 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [[[.,.],[.,.]],.]
 => 3 = 4 - 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [[[.,[.,.]],.],.]
 => 3 = 4 - 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[[[.,.],.],.],.]
 => 3 = 4 - 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
 => 0 = 1 - 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
 => 1 = 2 - 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[.,[.,[.,[.,.]]]],.]
 => 4 = 5 - 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[.,.],[[.,.],[.,.]]]
 => 1 = 2 - 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
 => 2 = 3 - 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[.,.],[[.,[.,.]],.]]
 => 1 = 2 - 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[.,[.,.]],[[.,.],.]]
 => 2 = 3 - 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[.,[.,[.,.]]],[.,.]]
 => 3 = 4 - 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
 => 2 = 3 - 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[[.,.],[.,.]],[.,.]]
 => 3 = 4 - 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[[.,[.,.]],.],[.,.]]
 => 3 = 4 - 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[[.,.],[.,[.,.]]],.]
 => 4 = 5 - 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[[.,[.,.]],[.,.]],.]
 => 4 = 5 - 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[[.,[.,[.,.]]],.],.]
 => 4 = 5 - 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[[.,.],.],[[.,.],.]]
 => 2 = 3 - 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[[.,.],[.,.]],[.,.]]
 => 3 = 4 - 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[[.,[.,.]],.],[.,.]]
 => 3 = 4 - 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[[.,.],[[.,.],.]],.]
 => 4 = 5 - 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[[.,[.,.]],[.,.]],.]
 => 4 = 5 - 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[[[.,.],.],.],[.,.]]
 => 3 = 4 - 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[[[.,.],.],[.,.]],.]
 => 4 = 5 - 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[[[.,.],[.,.]],.],.]
 => 4 = 5 - 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[[[.,[.,.]],.],.],.]
 => 4 = 5 - 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[[[[.,.],.],.],.],.]
 => 4 = 5 - 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [.,[.,[.,[.,[.,[.,.]]]]]]
 => 0 = 1 - 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [[.,.],[.,[.,[.,[.,.]]]]]
 => 1 = 2 - 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [[.,[.,.]],[.,[.,[.,.]]]]
 => 2 = 3 - 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [[.,[.,[.,.]]],[.,[.,.]]]
 => 3 = 4 - 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [[.,[.,[.,[.,.]]]],[.,.]]
 => 4 = 5 - 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [[.,[.,[.,[.,[.,.]]]]],.]
 => 5 = 6 - 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [[.,.],[[.,.],[.,[.,.]]]]
 => 1 = 2 - 1

Description

The size of the left subtree of a binary tree.

Matching statistic: St000439
(load all 4 compositions to match this statistic)

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000439: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1,0]
 => 2 = 1 + 1
[[1,2]]
 => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => 3 = 2 + 1
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => 3 = 2 + 1
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => 4 = 3 + 1
[[1,2,3,4]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[[1,2,4],[3]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[[1,2,3],[4]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[1,3],[2,4]]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => 3 = 2 + 1
[[1,2],[3,4]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => 4 = 3 + 1
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 4 = 3 + 1
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => 5 = 4 + 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => 4 = 3 + 1
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => 4 = 3 + 1
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => 5 = 4 + 1
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 4 = 3 + 1
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => 4 = 3 + 1
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => 5 = 4 + 1
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 4 + 1
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 5 + 1
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 3 = 2 + 1
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 5 = 4 + 1
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 6 = 5 + 1
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 7 = 6 + 1
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1

Description

The position of the first down step of a Dyck path.

Matching statistic: St000011

Mapping

Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[1]]
 => [1] => [1,0]
 => [1,0]
 => 1
[[1,2]]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[[1],[2]]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 2
[[1,2,3]]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[[1,3],[2]]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 2
[[1,2],[3]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[1],[2],[3]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[[1,2,3,4]]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,3],[2,4]]
 => [2,4,1,3] => [1,1,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0]
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,2],[3],[4]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1],[2],[3],[4]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,5]]
 => [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]
 => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 4
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3,4,5,6]]
 => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[[1,3,4,5,6],[2]]
 => [2,1,3,4,5,6] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2
[[1,2,4,5,6],[3]]
 => [3,1,2,4,5,6] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 3
[[1,2,3,5,6],[4]]
 => [4,1,2,3,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => 4
[[1,2,3,4,6],[5]]
 => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => 5
[[1,2,3,4,5],[6]]
 => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => 6
[[1,3,5,6],[2,4]]
 => [2,4,1,3,5,6] => [1,1,0,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,1,0,1,0,0]
 => 2

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.

The following 41 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St000382The first part of an integer composition. St000542The number of left-to-right-minima of a permutation. St000971The smallest closer of a set partition. St000991The number of right-to-left minima of a permutation. 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001497The position of the largest weak excedence of a permutation. St000141The maximum drop size of a permutation. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. 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. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St000061The number of nodes on the left branch of a binary tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001430The number of positive entries in a signed permutation. St000736The last entry in the first row of a semistandard tableau. St000739The first entry in the last row of a semistandard tableau. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000199The column of the unique '1' in the last row of the alternating sign matrix. 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. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000259The diameter of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001645The pebbling number of a connected graph. St000260The radius of a connected graph. St001631The number of simple modules

$S$ with

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

$A$ of the poset. St001896The number of right descents of a signed permutations. St000264The girth of a graph, which is not a tree. St001946The number of descents in a parking function. St001207The Lowey length of the algebra

$A/T$ when

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

$K[x]/(x^n)$ . St001330The hat guessing number of a graph. St001877Number of indecomposable injective modules with projective dimension 2.