Processing math: 29%

Your data matches 68 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
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
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 22 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
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 π of n, together with its rotations, obtained by conjugating with the long cycle (1,…,n). Drawing the labels 1 to n in this order on a circle, and the arcs (i,π(i)) as straight lines, the rotation of π is obtained by replacing each number i by (imod. 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
Description
The biggest entry in the block containing the 1.
Matching statistic: St000740
(load all 52 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
Description
The last entry of a permutation. This statistic is undefined for the empty permutation.
Matching statistic: St000839
(load all 23 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
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
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: St000193
(load all 4 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000193: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [[1]]
 => 1
[[1,2]]
 => [1,2] => [[1,0],[0,1]]
 => 1
[[1],[2]]
 => [2,1] => [[0,1],[1,0]]
 => 2
[[1,2,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 1
[[1,3],[2]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 2
[[1,2],[3]]
 => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 3
[[1],[2],[3]]
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 3
[[1,2,3,4]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 1
[[1,3,4],[2]]
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 2
[[1,2,4],[3]]
 => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 3
[[1,2,3],[4]]
 => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 4
[[1,3],[2,4]]
 => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 2
[[1,2],[3,4]]
 => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 3
[[1,4],[2],[3]]
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 3
[[1,3],[2],[4]]
 => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 4
[[1,2],[3],[4]]
 => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 4
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 4
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 1
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 2
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 4
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 5
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 2
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 3
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 2
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 3
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
 => 4
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 3
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 4
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 4
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 5
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 5
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
 => 5
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 3
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
 => 4
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
 => 4
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
 => 5
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 5
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 4
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 5
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
 => 5
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 5
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 5
Description
The row of the unique '1' in the first column of the alternating sign matrix.
Matching statistic: St000200
(load all 7 compositions to match this statistic)
Mapping
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000200: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [1] => [[1]]
 => 1
[[1,2]]
 => [1,2] => [[1,0],[0,1]]
 => 2
[[1],[2]]
 => [2,1] => [[0,1],[1,0]]
 => 1
[[1,2,3]]
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => 3
[[1,3],[2]]
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => 3
[[1,2],[3]]
 => [3,1,2] => [[0,1,0],[0,0,1],[1,0,0]]
 => 2
[[1],[2],[3]]
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => 1
[[1,2,3,4]]
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[1,3,4],[2]]
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => 4
[[1,2,4],[3]]
 => [3,1,2,4] => [[0,1,0,0],[0,0,1,0],[1,0,0,0],[0,0,0,1]]
 => 4
[[1,2,3],[4]]
 => [4,1,2,3] => [[0,1,0,0],[0,0,1,0],[0,0,0,1],[1,0,0,0]]
 => 3
[[1,3],[2,4]]
 => [2,4,1,3] => [[0,0,1,0],[1,0,0,0],[0,0,0,1],[0,1,0,0]]
 => 3
[[1,2],[3,4]]
 => [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
 => 2
[[1,4],[2],[3]]
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => 4
[[1,3],[2],[4]]
 => [4,2,1,3] => [[0,0,1,0],[0,1,0,0],[0,0,0,1],[1,0,0,0]]
 => 3
[[1,2],[3],[4]]
 => [4,3,1,2] => [[0,0,1,0],[0,0,0,1],[0,1,0,0],[1,0,0,0]]
 => 2
[[1],[2],[3],[4]]
 => [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
 => 1
[[1,2,3,4,5]]
 => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[[1,3,4,5],[2]]
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[[1,2,4,5],[3]]
 => [3,1,2,4,5] => [[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[[1,2,3,5],[4]]
 => [4,1,2,3,5] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 5
[[1,2,3,4],[5]]
 => [5,1,2,3,4] => [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[[1,3,5],[2,4]]
 => [2,4,1,3,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 5
[[1,2,5],[3,4]]
 => [3,4,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1]]
 => 5
[[1,3,4],[2,5]]
 => [2,5,1,3,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 4
[[1,2,4],[3,5]]
 => [3,5,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0]]
 => 4
[[1,2,3],[4,5]]
 => [4,5,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0]]
 => 3
[[1,4,5],[2],[3]]
 => [3,2,1,4,5] => [[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => 5
[[1,3,5],[2],[4]]
 => [4,2,1,3,5] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 5
[[1,2,5],[3],[4]]
 => [4,3,1,2,5] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 5
[[1,3,4],[2],[5]]
 => [5,2,1,3,4] => [[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[[1,2,4],[3],[5]]
 => [5,3,1,2,4] => [[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[[1,2,3],[4],[5]]
 => [5,4,1,2,3] => [[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
 => 3
[[1,4],[2,5],[3]]
 => [3,2,5,1,4] => [[0,0,0,1,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0]]
 => 4
[[1,3],[2,5],[4]]
 => [4,2,5,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0]]
 => 3
[[1,2],[3,5],[4]]
 => [4,3,5,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0]]
 => 2
[[1,3],[2,4],[5]]
 => [5,2,4,1,3] => [[0,0,0,1,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[1,0,0,0,0]]
 => 3
[[1,2],[3,4],[5]]
 => [5,3,4,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[1,0,0,0,0]]
 => 2
[[1,5],[2],[3],[4]]
 => [4,3,2,1,5] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1]]
 => 5
[[1,4],[2],[3],[5]]
 => [5,3,2,1,4] => [[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[1,0,0,0,0]]
 => 4
[[1,3],[2],[4],[5]]
 => [5,4,2,1,3] => [[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1],[0,1,0,0,0],[1,0,0,0,0]]
 => 3
[[1,2],[3],[4],[5]]
 => [5,4,3,1,2] => [[0,0,0,1,0],[0,0,0,0,1],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 2
[[1],[2],[3],[4],[5]]
 => [5,4,3,2,1] => [[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[1,0,0,0,0]]
 => 1
Description
The row of the unique '1' in the last column of the alternating sign matrix.
Matching statistic: St000736
Mapping
Mp00082: Standard tableaux —to Gelfand-Tsetlin pattern⟶ Gelfand-Tsetlin patterns
Mp00036: Gelfand-Tsetlin patterns —to semistandard tableau⟶ Semistandard tableaux
St000736: Semistandard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1]]
 => [[1]]
 => [[1]]
 => 1
[[1,2]]
 => [[2,0],[1]]
 => [[1,2]]
 => 2
[[1],[2]]
 => [[1,1],[1]]
 => [[1],[2]]
 => 1
[[1,2,3]]
 => [[3,0,0],[2,0],[1]]
 => [[1,2,3]]
 => 3
[[1,3],[2]]
 => [[2,1,0],[1,1],[1]]
 => [[1,3],[2]]
 => 3
[[1,2],[3]]
 => [[2,1,0],[2,0],[1]]
 => [[1,2],[3]]
 => 2
[[1],[2],[3]]
 => [[1,1,1],[1,1],[1]]
 => [[1],[2],[3]]
 => 1
[[1,2,3,4]]
 => [[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4]]
 => 4
[[1,3,4],[2]]
 => [[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2]]
 => 4
[[1,2,4],[3]]
 => [[3,1,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,4],[3]]
 => 4
[[1,2,3],[4]]
 => [[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4]]
 => 3
[[1,3],[2,4]]
 => [[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4]]
 => 3
[[1,2],[3,4]]
 => [[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4]]
 => 2
[[1,4],[2],[3]]
 => [[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3]]
 => 4
[[1,3],[2],[4]]
 => [[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4]]
 => 3
[[1,2],[3],[4]]
 => [[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4]]
 => 2
[[1],[2],[3],[4]]
 => [[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1],[2],[3],[4]]
 => 1
[[1,2,3,4,5]]
 => [[5,0,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4,5]]
 => 5
[[1,3,4,5],[2]]
 => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4,5],[2]]
 => 5
[[1,2,4,5],[3]]
 => [[4,1,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,4,5],[3]]
 => 5
[[1,2,3,5],[4]]
 => [[4,1,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,5],[4]]
 => 5
[[1,2,3,4],[5]]
 => [[4,1,0,0,0],[4,0,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3,4],[5]]
 => 4
[[1,3,5],[2,4]]
 => [[3,2,0,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,5],[2,4]]
 => 5
[[1,2,5],[3,4]]
 => [[3,2,0,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,5],[3,4]]
 => 5
[[1,3,4],[2,5]]
 => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2,5]]
 => 4
[[1,2,4],[3,5]]
 => [[3,2,0,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,4],[3,5]]
 => 4
[[1,2,3],[4,5]]
 => [[3,2,0,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4,5]]
 => 3
[[1,4,5],[2],[3]]
 => [[3,1,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4,5],[2],[3]]
 => 5
[[1,3,5],[2],[4]]
 => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3,5],[2],[4]]
 => 5
[[1,2,5],[3],[4]]
 => [[3,1,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2,5],[3],[4]]
 => 5
[[1,3,4],[2],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[1,1],[1]]
 => [[1,3,4],[2],[5]]
 => 4
[[1,2,4],[3],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[2,1,0],[2,0],[1]]
 => [[1,2,4],[3],[5]]
 => 4
[[1,2,3],[4],[5]]
 => [[3,1,1,0,0],[3,1,0,0],[3,0,0],[2,0],[1]]
 => [[1,2,3],[4],[5]]
 => 3
[[1,4],[2,5],[3]]
 => [[2,2,1,0,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2,5],[3]]
 => 4
[[1,3],[2,5],[4]]
 => [[2,2,1,0,0],[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,5],[4]]
 => 3
[[1,2],[3,5],[4]]
 => [[2,2,1,0,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,5],[4]]
 => 2
[[1,3],[2,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2,4],[5]]
 => 3
[[1,2],[3,4],[5]]
 => [[2,2,1,0,0],[2,2,0,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3,4],[5]]
 => 2
[[1,5],[2],[3],[4]]
 => [[2,1,1,1,0],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1,5],[2],[3],[4]]
 => 5
[[1,4],[2],[3],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[1,1,1],[1,1],[1]]
 => [[1,4],[2],[3],[5]]
 => 4
[[1,3],[2],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[1,1],[1]]
 => [[1,3],[2],[4],[5]]
 => 3
[[1,2],[3],[4],[5]]
 => [[2,1,1,1,0],[2,1,1,0],[2,1,0],[2,0],[1]]
 => [[1,2],[3],[4],[5]]
 => 2
[[1],[2],[3],[4],[5]]
 => [[1,1,1,1,1],[1,1,1,1],[1,1,1],[1,1],[1]]
 => [[1],[2],[3],[4],[5]]
 => 1
Description
The last entry in the first row of a semistandard tableau.
The following 58 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000739The first entry in the last row of a semistandard tableau. 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. St000051The size of the left subtree of a binary tree. St000439The position of the first down step of a Dyck path. 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. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000240The number of indices that are not small excedances. 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. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. 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. St001202Call 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. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001430The number of positive entries in a signed permutation. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000019The cardinality of the support of a permutation. St000141The maximum drop size of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000444The length of the maximal rise of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. 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. 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. 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. 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. St000678The number of up steps after the last double rise of a Dyck path. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000956The maximal displacement of a permutation. St000989The number of final rises of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000454The largest eigenvalue of a graph if it is integral. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St001896The number of right descents of a signed permutations. St001631The number of simple modules S with dim Ext^1(S,A)=1 in the incidence algebra A of the poset. 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). St001645The pebbling number of a connected graph. St000260The radius of a connected graph. St001877Number of indecomposable injective modules with projective dimension 2. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St001060The distinguishing index of a graph. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000264The girth of a graph, which is not a tree. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. 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.