A370665 Number of standard hexagonal Young tableaux with n cells.

1, 1, 1, 3, 2, 6, 7, 16, 19, 63, 83, 172, 485, 833

Offset: 0

Author

John Tyler Rascoe, Feb 24 2024

Keywords

• nonn
• more

Comments

A hexagonal Young diagram is a finite collection of hexagons or cells, arranged in center justified rows, with row lengths being either -1 or +1 of any adjacent rows above or below a given row, see illustration in links.

Reading the number of hexagons by row gives a integer composition (ordered partition) where differences between neighboring parts are in {-1,1}. These diagrams can also be drawn with tangent circles on a hexagonal grid oriented pointy side up, see illustration link in A173258.

A standard hexagonal Young tableau is then created by filling the cells of a hexagonal Young diagram with numbers {1..n} such that all rows and downward diagonals form increasing sequences. For every hexagonal young diagram there is at least one hexagonal Young tableau.

Examples

			The integer composition [2,3,2] of 7, corresponds to the hexagonal Young diagram:
  diagram      tableau
    0 0          1 2
   0 0 0  --->  3 4 5
    0 0          6 7
Then filling in the cells sequentially by rows gives the tableau having rows [[1,2], [3,4,5], [6,7]] right diagonals [[3,6], [1,4,7], [2,5]] and left diagonals [[1,3], [2,4,6], [5,7]]; all of which contain increasing sequences.
The a(5) = 6 hexagonal Young tableaux with 5 cells are:
 1 2 3 4 5    1 2     1 2      1 3    1 2 3    1 2 4
               3     3 4 5    2 4 5    4 5      3 5
              4 5
		

Links

• John Tyler Rascoe, Illustration of example.
• John Tyler Rascoe, Python program.
• Wikipedia, Young tableau.

Crossrefs

Cf. A173258 counts compositions where differences between neighboring parts are in {-1,1}.

Cf. A000085, A300121.

Programs

Python

# see linked program