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A370665 Number of standard hexagonal Young tableaux with n cells. 1
1, 1, 1, 3, 2, 6, 7, 16, 19, 63, 83, 172, 485, 833 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
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.
LINKS
John Tyler Rascoe, Illustration of example.
John Tyler Rascoe, Python program.
Wikipedia, Young tableau.
EXAMPLE
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
PROG
(Python) # see linked program
CROSSREFS
Cf. A173258 counts compositions where differences between neighboring parts are in {-1,1}.
Sequence in context: A329691 A293204 A273344 * A127717 A210236 A193998
KEYWORD
nonn,more
AUTHOR
John Tyler Rascoe, Feb 24 2024
STATUS
approved

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Last modified June 14 03:16 EDT 2024. Contains 373392 sequences. (Running on oeis4.)