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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 Table of n, a(n) for n=0..13. 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}. Cf. A000085, A300121. Sequence in context: A329691 A293204 A273344 * A127717 A210236 A193998 Adjacent sequences: A370662 A370663 A370664 * A370666 A370667 A370668 KEYWORD nonn,more AUTHOR John Tyler Rascoe, Feb 24 2024 STATUS approved

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