cp's OEIS Frontend

A370665 Number of standard hexagonal Young tableaux with n cells.

%I A370665 #9 Feb 25 2024 09:10:24
%S A370665 1,1,1,3,2,6,7,16,19,63,83,172,485,833
%N A370665 Number of standard hexagonal Young tableaux with n cells.
%C A370665 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.
%C A370665 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.
%C A370665 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.
%H A370665 John Tyler Rascoe, <a href="/A370665/a370665.png">Illustration of example</a>.
%H A370665 John Tyler Rascoe, <a href="/A370665/a370665_1.py.txt">Python program</a>.
%H A370665 Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableaux">Young tableau</a>.
%e A370665 The integer composition [2,3,2] of 7, corresponds to the hexagonal Young diagram:
%e A370665   diagram      tableau
%e A370665     0 0          1 2
%e A370665    0 0 0  --->  3 4 5
%e A370665     0 0          6 7
%e A370665 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.
%e A370665 The a(5) = 6 hexagonal Young tableaux with 5 cells are:
%e A370665  1 2 3 4 5    1 2     1 2      1 3    1 2 3    1 2 4
%e A370665                3     3 4 5    2 4 5    4 5      3 5
%e A370665               4 5
%o A370665 (Python) # see linked program
%Y A370665 Cf. A173258 counts compositions where differences between neighboring parts are in {-1,1}.
%Y A370665 Cf. A000085, A300121.
%K A370665 nonn,more
%O A370665 0,4
%A A370665 _John Tyler Rascoe_, Feb 24 2024