login
A299968
Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.
16
1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296
OFFSET
0,3
COMMENTS
A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..50 (first 46 terms from Ludovic Schwob)
D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics, Vol. 34, No. 3 (1970), 709-727.
Wikipedia, Young tableau
FORMULA
a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023
EXAMPLE
The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3 1 2 4 1 3 4 1 2 3 1 2 3
4 3 2 2 3
.
1 2 1 3 1 2
3 4 2 4 2 3
.
1 2 1 3 1 2 1 4 1 3
3 2 2 2 2
4 4 3 3 3
.
1
2
3
4
MATHEMATICA
unddis[y_]:=DeleteCases[y-#, 0]&/@Tuples[Table[If[y[[i]]>Append[y, 0][[i+1]], {0, 1}, {0}], {i, Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]}, If[Length[sam]===0, If[Total[y]===0, {{}}, {}], Join@@Table[Prepend[#, y]&/@dos[sam[[k]]], {k, 1, Length[sam]}]]];
Table[Sum[Length[dos[y]], {y, IntegerPartitions[n]}], {n, 1, 8}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 26 2018
EXTENSIONS
More terms from Ludovic Schwob, Sep 23 2023
STATUS
approved