OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Rows n=0..50 of triangle, flattened
Richard P. Stanley, Some Combinatorial Properties of Hook Lengths, Contents, and Parts of Partitions arXiv:0807.0383 [math.CO], 2009.
FORMULA
E.g.f.: Product_{i=1..n} (1 - x^i)^(-1 - t).
EXAMPLE
Table starts as:
1;
1, 1;
4, 5, 1;
18, 29, 12, 1;
120, 218, 119, 22, 1;
840, 1814, 1285, 345, 35, 1;
7920, 18144, 14674, 5205, 805, 51, 1;
...
MATHEMATICA
w=9; MapIndexed[ CoefficientList[#1, t] Tr[#2-1]! &, CoefficientList[Series[Product[(1-x^i)^(-1-t), {i, w}], {x, 0, w}], x]];
or alternatively:
CoefficientList[#, t] & /@ Table[1/n! Tr[(NumberOfTableaux[#1]^2 Apply[Times, t + Flatten[hooklength[#1]]^2] &) /@ Partitions[n]], {n, 0, 9}]
or alternatively:
Table[1/n!Tr[NumberOfTableaux[#]^2 f[ Flatten[hooklength[#]]^2, e, k, n ]&/@ Partitions[n] ], {n, 0, 9}, {k, 0, n}]
with e and f defined as:
e[n_, v_]:= Tr[Times @@@ Select[Subsets[Table[Subscript[x, j], {j, v}]], Length[#]==n&]];
f[li_List, fun_, par_, k_]:=fun[par, k]/.Thread[Array[Subscript[x, #1]&, Length[li]]->li];
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wouter Meeussen, Mar 24 2012
STATUS
approved