OFFSET
0,2
COMMENTS
a(n-1,k) is conjectured to also be the count of monomials (or terms) in the Schur polynomials of k variables and degree n, summed over all partitions of n in at most k parts (zero-padded to length k).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Wikipedia, Schur Polynomial
EXAMPLE
a(3)=85 since the Schur polynomial of 5 variables and degree 4 starts off as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.
MATHEMATICA
Tr[toz/@(Function[q, PadRight[q, k]]/@ (TransposePartition/@ Partitions[n, k]))/. w[arg__] -> 1 ]; with toz[p_]:=Block[{a, q, e, w}, u1=Expand[q Together[Expand[schur[p]]] +q a]/. Plus-> List ; u2=u1/. Times->w /. q->Sequence[]/. w[i_Integer, r__]-> i w[r] /. x[_]^(e_:1) ->e ; u3=Plus@@ u2/. w[arg__]:> Reverse@ Sort@ w[arg] /. w[a]->0 ]; and schur[p_]:=Block[{le=Length[p], n=Tr[p]}, Together[Expand[Factor[Det[Outer[ #2^#1&, p+le-Range[le] , Array[x, le]]]]/Factor[Det[Outer[ #2^#1&, Range[le-1, 0, -1] , Array[x, le]]]] ]] ]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wouter Meeussen, Oct 24 2010
STATUS
approved