OFFSET
0,3
COMMENTS
Also number of normal set multipartitions of weight n. These are defined as multisets of sets that together partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. Set multipartitions are involved in the expansion of elementary symmetric functions in terms of augmented monomial symmetric functions. - Gus Wiseman, Oct 22 2015
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..230
P. J. Cameron, T. Prellberg and D. Stark, Asymptotics for incidence matrix classes, arXiv:math/0510155 [math.CO], 2005-2006.
M. Klazar, Extremal problems for ordered hypergraphs, arXiv:math/0305048 [math.CO], 2003.
Gus Wiseman, Four symmetric function identities
EXAMPLE
The a(3) = 10 normal set multipartitions are: {1,1,1}, {1,12}, {1,1,2}, {2,12}, {1,2,2}, {123}, {1,23}, {2,13}, {3,12}, {1,2,3}.
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(b(n-i*j,
min(n-i*j, i-1), k)*binomial(binomial(k, i)+j-1, j), j=0..n/i)))
end:
a:= n-> add(add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..24); # Alois P. Heinz, Sep 13 2019
MATHEMATICA
MSOSA[s_List] :=
MSOSA[s] = If[Length[s] === 0, {{}}, Module[{sbs, fms},
sbs = Rest[Subsets[Union[s]]];
fms =
Function[r,
Append[#, r] & /@
MSOSA[Fold[DeleteCases[#1, #2, {1}, 1] &, s, r]]] /@ sbs;
Select[Join @@ fms, OrderedQ]
]];
mmallnorm[n_Integer] :=
Function[s, Array[Count[s, y_ /; y <= #] + 1 &, n]] /@
Subsets[Range[n - 1] + 1];
Array[Plus @@ Length /@ MSOSA /@ mmallnorm[#] &, 9]
(* Gus Wiseman, Oct 22 2015 *)
PROG
(PARI)
R(n, k)={Vec(-1 + 1/prod(j=1, k, (1 - x^j + O(x*x^n))^binomial(k, j) ))}
seq(n) = {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Sep 23 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladeta Jovovic, Mar 27 2006
EXTENSIONS
a(0)=1 prepended and more terms added by Alois P. Heinz, Sep 13 2019
STATUS
approved