OFFSET
1,2
COMMENTS
A multiset is normal if it spans an initial interval of positive integers. It is aperiodic if its multiplicities are relatively prime.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..500
FORMULA
a(n) = Sum_{k=1..n} Sum_{d|k} mu(k/d) * Sum_{i=1..d} binomial(d-1, i-1)*binomial(n-k+i, i). - Andrew Howroyd, Sep 18 2018
G.f.: Sum_{d>=1} mu(d)*x^d/((1 - x - x^d*(2-x))*(1-x)). - Andrew Howroyd, Feb 04 2021
EXAMPLE
The a(4) = 26 aperiodic multisets:
(1), (2), (3), (4),
(12), (13), (14), (23), (24), (34),
(112), (113), (122), (123), (124), (133), (134), (223), (233), (234),
(1112), (1123), (1222), (1223), (1233), (1234).
MATHEMATICA
allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];
Table[Length[Select[Union@@Rest/@Subsets/@allnorm[n], GCD@@Length/@Split[#]===1&]], {n, 10}]
PROG
(PARI) seq(n)={Vec(sum(d=1, n, moebius(d)*x^d/(1 - x - x^d*(2-x)) + O(x*x^n))/(1-x))} \\ Andrew Howroyd, Feb 04 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 03 2018
EXTENSIONS
Terms a(13) and beyond from Andrew Howroyd, Sep 18 2018
STATUS
approved