OFFSET
0,3
COMMENTS
A multiset partition is square if its length is equal to its number of distinct atoms.
EXAMPLE
The a(1) = 1 through a(6) = 20 square partitions:
{{1}} {{2}} {{3}} {{4}} {{5}} {{6}}
{{1,1}} {{1,1,1}} {{2,2}} {{1},{4}} {{3,3}}
{{1},{2}} {{1},{3}} {{2},{3}} {{1},{5}}
{{1,1,1,1}} {{1},{1,3}} {{2,2,2}}
{{1},{1,2}} {{1},{2,2}} {{2},{4}}
{{2},{1,1}} {{2},{1,2}} {{1},{1,4}}
{{3},{1,1}} {{4},{1,1}}
{{1,1,1,1,1}} {{1},{1,1,3}}
{{1},{1,1,2}} {{1,1},{1,3}}
{{1,1},{1,2}} {{1},{1,2,2}}
{{2},{1,1,1}} {{1,1},{2,2}}
{{1,2},{1,2}}
{{1},{2},{3}}
{{2},{1,1,2}}
{{3},{1,1,1}}
{{1,1,1,1,1,1}}
{{1},{1,1,1,2}}
{{1,1},{1,1,2}}
{{1,2},{1,1,1}}
{{2},{1,1,1,1}}
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n], Length[#]==Length[Union@@#]&]], {n, 8}]
PROG
(Python)
from itertools import groupby
from sympy.combinatorics.partitions import IntegerPartition
from sympy.utilities.iterables import partitions, multiset_partitions
def square(ms): return len(ms) == len(set(e for s in ms for e in s))
def msp(n): # generator of multiset partitions of integer partitions of n
yield from (m for p in partitions(n) for m in multiset_partitions(IntegerPartition(p).partition))
def a(n): return sum(1 for ms in msp(n) if square(ms))
print([a(n) for n in range(20)]) # Michael S. Branicky, Oct 12 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 11 2018
EXTENSIONS
a(13)-a(38) from Michael S. Branicky, Oct 12 2025
STATUS
approved