cp's OEIS Frontend

A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.

%I A320451 #12 Feb 04 2022 11:21:17
%S A320451 1,1,3,5,8,7,19,11,24,26,38,28,85,46,89,99,146,110,246,163,326,305,
%T A320451 416,376,816,591,903,971,1450,1295,2517,1916,3045,3141,4042,4117,7073,
%U A320451 5736,8131,9026,12658,11514,19459,16230,24638,27129,33747,32279,55778,45761,71946
%N A320451 Number of multiset partitions of uniform integer partitions of n in which all parts have the same length.
%C A320451 An integer partitions is uniform if all parts appear with the same multiplicity.
%C A320451 Terms can be computed by the formula: Sum_{d|n} Sum_{i>=1} P(n/d,i) * Sum_{h|i*d} M(i*d/h, i, h, d) where P(n,k) is the number of partitions of n into k distinct parts and M(h,w,r,s) is the number of nonnegative integer h X w matrices up to row permutations with all row sums equal to r and all column sums equal to s. The cases of M(h,w,w,h) and M(n,n,k,k) are enumerated by the arrays A257462 and A257463. - _Andrew Howroyd_, Feb 04 2022
%H A320451 Andrew Howroyd, <a href="/A320451/b320451.txt">Table of n, a(n) for n = 0..150</a>
%H A320451 A. David Christopher and M. Davamani Christober, <a href="http://emis.impa.br/EMIS/journals/GMN/yahoo_site_admin/assets/docs/1_GMN-2492-V13N2.77213831.pdf">Relatively Prime Uniform Partitions</a>, Gen. Math. Notes, Vol. 13, No. 2, December, 2012, pp. 1-12.
%e A320451 The a(9) = 26 multiset partitions:
%e A320451   {{9}}
%e A320451   {{1,8}}
%e A320451   {{2,7}}
%e A320451   {{3,6}}
%e A320451   {{4,5}}
%e A320451   {{1,2,6}}
%e A320451   {{1,3,5}}
%e A320451   {{1},{8}}
%e A320451   {{2,3,4}}
%e A320451   {{2},{7}}
%e A320451   {{3,3,3}}
%e A320451   {{3},{6}}
%e A320451   {{4},{5}}
%e A320451   {{1},{2},{6}}
%e A320451   {{1},{3},{5}}
%e A320451   {{2},{3},{4}}
%e A320451   {{3},{3},{3}}
%e A320451   {{1,1,1,2,2,2}}
%e A320451   {{1,1,1},{2,2,2}}
%e A320451   {{1,1,2},{1,2,2}}
%e A320451   {{1,1},{1,2},{2,2}}
%e A320451   {{1,2},{1,2},{1,2}}
%e A320451   {{1,1,1,1,1,1,1,1,1}}
%e A320451   {{1,1,1},{1,1,1},{1,1,1}}
%e A320451   {{1},{1},{1},{2},{2},{2}}
%e A320451   {{1},{1},{1},{1},{1},{1},{1},{1},{1}}
%t A320451 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A320451 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A320451 Table[Length[Select[Join@@mps/@IntegerPartitions[n],And[SameQ@@Length/@Split[Sort[Join@@#]],SameQ@@Length/@#]&]],{n,10}]
%Y A320451 Cf. A001970, A047966, A047968, A050342, A089259, A258466, A261049, A305551, A319056, A319066, A320328, A320330, A320331.
%K A320451 nonn
%O A320451 0,3
%A A320451 _Gus Wiseman_, Oct 12 2018
%E A320451 Terms a(11) and beyond from _Andrew Howroyd_, Feb 04 2022