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 A318396 Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, the first having block sizes y and the second v. 10
 1, 1, 3, 6, 15, 28, 64, 116, 238, 430, 818, 1426, 2618, 4439, 7775, 12993, 22025, 35946, 59507, 95319, 154073, 243226, 385192, 598531, 933096, 1429794, 2193699, 3322171, 5027995, 7524245, 11253557, 16661211, 24637859, 36130242, 52879638, 76830503, 111422013, 160505622 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. a(n) is also the number of combinatory separations (see A269134 for definition) of strongly normal multisets of size n into normal sets. From Andrew Howroyd, Oct 31 2019: (Start) Also, the number of distinct unordered row and column sums of binary matrices without empty columns or rows and with a total of n ones. Only matrices in which both row and columns sums are weakly increasing need to be considered. By the Gale-Ryser theorem this is equivalent to the number of pairs of integer partitions (y,v) of n with y dominating v. (End) LINKS Andrew Howroyd, Table of n, a(n) for n = 0..100 Manfred Krause, A simple proof of the Gale-Ryser theorem, American Mathematical Monthly, 1996. Wikipedia, Gale-Ryser theorem EXAMPLE The a(4) = 15 pairs of integer partitions:      4, 1111     22, 22     22, 211     22, 1111     31, 211     31, 1111    211, 22    211, 31    211, 211    211, 1111   1111, 4   1111, 22   1111, 31   1111, 211   1111, 1111 The a(4) = 15 combinatory separations:   1111<={1,1,1,1}   1112<={1,1,12}   1112<={1,1,1,1}   1122<={12,12}   1122<={1,1,12}   1122<={1,1,1,1}   1123<={1,123}   1123<={12,12}   1123<={1,1,12}   1123<={1,1,1,1}   1234<={1234}   1234<={1,123}   1234<={12,12}   1234<={1,1,12}   1234<={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]]]]; strnorm[n_]:=Flatten[MapIndexed[Table[#2, {#1}]&, #]]&/@IntegerPartitions[n]; normize[m_]:=m/.Rule@@@Table[{Union[m][[i]], i}, {i, Length[Union[m]]}]; Table[Length[Select[Union@@Table[{m, Sort[normize/@#]}&/@mps[m], {m, strnorm[n]}], And@@UnsameQ@@@#[[2]]&]], {n, 6}] PROG (PARI) IsDom(p, q)=if(#q<#p, 0, my(s=0, t=0); for(i=0, #p-1, s+=p[#p-i]; t+=q[#q-i]; if(t>s, return(0))); 1) a(n)={if(n<1, n==0, my(s=0); forpart(p=n, forpart(q=n, s+=IsDom(p, q), [1, p[#p]], [#p, n])); s)} \\ Andrew Howroyd, Oct 31 2019 (PARI) \\ faster version. a(n)={local(Cache=Map());   my(recurse(b, c, s, t)=my(hk=Vecsmall([b, c, s, t]), z);      if(!mapisdefined(Cache, hk, &z),        z = if(s, sum(i=1, min(s, b), sum(j=1, min(t-s+i, c), self()(i, j, s-i, t-j))),            if(t, sum(j=1, min(t, c), self()(b, j, s, t-j)), 1));        mapput(Cache, hk, z)); z);   recurse(n, n, n, n) } \\ Andrew Howroyd, Oct 31 2019 CROSSREFS Cf. A000041, A000110, A001247, A007716, A008277, A029894, A049311, A059849, A116540, A181939, A265947, A269134, A318393, A318394, A327913. Sequence in context: A185378 A285563 A285543 * A034953 A086737 A063834 Adjacent sequences:  A318393 A318394 A318395 * A318397 A318398 A318399 KEYWORD nonn AUTHOR Gus Wiseman, Aug 25 2018 EXTENSIONS Terms a(9) and beyond from Andrew Howroyd, Oct 31 2019 STATUS approved

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Last modified December 16 03:14 EST 2019. Contains 330013 sequences. (Running on oeis4.)