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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
Manfred Krause, A simple proof of the Gale-Ryser theorem, American Mathematical Monthly, 1996.
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
Sequence in context: A185378 A285563 A285543 * A034953 A086737 A063834
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 March 29 10:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)