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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.)