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A318562 Number of combinatory separations of strongly normal multisets of weight n with strongly normal parts. 5
1, 4, 10, 32, 80, 239, 605, 1670, 4251 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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. The type of a multiset of integers is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223.

A pair h<={g_1,...,g_k} is a combinatory separation iff there exists a multiset partition of h whose multiset of block-types is {g_1,...,g_k}. For example, the (headless) combinatory separations of the multiset 1122 are {1122}, {1,112}, {1,122}, {11,11}, {12,12}, {1,1,11}, {1,1,12}, {1,1,1,1}. This list excludes {12,11} because one cannot partition 1122 into two blocks where one block has two distinct elements and the other block has two equal elements.

LINKS

Table of n, a(n) for n=1..9.

EXAMPLE

The a(3) = 10 combinatory separations:

  111<={111}

  111<={1,11}

  111<={1,1,1}

  112<={112}

  112<={1,11}

  112<={1,12}

  112<={1,1,1}

  123<={123}

  123<={1,12}

  123<={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]]}];

strnormQ[m_]:=OrderedQ[Length/@Split[m], GreaterEqual];

Table[Length[Select[Union@@Table[{m, Sort[normize/@#]}&/@mps[m], {m, strnorm[n]}], And@@strnormQ/@#[[2]]&]], {n, 6}]

CROSSREFS

Cf. A000041, A007716, A255906, A265947, A269134.

Cf. A317533, A317791, A318396, A318559, A318563, A318564, A318565, A318566.

Sequence in context: A135831 A213106 A015796 * A034717 A295404 A001673

Adjacent sequences:  A318559 A318560 A318561 * A318563 A318564 A318565

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Aug 29 2018

STATUS

approved

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Last modified June 16 21:55 EDT 2021. Contains 345080 sequences. (Running on oeis4.)