A336137 - OEIS

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A336137

Number of set partitions of the binary indices of n with equal block-sums.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1

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OFFSET

0,8

COMMENTS

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

LINKS

Table of n, a(n) for n=0..86.

Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

EXAMPLE

The a(n) set partitions for n = 7, 59, 119, 367, 127:

{123} {12456} {123567} {1234679} {1234567}

{12}{3} {126}{45} {1236}{57} {12346}{79} {1247}{356}

{15}{24}{6} {156}{237} {1249}{367} {1256}{347}

{17}{26}{35} {1267}{349} {1346}{257}

{169}{2347} {167}{2345}

{16}{25}{34}{7}

The binary indices of 382 are {2,3,4,5,6,7,9}, with equal block-sum set partitions:

{{2,7},{3,6},{4,5},{9}}

{{2,4,6},{3,9},{5,7}}

{{2,7,9},{3,4,5,6}}

{{2,3,4,9},{5,6,7}}

{{2,3,6,7},{4,5,9}}

{{2,4,5,7},{3,6,9}}

so a(382) = 7.

MATHEMATICA

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

Table[Length[Select[sps[bpe[n]], SameQ@@Total/@#&]], {n, 0, 100}]

CROSSREFS

These set partitions are counted by A035470.

The version for twice-partitions is A279787.

The version for partitions of partitions is A305551.

The version for factorizations is A321455.

The version for normal multiset partitions is A326518.

The version for distinct block-sums is A336138.

Set partitions of binary indices are A050315.

Normal multiset partitions with equal lengths are A317583.

Normal multiset partitions with equal averages are A326520.

Multiset partitions with equal block-sums are ranked by A326534.

Cf. A000110, A007837, A032011, A038041, A271619, A275780, A322794.

Sequence in context: A088530 A058060 A338160 * A371735 A088323 A003652

Adjacent sequences: A336134 A336135 A336136 * A336138 A336139 A336140

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 12 2020

STATUS

approved