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A299408 Number of partitions of n whose minimal excluded multiplicity is even. 1
0, 1, 1, 2, 2, 4, 6, 9, 11, 17, 24, 33, 41, 59, 75, 99, 128, 168, 214, 278, 347, 438, 559, 696, 866, 1084, 1346, 1653, 2044, 2500, 3070, 3741, 4559, 5507, 6712, 8061, 9746, 11699, 14076, 16794, 20141, 23943, 28588, 33904, 40291, 47593, 56461, 66462, 78495 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

This is analogous to the minimal excludant of Fraenkel.

LINKS

Table of n, a(n) for n=0..48.

A. S. Fraenkel, U. Peled, Harnessing the Unwieldy MEX Function, In: Games of No Chance 4, Proc. BIRS Workshop on Combinatorial Games, 2008, Banff, Alberta, Canada, MSRI Publ. (R. J. Nowakowski, ed.), Vol. 63, Cambridge University Press, Cambridge, 77-94, 2015.

FORMULA

a(n) = A000041(n) - A300183(n).

EXAMPLE

For the partition 1+1+2; 2 has multiplicity 1; 1 has multiplicity 2, so the smallest absent multiplicity is 3, which is odd. So this partition is not counted.

a(4) = 2: [3,1], [4].

a(5) = 4: [2,1,1,1], [3,2], [4,1], [5].

a(6) = 6: [2,1,1,1,1], [3,1,1,1], [3,2,1], [4,2], [5,1], [6].

MATHEMATICA

<< Combinatorica`;

nend = 15;

For[n = 1, n <= nend, n++, count[n] = 0;

  For[i = 1, i <= PartitionsP[n], i++, partition = Partitions[n][[i]];

    set = {};

   tally = Tally[partition];

   For[j = 1, j <= Length[tally], j++,

    set = Union[set, {tally[[j]][[2]]}]];

   comp = Complement[Range[Max[set] + 1], set];

   If[EvenQ[Min[comp]], count[n]++]

   ]];

Table[count[i], {i, 1, nend}]

CROSSREFS

Cf. A000041, A300183.

Sequence in context: A231187 A055529 A222735 * A319381 A035564 A240065

Adjacent sequences:  A299405 A299406 A299407 * A299409 A299410 A299411

KEYWORD

nonn

AUTHOR

David S. Newman, Feb 20 2018

EXTENSIONS

More terms from Alois P. Heinz, Feb 26 2018

STATUS

approved

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)