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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
OFFSET
0,4
COMMENTS
This is analogous to the minimal excludant of Fraenkel.
LINKS
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
Sequence in context: A055529 A337723 A222735 * A338937 A319381 A340496
KEYWORD
nonn
AUTHOR
David S. Newman, Feb 20 2018
EXTENSIONS
More terms from Alois P. Heinz, Feb 26 2018
STATUS
approved