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Number of partitions of n whose minimal excluded multiplicity is even.
1

%I #29 Feb 27 2018 19:08:26

%S 0,1,1,2,2,4,6,9,11,17,24,33,41,59,75,99,128,168,214,278,347,438,559,

%T 696,866,1084,1346,1653,2044,2500,3070,3741,4559,5507,6712,8061,9746,

%U 11699,14076,16794,20141,23943,28588,33904,40291,47593,56461,66462,78495

%N Number of partitions of n whose minimal excluded multiplicity is even.

%C This is analogous to the minimal excludant of Fraenkel.

%H A. S. Fraenkel, U. Peled, <a href="http://www.wisdom.weizmann.ac.il/~fraenkel/Papers/Harnessing.The.Unwieldy.MEX.Function_2.pdf">Harnessing the Unwieldy MEX Function</a>, 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.

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

%e 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.

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

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

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

%t << Combinatorica`;

%t nend = 15;

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

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

%t set = {};

%t tally = Tally[partition];

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

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

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

%t If[EvenQ[Min[comp]], count[n]++]

%t ]];

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

%Y Cf. A000041, A300183.

%K nonn

%O 0,4

%A _David S. Newman_, Feb 20 2018

%E More terms from _Alois P. Heinz_, Feb 26 2018