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Number of partitions p of n such that neither floor(mean(p)) nor ceiling(mean(p)) is a part.
5

%I #12 Apr 26 2014 21:05:38

%S 1,0,0,0,1,1,3,3,6,8,11,12,27,23,33,51,68,67,114,111,186,217,242,277,

%T 502,501,571,760,1014,1021,1649,1549,2195,2506,2777,3712,5275,4919,

%U 5800,7259,10389,9858,13987,13846,18261,23029,23314,26523,40250,39613,49286

%N Number of partitions p of n such that neither floor(mean(p)) nor ceiling(mean(p)) is a part.

%e a(6) counts these 3 partitions: 51, 42, 411.

%t z = 30; f[n_] := f[n] = IntegerPartitions[n];

%t t1 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241340 *)

%t t2 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241341 *)

%t t3 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241342 *)

%t t4 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241343 *)

%t t5 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241344 *)

%Y Cf. A241340, A241341, A241342, A241344.

%K nonn,easy

%O 0,7

%A _Clark Kimberling_, Apr 20 2014