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Number of partitions p of n such that ceiling(mean(p)) is a part and floor(mean(p)) is not.
+10
5
0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 6, 2, 13, 16, 14, 23, 41, 38, 73, 58, 94, 152, 196, 137, 271, 384, 422, 481, 751, 624, 1149, 1142, 1558, 2096, 2120, 2116, 3748, 4477, 5075, 4788, 7840, 7543, 11227, 11772, 13122, 18916, 22408, 19619, 29862, 32604, 41688
EXAMPLE
a(10) counts these 4 partitions: 541, 5311, 442, 3331.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241340 *) t2 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241341 *) t3 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241342 *) t4 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241343 *) t5 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241344 *)
Number of partitions p of n such that floor(mean(p)) is a part and ceiling(mean(p)) is not.
+10
5
0, 0, 0, 0, 0, 1, 1, 3, 4, 6, 9, 16, 11, 29, 36, 38, 51, 89, 81, 145, 134, 191, 278, 369, 290, 520, 678, 768, 875, 1320, 1161, 1961, 2009, 2624, 3453, 3733, 3650, 6131, 7244, 8187, 8097, 12563, 12301, 17770, 18725, 20962, 29260, 34902, 31199, 46507, 50889
EXAMPLE
a(10) counts these 9 partitions: 631, 6211, 532, 5221, 511111, 4222, 4111111, 331111, 31111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241340 *) t2 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241341 *) t3 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241342 *) t4 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241343 *) t5 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241344 *)
Number of partitions p of n such that neither floor(mean(p)) nor ceiling(mean(p)) is a part.
+10
5
1, 0, 0, 0, 1, 1, 3, 3, 6, 8, 11, 12, 27, 23, 33, 51, 68, 67, 114, 111, 186, 217, 242, 277, 502, 501, 571, 760, 1014, 1021, 1649, 1549, 2195, 2506, 2777, 3712, 5275, 4919, 5800, 7259, 10389, 9858, 13987, 13846, 18261, 23029, 23314, 26523, 40250, 39613, 49286
EXAMPLE
a(6) counts these 3 partitions: 51, 42, 411.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241340 *) t2 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241341 *) t3 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241342 *) t4 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241343 *) t5 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241344 *)
Number of partitions p of n such that floor(mean(p)) or ceiling(mean(p)) is a part.
+10
5
0, 1, 2, 3, 4, 6, 8, 12, 16, 22, 31, 44, 50, 78, 102, 125, 163, 230, 271, 379, 441, 575, 760, 978, 1073, 1457, 1865, 2250, 2704, 3544, 3955, 5293, 6154, 7637, 9533, 11171, 12702, 16718, 20215, 23926, 26949, 34725, 39187, 49415, 56914, 66105, 82244, 98231
EXAMPLE
a(6) counts these 8 partitions: 6, 33, 321, 3111, 222, 21111, 111111.
MATHEMATICA
z = 30; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241340 *) t2 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241341 *) t3 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241342 *) t4 = Table[Count[f[n], p_ /; ! MemberQ[p, Floor[Mean[p]]] && ! MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241343 *) t5 = Table[Count[f[n], p_ /; MemberQ[p, Floor[Mean[p]]] || MemberQ[p, Ceiling[Mean[p]]]], {n, 0, z}] (* A241344 *)
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