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A240311
Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (maximal part of p).
5
1, 1, 1, 2, 4, 5, 8, 10, 16, 22, 32, 42, 59, 76, 103, 134, 179, 228, 301, 382, 494, 623, 796, 995, 1259, 1564, 1957, 2419, 3005, 3690, 4552, 5562, 6815, 8288, 10095, 12218, 14808, 17842, 21514, 25823, 30999, 37058, 44306, 52775, 62851, 74613, 88538, 104764
OFFSET
0,4
FORMULA
a(n) = A240310(n) + A240312(n) for n >= 0.
a(n) + A240314(n) = A000041(n) for n >= 0.
EXAMPLE
a(6) counts these 8 partitions: 6, 51, 42, 411, 33, 3111, 321, 2211.
MATHEMATICA
z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]] (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] < Max[p]], {n, 0, z}] (* A240310 *)
Table[Count[f[n], p_ /; m[p] <= Max[p]], {n, 0, z}] (* A240311 *)
Table[Count[f[n], p_ /; m[p] == Max[p]], {n, 0, z}] (* A240312 *)
Table[Count[f[n], p_ /; m[p] >= Max[p]], {n, 0, z}] (* A240313 *)
Table[Count[f[n], p_ /; m[p] > Max[p]], {n, 0, z}] (* A240314 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 05 2014
STATUS
approved