A240177 -id:A240177 - cp's OEIS frontend

A240175 Number of partitions of n such that the least part is less than its multiplicity.

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 24, 32, 47, 60, 84, 110, 148, 191, 254, 323, 423, 535, 687, 864, 1100, 1371, 1726, 2141, 2669, 3290, 4075, 4990, 6136, 7481, 9137, 11087, 13471, 16264, 19659, 23641, 28438, 34060, 40801, 48676, 58074, 69049, 82064, 97246

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 12 partitions:  611, 5111, 4211, 41111, 3311, 32111, 311111, 2222, 22211, 221111, 2111111, 11111111.

Cf. A188216, A096403, A240176, A240177.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Min[p]]], {n, 0, z}](* A240175 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Min[p]]], {n, 0, z}](* A188216 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Min[p]]], {n, 0, z}](* A096403 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Min[p]]], {n, 0, z}] (* A240176 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240177 *)

a(n) = A188216(n) - A096403(n), for n >= 0.

A240176 Number of partitions of n such that (least part) > (multiplicity of least part).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 10, 13, 15, 21, 25, 31, 39, 50, 59, 75, 89, 111, 134, 164, 194, 240, 285, 344, 410, 493, 582, 699, 824, 981, 1157, 1369, 1606, 1901, 2223, 2613, 3054, 3579, 4166, 4871, 5658, 6590, 7645, 8877, 10264, 11900, 13733, 15868

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 5 partitions:  8, 61, 53, 44, 332.

Cf. A188216, A096403, A240175, A240177.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Min[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240175 *)
t2 = Table[Count[f[n], p_ /; Min[p] <= Count[p, Min[p]]], {n, 0, z}] (* A188216 *)
t3 = Table[Count[f[n], p_ /; Min[p] == Count[p, Min[p]]], {n, 0, z}] (* A096403 *)
t4 = Table[Count[f[n], p_ /; Min[p] > Count[p, Min[p]]], {n, 0, z}] (* A240176 *)
t5 = Table[Count[f[n], p_ /; Min[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240177 *)

a(n) + A240175(n) + A096403(n) = A000041(n), for n >= 0.

cp's OEIS Frontend

A240175 Number of partitions of n such that the least part is less than its multiplicity.

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