A240178 -id:A240178 - cp's OEIS frontend

A240179 Number of partitions of n such that (least part) <= (multiplicity of greatest part).

0, 1, 1, 2, 4, 5, 8, 11, 17, 23, 33, 43, 61, 79, 108, 140, 187, 238, 314, 397, 513, 648, 826, 1032, 1307, 1622, 2029, 2508, 3113, 3821, 4713, 5754, 7048, 8569, 10431, 12618, 15290, 18413, 22193, 26628, 31954, 38184, 45639, 54340, 64694, 76780, 91077, 107732

Offset: 0

Clark Kimberling, Apr 02 2014

Also, for n >= 0, a(n) is the number of partitions of n such that (greatest part) >= (multiplicity of least part). For n >=1, a(n) is also the number of partitions of n such that (least part) >= (multiplicity of greatest part), as well as the number of partitions p of n such that min(p) = min(c(p)), where c = conjugate..

			a(6) counts these 8 partitions:  51, 411, 321, 3111, 222, 2211, 21111, 111111.

Cf. A240178, A240180, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}]  (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

a(n) = A240178(n) + A240180(n), for n >= 0.

2*a(n) + A240180(n) = A000041(n) for n >= 0.

A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

Original entry on oeis.org

0, 2, 3, 4, 6, 10, 14, 21, 28, 40, 53, 74, 97, 131, 171, 225, 290, 377, 480, 616, 779, 987, 1238, 1556, 1935, 2411, 2981, 3685, 4527, 5562, 6793, 8295, 10081, 12241, 14805, 17890, 21538, 25906, 31062, 37201, 44429, 53004, 63070, 74964, 88898, 105297

Offset: 1

Clark Kimberling, Apr 02 2014

Let # denote "number of" and c(p) = conjugate of partitionp.  Then
A240057(n) = # p such that min(p) not = max(c(p));
A039899(n) = # p such that min(p) < max(c(p));
A039900(n) = # p such that min(p) <= max(c(p));
A006141(n) = # p such that min(p) = max(c(p));
A003114(n) = # p such that min(p) > max(c(p));
A003016(n) = # p such that min(p) >= max(c(p));
A064173(n) = # p such that max(p) < max(c(p));
A064174(n) = # p such that max(p) <= max(c(p));
A047993(n) = # p such that max(p) = max(c(p)).
See A240178 for related sequences. - Clark Kimberling, Apr 11 2014

			a(9) = 28 counts all the 30 partitions of 9 except 333 and 2211111.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A003106, A003114, A006141, A039899, A039900.

b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i)))))
    end:
a:= n->combinat[numbpart](n)-add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..50);  # Alois P. Heinz, Apr 03 2014

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Max[p]]], {n, 0, z}]  (* A003106 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Max[p]]], {n, 0, z}] (* A003114 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Max[p]]], {n, 0, z}] (* A006141 *)
tt = Table[Count[f[n], p_ /; Max[p] != Count[p, Max[p]]], {n, 0, z}] (* A240057 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Max[p]]], {n, 0, z}] (* A039899 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Max[p]]], {n, 0, z}] (* A039900 *)
(* second program: *)
b[n_, i_] := b[n, i] = If[n < 0, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]]];
a[n_] := PartitionsP[n] - Sum[b[n - j^2, j - 1], {j, 0, Sqrt[n]}];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

a(n) + A006141(n) = A000041(n) for n > 0.

A240180 Number of partitions of n such that (least part) = (multiplicity of greatest part).

Original entry on oeis.org

0, 1, 0, 1, 3, 3, 5, 7, 12, 16, 24, 30, 45, 57, 81, 104, 143, 179, 243, 304, 399, 504, 650, 809, 1039, 1286, 1622, 2006, 2508, 3077, 3822, 4666, 5747, 6995, 8552, 10353, 12603, 15189, 18371, 22071, 26570, 31785, 38104, 45419, 54213, 64426, 76596, 90710

Offset: 0

Clark Kimberling, Apr 02 2014

Also the number of partitions p of n such that min(p) = min(conjugate(p)). Example:a(7) counts these 7 partitions: 61, 511, 421, 4111, 3211, 31111, 211111, of which the respective conjugates are 211111, 31111, 3211, 4111, 421, 511, 61. - Clark Kimberling, Apr 11 2014

			a(6) counts these 5 partitions:  51, 411, 321, 3111, 21111.

Cf. A240178, A240179, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Max[p]]], {n, 0, z}]  (* A240178 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Max[p]]], {n, 0, z}] (* A240179 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Max[p]]], {n, 0, z}] (* A240180 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240178, n>0 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Max[p]]], {n, 0, z}] (* A240179, n>0 *)

a(n) = A240179(n) - A240178(n), for n >= 0.

a(n) + 2*A240178(n) = A000041(n) for n >= 0.

A240182 Number of partitions of n such that (greatest part) <= (multiplicity of least part).

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 5, 4, 8, 10, 13, 15, 25, 25, 37, 46, 61, 70, 97, 112, 150, 177, 224, 270, 347, 407, 508, 611, 754, 895, 1106, 1304, 1594, 1892, 2283, 2708, 3262, 3835, 4595, 5421, 6452, 7574, 8993, 10530, 12445, 14564, 17123, 19992, 23465, 27302, 31931

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 8 partitions:  41111, 32111, 311111, 2222, 22211, 221111, 2111111, 11111111.

Cf. A240178, A240183, A240184, A240179, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240178 except for n=0 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)

a(n) = A240178(n) + A240183(n), for n >= 1.

a(n) + A240179(n) = A000041(n) for n >= 0.

A240183 Number of partitions of n such that (greatest part) = (multiplicity of least part).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 2, 0, 3, 3, 4, 2, 9, 3, 10, 10, 17, 11, 26, 19, 36, 33, 48, 47, 79, 71, 101, 109, 149, 151, 215, 216, 293, 318, 404, 443, 575, 611, 773, 864, 1068, 1175, 1458, 1609, 1964, 2210, 2642, 2970, 3577, 3995, 4753, 5369, 6332, 7138, 8414, 9476

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 3 partitions:  41111, 32111, 22211.

Cf. A240178, A240182, A240184, A240179, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240178 except for n=0 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)

A240178(n) + a(n) + A240184(n) = A000041(n) for n >= 0.

A240184 Number of partitions of n such that (greatest part) > (multiplicity of least part).

Original entry on oeis.org

0, 0, 1, 2, 2, 5, 6, 11, 14, 20, 29, 41, 52, 76, 98, 130, 170, 227, 288, 378, 477, 615, 778, 985, 1228, 1551, 1928, 2399, 2964, 3670, 4498, 5538, 6755, 8251, 10027, 12175, 14715, 17802, 21420, 25764, 30886, 37009, 44181, 52731, 62730, 74570, 88435, 104762

Offset: 0

Clark Kimberling, Apr 02 2014

			a(6) counts these 6 partitions:  6, 51, 42, 411, 33, 321.

Cf. A240178, A240182, A240183, A240179, A000041.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
t1 = Table[Count[f[n], p_ /; Max[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240178 *)
t2 = Table[Count[f[n], p_ /; Max[p] <= Count[p, Min[p]]], {n, 0, z}] (* A240182 *)
t3 = Table[Count[f[n], p_ /; Max[p] == Count[p, Min[p]]], {n, 0, z}] (* A240183 *)
t4 = Table[Count[f[n], p_ /; Max[p] > Count[p, Min[p]]], {n, 0, z}] (* A240184 *)
t5 = Table[Count[f[n], p_ /; Max[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240179 *)

A240178(n) + A240183(n) + a(n ) = A000041(n) for n >= 1.

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A240179 Number of partitions of n such that (least part) <= (multiplicity of greatest part).

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A240057 Number of partitions of n such that (greatest part) is not = (multiplicity of greatest part).

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A240180 Number of partitions of n such that (least part) = (multiplicity of greatest part).

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A240182 Number of partitions of n such that (greatest part) <= (multiplicity of least part).

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A240183 Number of partitions of n such that (greatest part) = (multiplicity of least part).

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A240184 Number of partitions of n such that (greatest part) > (multiplicity of least part).

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