cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A240178 Number of partitions of n such that (least part) < (multiplicity of greatest part).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 59, 71, 93, 114, 144, 176, 223, 268, 336, 407, 502, 605, 744, 891, 1088, 1301, 1574, 1879, 2265, 2687, 3224, 3822, 4557, 5384, 6399, 7535, 8921, 10481, 12354, 14481, 17022, 19888, 23307, 27178, 31745
Offset: 0

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Comments

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.

Examples

			a(6) counts these 3 partitions:  222, 2211, 111111.

Crossrefs

Cf. A240179, A240180.

Programs

  • Mathematica
    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 *)

Formula

a(n) = A240179(n) - A240180(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

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A240178, A240179, A000041.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Clark Kimberling, Apr 02 2014

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

A240178(n) + A240183(n) + a(n ) = A000041(n) for n >= 1.
Showing 1-5 of 5 results.

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