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 1-7 of 7 results.

A239964 Number of partitions of n such that (number of distinct parts) = maximal multiplicity of the parts.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 3, 4, 6, 5, 11, 12, 15, 21, 26, 30, 41, 52, 56, 82, 99, 118, 144, 189, 221, 279, 327, 405, 491, 600, 699, 860, 1038, 1230, 1470, 1754, 2106, 2487, 2970, 3489, 4148, 4883, 5779, 6763, 8024, 9284, 11006, 12780, 15029, 17452, 20405, 23660
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

			a(8) counts these partitions:  8, 611, 422, 332, 3311, 32111.

Crossrefs

Cf. A239948, A239966.

Programs

  • Mathematica
    z = 58; d[p_] := d[p] = Length[DeleteDuplicates[p]]; m[p_] := Max[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] == m[p]], {n, 0, z}]

A239950 Number of partitions of n such that (number of distinct parts) = least part.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 3, 4, 4, 6, 6, 9, 8, 14, 11, 19, 18, 25, 24, 37, 31, 50, 46, 61, 64, 86, 79, 112, 115, 136, 149, 190, 184, 239, 255, 293, 329, 382, 408, 489, 531, 595, 675, 772, 827, 952, 1066, 1176, 1320, 1468, 1627, 1827, 2030, 2219, 2493, 2769, 3053
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Comments

Also for n>0 the number of partitions of n such that (number of distinct parts) = multiplicity of the greatest part (by conjugation of the partition table). - Joerg Arndt, Apr 28 2014

Examples

			a(8) counts these 4 partitions :  62, 422, 332, 11111111.

Links

Crossrefs

Cf. A239948, A239949, A239951, A239952.

Programs

  • Maple
    b:= proc(n, i, d) option remember; `if`(min(i, n) b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[Min[i, n]Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Formula

A239948(n) + a(n) + A239951(n) = A000041(n) for n >= 0.

A239949 Number of partitions of n such that (number of distinct parts) <= least part.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 8, 8, 12, 12, 18, 17, 26, 25, 36, 36, 50, 50, 69, 69, 93, 95, 123, 129, 164, 171, 215, 229, 278, 300, 365, 387, 468, 507, 595, 652, 760, 830, 966, 1055, 1214, 1336, 1530, 1674, 1910, 2104, 2380, 2617, 2953, 3253, 3656, 4019, 4504
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

			a(8) counts these 8 partitions:  8, 62, 53, 44, 422, 332, 2222, 11111111.

Links

Crossrefs

Cf. A239948, A239950, A239951, A239952.

Programs

  • Maple
    b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i <= d, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Formula

a(n) + A239951(n) = A000041(n) for n >= 0.

A239951 Number of partitions of n such that (number of distinct parts) > least part.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 6, 10, 14, 22, 30, 44, 59, 84, 109, 151, 195, 261, 335, 440, 558, 723, 909, 1160, 1452, 1829, 2272, 2839, 3503, 4336, 5326, 6542, 7984, 9756, 11842, 14376, 17382, 20985, 25255, 30355, 36372, 43528, 51960, 61925, 73645, 87460, 103648, 122650
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

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

Links

Crossrefs

Cf. A239948, A239949, A239950, A239952.

Programs

  • Maple
    b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d, 0,
      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> combinat[numbpart](n) -b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i<=d, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Formula

a(n) + A239949(n) = A000041(n) for n >= 0.

A239952 Number of partitions of n such that (number of distinct parts) >= least part.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 8, 13, 18, 26, 36, 50, 68, 92, 123, 162, 214, 279, 360, 464, 595, 754, 959, 1206, 1513, 1893, 2358, 2918, 3615, 4451, 5462, 6691, 8174, 9940, 12081, 14631, 17675, 21314, 25637, 30763, 36861, 44059, 52555, 62600, 74417, 88287, 104600, 123716
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

			a(6) counts these 8 partitions:  51, 42, 411, 321, 3111, 2211, 21111, 111111.

Links

Crossrefs

Cf. A239948, A239949, A239950, A239951.

Programs

  • Maple
    b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d+1, 0,
      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> combinat[numbpart](n) -b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i<=d+1, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Formula

a(n) + A239948(n) = A000041(n) for n >= 0.

A239953 Number of partitions of n such that twice the least part is the number of distinct parts.

Original entry on oeis.org

0, 0, 0, 1, 2, 4, 5, 8, 9, 12, 14, 17, 18, 23, 25, 28, 33, 39, 44, 54, 61, 77, 92, 112, 131, 167, 194, 246, 280, 352, 401, 501, 562, 697, 779, 939, 1055, 1274, 1401, 1684, 1846, 2186, 2408, 2825, 3103, 3617, 3969, 4583, 5045, 5801, 6367, 7304, 8050, 9150
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

			a(6) counts these 5 partitions :  51, 42, 3111, 22111, 21111.

Links

Crossrefs

Cf. A239948.

Programs

  • Maple
    b:= proc(n, i, d) option remember; `if`(2*min(i, n) b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 02 2014
  • Mathematica
    z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; Table[Count[ IntegerPartitions[n], p_ /; d[p] == 2 Min[p]], {n, 0, z}]  (* A239953 *)
(* Second program: *)
b[n_, i_, d_] := b[n, i, d] = If[2*Min[i, n] < d + 1, 0,
     If[Mod[n, i] == 0 && 2*i == d + 1, 1, b[n, i - 1, d] +
     Sum[b[n - i*j, i - 1, d + 1], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

A239966 Number of partitions of n such that (number of distinct parts) = minimal multiplicity of the parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 2, 4, 3, 6, 5, 8, 6, 13, 6, 15, 10, 19, 12, 24, 15, 33, 22, 38, 28, 52, 39, 61, 51, 78, 66, 94, 85, 118, 103, 140, 130, 168, 165, 194, 190, 244, 230, 274, 285, 328, 327, 394, 386, 449, 485, 522, 540, 646, 639, 712, 790, 846, 880, 1025
Offset: 0

Views

Author

Clark Kimberling, Mar 30 2014

Keywords

Examples

			a(8) counts these 4 partitions :  8, 3311, 22211, 221111.

Crossrefs

Cf. A239964, A239948.

Programs

  • Mathematica
    z = 61; d[p_] := d[p] = Length[DeleteDuplicates[p]]; m[p_] := Min[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] == m[p]], {n, 0, z}]
Showing 1-7 of 7 results.

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