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 1-3 of 3 results.

A237685 Number of partitions of n having depth 1; see Comments.

Original entry on oeis.org

0, 1, 1, 2, 4, 6, 9, 11, 20, 25, 37, 47, 67, 85, 122, 142, 200, 259, 330, 412, 538, 663, 846, 1026, 1309, 1598, 2013, 2432, 3003, 3670, 4467, 5383, 6591, 7892, 9544, 11472, 13768, 16424, 19686, 23392, 27802, 33011, 39094, 46243, 54700, 64273, 75638, 88765
Offset: 1

Views

Author

Clark Kimberling, Feb 19 2014

Keywords

Comments

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is introduced here as the depth of P. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

Examples

			The 11 partitions of 6 are partitioned by depth as follows:
  depth 0: 6, 51, 42, 321;
  depth 1: 411, 33, 222, 2211, 21111, 11111;
  depth 2: 3111.
Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.

Crossrefs

Cf. A237750, A237978, A366063, A000009, A000041.

Programs

  • Mathematica
    z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
Table[Count[c[n], 1], {n, 1, z}] (* this sequence *)
Table[Count[c[n], 2], {n, 1, z}] (* A237750 *)
Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
(* Peter J. C. Moses, Feb 19 2014 *)

A237750 Number of partitions of n having depth 2; see Comments.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 1, 4, 2, 7, 6, 13, 15, 25, 26, 46, 53, 74, 92, 136, 157, 218, 274, 356, 443, 583, 703, 899, 1125, 1447, 1746, 2182, 2661, 3331, 4077, 4997, 6066, 7432, 8984, 10904, 13212, 15845, 19161, 22932, 27526, 32968, 39351, 46778, 55791, 66272, 78480
Offset: 1

Views

Author

Clark Kimberling, Feb 19 2014

Keywords

Comments

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P),..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is introduced here as the depth of P. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

Examples

			The 11 partitions of 6 are partitioned by depth as follows:
  depth 0: 6, 51, 42, 321;
  depth 1: 411, 33, 222, 2211, 21111, 11111;
  depth 2: 3111.
Thus, a(6) = 6, A000009(6) = 4, A237750(6) = 1, A237978(6) = 0.

Crossrefs

Cf. A237685, A237978, A366063, A000009, A000041.

Programs

  • Mathematica
    z = 60; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
Table[Count[c[n], 1], {n, 1, z}] (* A237685 *)
Table[Count[c[n], 2], {n, 1, z}] (* this sequence *)
Table[Count[c[n], 3], {n, 1, z}] (* A237978 *)
(* Peter J. C. Moses, Feb 19 2014 *)

A366063 Irregular triangle read by rows: T(n,k) is the number of partitions of n that have depth k.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 1, 3, 4, 0, 4, 6, 1, 5, 9, 1, 6, 11, 4, 1, 8, 20, 2, 0, 10, 25, 7, 0, 12, 37, 6, 1, 15, 47, 13, 2, 18, 67, 15, 1, 22, 85, 25, 3, 27, 122, 26, 1, 32, 142, 46, 10, 1, 38, 200, 53, 6, 0, 46, 259, 74, 6, 0, 54, 330, 92, 13, 1, 64, 412, 136
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2023

Keywords

Comments

Suppose that P is a partition of n. Let x(1), x(2),...,x(k) be the distinct parts of P, and let m(i) be the multiplicity of x(i) in P. Let f(P) be the partition [m(1)*x(1), m(2)*x(2),...,x(k)*m(k)] of n. Define c(0,P) = P, c(1,P) = f(P), ..., c(n,P) = f(c(n-1,P)), and define d(P) = least n such that c(n,P) has no repeated parts; d(P) is as the depth of P, as defined in A237685. Clearly d(P) = 0 if and only if P is a strict partition, as in A000009. Conjecture: if d >= 0, then 2^d is the least n that has a partition of depth d.

Examples

			First 20 rows:
   1
   1      1
   2      1
   2      2     1
   3      4     0
   4      6     1
   5      9     1
   6     11     4    1
   8     20     2    0
  10     25     7    0
  12     37     6    1
  15     47    13    2
  18     67    15    1
  22     85    25    3
  27    122    26    1
  32    142    46   10    1
  38    200    53    6    0
  46    259    74    6    0
  54    330    92   13    1
  64    412   136   15    0

Crossrefs

Cf. A000009, A000041, A237685 (column 1), A237750 (column 2), A237978 (column 3), A225485 (frequency depth array).

Programs

  • Mathematica
    z = 36; c[n_] := c[n] = Map[Length[FixedPointList[Sort[Map[Total, Split[#]], Greater] &, #]] - 2 &, IntegerPartitions[n]]
t = Table[Count[c[n], k], {n, 1, z}, {k, 0, Floor[Log[2, n]]}]
TableForm[t] (* this sequence as an array *)
Flatten[t]   (* this sequence *)
Showing 1-3 of 3 results.

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