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.

A240675 Number of partitions p of n such that exactly one number is in both p and its conjugate.

Original entry on oeis.org

1, 0, 0, 3, 4, 6, 8, 8, 9, 22, 22, 34, 50, 60, 74, 105, 120, 144, 186, 234, 280, 358, 440, 524, 665, 782, 954, 1150, 1354, 1630, 1944, 2258, 2666, 3170, 3728, 4365, 5128, 5976, 6978, 8144, 9488, 10952, 12700, 14716, 16932, 19558, 22434, 25764, 29505, 33782
Offset: 1

Views

Author

Clark Kimberling, Apr 12 2014

Keywords

Comments

Second column of the array at A240181. Multiplicities greater than 1 are not counted; e.g. there is exactly one number that is in both {4,1,1} and {3,1,1,1}.

Examples

			a(6) counts these 6 partitions:  51, 42, 411, 3111, 2211, 21111, of which the respective conjugates are 21111, 2211, 3111, 411, 42, 51.

Links

Crossrefs

Cf. A240674, A240181.

Programs

  • Mathematica
    z = 30; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; c[p_] := c[p] = Table[Count[#, ?(# >= i &)], {i, First[#]}] &[p]; b[n] := b[n] = Table[Intersection[p[n, k], c[p[n, k]]], {k, 1, PartitionsP[n]}]; Table[Count[Map[Length, b[n]], 0], {n, 1, z}] (* A240674 *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)
  • PARI
    conjug(v) = {my(m = matrix(#v, vecmax(v))); for (i=1, #v, for (j=1, v[i], m[i, j] = 1;);); vector(vecmax(v), i, sum(j=1, #v, m[j, i]));}
a(n) = {my(v = partitions(n)); my(nb = 0); for (k=1, #v, if (#setintersect(Set(v[k]), Set(conjug(v[k]))) == 1, nb++);); nb;} \\ Michel Marcus, Jun 02 2015

Extensions

More terms from Manfred Scheucher, Jun 01 2015

A240450 Greatest number of distinct numbers in the intersection of p and its conjugate, as p ranges through the partitions of n.

Original entry on oeis.org

2, 1, 3, 2, 3, 4, 3, 4, 3, 5, 4, 5, 4, 5, 6, 5, 6, 5, 6, 5, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 7, 8, 7, 9, 8, 9, 8, 9
Offset: 1

Views

Author

Clark Kimberling, Apr 12 2014

Keywords

Comments

Number of terms in row n of the array at A240181.
To match the definition, all terms need to be decreased by 1 (because the rows in A240181 start with k=0). So this appears to be an incorrect duplicate of A067731. - Joerg Arndt, Jul 30 2017

Crossrefs

Cf. A067731, A240181.

Programs

  • Mathematica
    z = 30; conjugatePartition[part_] := Table[Count[#, _?(# >= i &)], {i, First[#]}] &[part]; c = Map[BinCounts[#, {0, 1 + Max[#]}] &[Map[Length, Map[Intersection[#, conjugatePartition[#]] &, IntegerPartitions[#]]]] &, Range[z]]; Flatten[c]  (* A240181 *)
Table[Length[c[[n]]], {n, 1, z}] (* A240450 *) (* Peter J. C. Moses, Apr 10 2014 *)

A240674 Number of partitions p of n that are disjoint from their conjugate.

Original entry on oeis.org

1, 0, 2, 2, 2, 2, 4, 4, 8, 10, 10, 14, 18, 18, 26, 30, 36, 44, 60, 64, 82, 96, 114, 130, 164, 176, 222, 248, 296, 338, 406, 450, 550, 620, 726, 816, 968, 1074, 1270, 1418, 1648, 1836, 2150, 2382, 2758, 3080, 3534, 3942, 4538, 5034, 5778, 6416, 7312, 8136, 9258
Offset: 0

Views

Author

Clark Kimberling, Apr 12 2014

Keywords

Comments

First column of the array at A240181.

Examples

			a(6) counts these 4 partitions:  6, 33, 222, 111111, of which the respective conjugates are 111111, 222, 33, 6.

Crossrefs

Cf. A240675, A114701, A240181.

Programs

  • Mathematica
    z = 30; p[n_, k_] := p[n, k] = IntegerPartitions[n][[k]]; c[p_] := c[p] = Table[Count[#, ?(# >= i &)], {i, First[#]}] &[p]; b[n] := b[n] = Table[Intersection[p[n, k], c[p[n, k]]], {k, 1, PartitionsP[n]}]; Table[Count[Map[Length, b[n]], 0], {n, 1, z}] (* this sequence *)
Table[Count[Map[Length, b[n]], 1], {n, 1, z}]   (* A240675 *)

Formula

a(n) = 2*A114701(n), for n >= 1.

Extensions

Name corrected by Clark Kimberling, Sep 28 2023
a(0)=1 prepended by Alois P. Heinz, Jul 19 2024
Showing 1-3 of 3 results.

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