A240181 -id:A240181

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A240675

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

+10
3

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

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OFFSET

1,4

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}.

LINKS

Manfred Scheucher, Table of n, a(n) for n = 1..65

Manfred Scheucher, Sage Script

EXAMPLE

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

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 *)

PROG

(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

CROSSREFS

Cf. A240674, A240181.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 12 2014

EXTENSIONS

More terms from Manfred Scheucher, Jun 01 2015

STATUS

approved

A240450

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..40.

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 *)

CROSSREFS

Cf. A067731, A240181.

KEYWORD

nonn,more

AUTHOR

Clark Kimberling, Apr 12 2014

STATUS

approved

A240674

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

First column of the array at A240181.

LINKS

Table of n, a(n) for n=0..54.

FORMULA

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

EXAMPLE

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

MATHEMATICA

Table[Count[Map[Length, b[n]], 1], {n, 1, z}] (* A240675 *)

CROSSREFS

Cf. A240675, A114701, A240181.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 12 2014

EXTENSIONS

Name corrected by Clark Kimberling, Sep 28 2023

a(0)=1 prepended by Alois P. Heinz, Jul 19 2024

STATUS

approved

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