A240674 -id:A240674

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A240181

Array: t(n,k) is the number of partitions p of n such that the number of distinct numbers in the intersection of p and its conjugate is k, for k >= 0, n >= 1.

+10
4

0, 1, 2, 2, 0, 1, 2, 3, 2, 4, 1, 4, 6, 0, 1, 4, 8, 3, 8, 8, 5, 1, 10, 9, 11, 10, 22, 8, 1, 1, 14, 22, 17, 3, 18, 34, 19, 5, 1, 18, 50, 21, 12, 26, 60, 34, 13, 2, 30, 74, 52, 19, 0, 1, 36, 105, 57, 29, 4, 44, 120, 93, 34, 5, 1, 60, 144, 128, 40, 13, 64, 186

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

OFFSET

1,3

COMMENTS

First two columns are A240674 and A240675. Sum of numbers in row n is A000041(n), for n >= 1. Number of numbers in row n is A240450(n).

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..200

EXAMPLE

First 15 rows:

0 ... 1

2 ... 0 ... 1

2 ... 3

2 ... 4 ... 1

4 ... 6 ... 0 ...1

4 ... 8 ... 3

8 ... 8 ... 5 ... 1

10 .. 9 ... 11

10 .. 22 .. 8 ... 1 ... 1

14 .. 22 .. 17 .. 3

18 .. 34 .. 19 .. 5 ... 1

18 .. 50 .. 21 .. 12

26 .. 60 .. 34 .. 13 .. 2

30 .. 74 .. 52 .. 19 .. 0 .. 1

In the following table, p and c(p) denote a partition of 6 and its conjugate:

p ........ c(p)

6 ........ 111111

51 ....... 21111

42 ....... 2211

411 ...... 3111

33 ....... 222

321 ...... 321

3111 ..... 411

222 ...... 33

2211 ..... 42

21111 .... 51

111111 ... 6

Let I(p) be number of numbers in the intersection of c and c(p); Then I(p) = 0 for 4 choices of p, I(p) = 1 for 6 choices, I(p) = 2 for 0 choices, and I(p) = 3 for 1 choice. Thus, row 6 is 4 6 0 1.

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] (* this sequence *)

Table[Length[c[[n]]], {n, 1, z}] (* A240450 *) (* Peter J. C. Moses, Apr 10 2014 *)

CROSSREFS

Cf. A240674, A240675, A240450.

KEYWORD

nonn,easy,tabf

AUTHOR

Clark Kimberling, Apr 12 2014

EXTENSIONS

Name corrected by Clark Kimberling, Sep 28 2023

STATUS

approved

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

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

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.