OFFSET

1,2

COMMENTS

In other words: row n lists in nonincreasing order the terms of the first n rows of triangle A176206.

Conjecture: all divisors of all terms in row n are also all parts of all partitions of all positive integers <= n.

A336811 contains the most elementary conjecture about the correspondence divisors/partitions.

EXAMPLE

Triangle begins:

1;

2, 1, 1;

3, 2, 2, 1, 1, 1, 1;

4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1;

5, 4, 4, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

...

For n = 4, by definition the length of row 6 is A014153(4-1) = A014153(3) = 14, so the row 4 of triangle has 14 terms. Since every column lists the positive integers A000027 so the row 4 is [4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1].

Then we have that the divisors of the numbers of the 4th row are:

.

4th row of the triangle ----------> 4 3 3 2 2 2 2 1 1 1 1 1 1 1

2 1 1 1 1 1 1

1

.

There are fourteen 1's, five 2's, two 3's and one 4.

In total there are 14 + 5 + 2 + 1 = 22 divisors.

On the other hand all partitions of all positive integers <= 4 are as shown below:

.

. Partition Partitions Partitions Partitions

. of 1 of 2 of 3 of 4

.

. 4

. 2 2

. 3 3 1

. 2 2 1 2 1 1

. 1 1 1 1 1 1 1 1 1 1

.

In these partitions there are fourteen 1's, five 2's, two 3's and one 4.

In total there are 14 + 5 + 2 + 1 = A284870(4) = 22 parts.

Finally in accordance with the conjecture we can see that all divisors of all numbers in the 4th row of the triangle are the same positive integers as all parts of all partitions of all positive integers <= 4.

CROSSREFS

KEYWORD

nonn,tabf

AUTHOR

Omar E. Pol, Jan 14 2021

STATUS

approved