User talk:Johannes W. Meijer

A178867

Hi Johannes,
I have a question concerning your number triangle A178867:
Your triangle (version 3) differs only in the order of elements in each row from A036040 (version 1) and A080575 (version 2). In your triangle the columns correspond to integer partitions, and your order of these integer partitions seems to be the same that I have defined in A194602. Is that correct, and do you have a source for this ordering? If so, I would like to add it to my sequence.
By the way: I think the title is quite strange. That's how I would call it:
Refined triangle A124323: T(n,k) is the number of n-set partitions of type k (index number of A194602).
(Compare name of my triangle A181897.)
This illustration is related to this triangle - although the set partitions are only shown up to rotation and reflection: http://commons.wikimedia.org/wiki/File:Partition_necklaces_by_integer_partition.svg
Greetings, Tilman Piesk 21:01, 5 April 2012 (UTC)

Hi Tilman,

I am not sure whether A178867 has anything to do with A194602.

If you wish I can send you an excel file with my notes about A178867.

Please send me an email adress to which I can send the excel file.

The title of A178867 is the similar to the titles of A036040 and A080575.

Greetings,

Hans Meijer.

• The columns in your triangle A178867 correspond to integer partitions (as in this triangle).
• My sequence A194602 defines an order of all finite integer partitions (see this table).

I have just realized, that our orders are equal only for the first 12 integer partitions:

A178867:
1,
1,  1,
1,  3,  1,
1,  6,  4,  3,  1,
1, 10, 10, 15,  5, 10,  1,
1, 15, 20, 45, 15, 60,  6, 15, 15, 10,  1,
1, 21, 35,105, 35,210, 21,105,105, 70,  7,105, 35, 21,  1,
1, 28, 56,210, 70,560, 56,420,420,280, 28,840,280,168,  8,280,210,105, 56, 35, 28  1

My version (integer partitions ordered as defined in A194602):
...
1, 21, 35,105, 35,210, 21,105,105, 70,  7,105, 21, 35,  1,
1, 28, 56,210, 70,560, 56,420,420,280, 28,840,168,280,  8,105,210,280, 28, 56, 35  1

So I have to put my question differently: Which order of integer partitions is used in the definition of your triangle A178867?
Your triangle is only well-defined when the order of integer partitions is well-defined. At the moment I can't figure it out when I look at the description.

Even more confusing: You link to Generalized Birthday Problem, to A178866, and from there to The Collector's Problem - and the order of integer partitions shown on these pages is different from the one you have used for your triangle.
There the first integer partitions are (), (2), (2,2), (3),... but in our triangles it's (), (2), (3), (2,2),...

As I don't ask you for anything special I don't think I should try to extract it from a file you send me.
Which order you refer to should be comprehensibly explained in the description of your triangle.

Thanks in advance, Tilman Piesk 19:07, 6 April 2012 (UTC)

My five cents

Tilman asks: "Which order of integer partitions is used in the definition of your triangle A178867?"

Good question! However, the order of integer partitions is a very delicate matter ;-))

It might well be the case that this order has no 'official' name; I did not check this though.

Maybe it helps you if I show you the used order explicitly. At least this reflects the way

I understand the program of Johannes, which is not very easy. So please take this comment with a pinch of salt.

[[1]] [[2],[1,1]] [[1,1,1],[3],[1,2]] [[2,2],[1,3],[4],[1,1,2],[1,1,1,1]] [[1,1,1,1,1],[1,1,1,2],[1,2,2],[1,1,3],[2,3],[1,4],[5]] [[3,3],[1,1,4],[2,4],[1,5],[6],[1,1,1,1,2],[1,1,2,2],[2,2,2],[1,1,1,3],[1,2,3],[1,1,1,1,1,1]] [[1,1,1,1,1,1,1],[1,1,1,1,1,2],[7],[1,1,2,3],[2,2,3],[1,3,3],[1,1,1,4],[1,2,4],[3,4],[1,1,5],[2,5],[1,6],[1,1,1,2,2],[1,2,2,2],[1,1,1,1,3]] [[1,1,6],[8],[1,7],[1,1,1,1,1,3],[1,1,1,2,3],[1,2,2,3],[1,1,3,3],[2,3,3],[1,1,1,1,4],[1,1,2,4],[2,2,4],[1,3,4],[4,4],[1,1,1,5],[1,2,5],[3,5],[2,6],[1,1,1,1,1,1,1,1],[1,1,1,1,1,1,2],[1,1,1,1,2,2],[1,1,2,2,2],[2,2,2,2]] [[1,2,2,4],[1,1,3,4],[2,3,4],[1,4,4],[1,1,1,1,5],[1,1,2,5],[2,2,5],[1,3,5],[4,5],[1,1,1,6],[1,2,6],[3,6],[1,1,7],[2,7],[1,8],[9],[1,1,1,1,1,1,3],[1,1,1,1,2,3],[1,1,2,2,3],[2,2,2,3],[1,1,1,3,3],[1,2,3,3],[3,3,3],[1,1,1,1,1,4],[1,1,1,2,4],[1,1,1,1,1,1,1,1,1],[1,1,1,1,1,1,1,2],[1,1,1,1,1,2,2],[1,1,1,2,2,2],[1,2,2,2,2]]

Peter Luschny 20:53, 6 April 2012 (UTC)

That may or may not be what the program says. But with these partitions the triangle would look completely different.

The advantage of Johannes' triangle over A036040 and A080575 is that entries in the same column correspond to the same integer partition up to ones.

Thanks to that not only the rows make sense but also the columns. These are the columns from left to right:

A000012,   A000217,   A000292,   3*A000332, A000332,   10*A000389, A000389,   15*A000579, 15*A000579, 10*A000579, A000579,   105*A000580, ...

The columns that start with a 1 are the columns of A184049.

This is clearly not the case for the partitions in your list above (which looks like complete chaos to me).

We can ignore the ones:

Beginning of Johannes' order: (), (2), (3), (2,2), (4), (3,2), (5), (2,2,2), (4,2), (3,3), (6), (3,2,2), (4,3), (5,2), (7), (3,3,2), (4,2,2), (2,2,2,2), (5,3), (4,4), (6,2), (8) Beginning of my order defined in A194602: (), (2), (3), (2,2), (4), (3,2), (5), (2,2,2), (4,2), (3,3), (6), (3,2,2), (5,2), (4,3), (7), (2,2,2,2), (4,2,2), (3,3,2), (6,2), (5,3), (4,4), (8)

The question is if there is a clear definition of Johannes' order - i.e. of a bijection between all integers and all integer partitions up to ones.

If that's the case I propose that we add the permutation of one ordering into the other as a sequence.
Tilman Piesk 22:21, 6 April 2012 (UTC)

I just got an idea how Johannes may have defined the succession of the columns.

To put it informally: The treads of the steps are weakly decreasing.

In this case the order of the columns doesn't uniquely define an order of partitions, because some columns are equal (which at first I didn't realize). So possibly Johannes doesn't care whether the two 15*A000579 columns correspond to (2,2,2), (4,2) or to (4,2), (2,2,2).

If my assumption is correct my question is answered. However, I think the description of A178867 is quite inaccessible. I didn't get the information in the formula section, because the headline refers to the triangle columns as the "matrix rows".

Tilman Piesk 12:19, 7 April 2012 (UTC)

Hi Tilman, You correctly state that I didn´t really care about defining a unique order (with Maple I chose for 'a weakly decreasing ordering'). From what I read elsewhere, see for the Hindenburg ordering e.g. Peter Luschny's website and see also oeis.org/wiki/Orderings_of_partitions, I thought it wise not to try to define my own ordering. I only wanted to show how one could generate all multinomial coefficients, see A178667, with the Basic Multinomial Coefficients, see A178666. Thanks for pointing out the row-column error. I corrected it. Once again I like to repeat my offer to send you an Excel file with some more information. Best regards. Johannes Meijer.