# [seqfan] Re: Partitions and A101509

Maximilian Hasler maximilian.hasler at gmail.com
Wed Jan 14 22:18:33 CET 2009

```comparing to A120733 I suppose it is meant :

a(n) = Number of matrices with positive integer entries such that sum
of all entries is equal to n.

n=1: [1]
n=2: [2],[1,1], [1;1]
...

The matrices must be of dimensions r x c <= n
a(n) is the sum for k=1 to n of the product of

sum of ways to write k as a product = number of divisors of k = tau(k)

times

the number of decompositions of n in k terms > 0 (respecting the order)
= number of strictly increasing functions from [1,k] to [1,n] such that f(k)=n
= number of ways to chose k-1 (different) numbers among n-1 (without order)
= binomial(n-1,k-1)

a(n) = sum( k=1,n, tau(k)*binomial(n-1,k-1))

this is indeed equal to A101509.

Maximilian

> On Wed, Jan 14, 2009 at 4:04 PM,  <franktaw at netscape.net> wrote:
>> I don't understand your definition.  Can you clarify; perhaps show the
>> 3
>> matrices you claim for n=2?
>>
>>
>> P.s. The word you want is "where", not "were".
>>
>> -----Original Message-----
>> From: Lior Manor <lior.manor at gmail.com>
>>
>> Hi,
>>
>> Consider the number of possible partitions of n were the partitions are
>> not
>> a set of numbers but are elements of a matrix. Zeros are not allowed.
>> (One
>> variation were zeros are allowed is A120733).
>>
>> By manual calculation I get the first few elements to be: 1, 3, 7, 16,
>> 35,
>> 75, 159, 334
>> This is the beginning of A101509.
>>
>> Is it trivial that these are the same sequences?
>>
>> Lior
>>
>>
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>

```