# [seqfan] Re: Breaking news on partition numbers.

Georgi Guninski guninski at guninski.com
Tue Jan 25 17:03:48 CET 2011

```On Mon, Jan 24, 2011 at 06:31:23PM +0200, Georgi Guninski wrote:
> On Mon, Jan 24, 2011 at 05:04:57PM +0200, Georgi Guninski wrote:
> > On Mon, Jan 24, 2011 at 06:22:57AM -0500, RGWv wrote:
> > > ?Neil,
> > >
> > > They are fairly short.
> > >
> > > Bob.
> > >
> > > number of partitions of n having no element smaller than k.
> > > f[1, 1] = 1; f[n_, k_] := f[n, k] = If[n < 0, 0, If[k > n, 0, If[k
> > > == n, 1, f[n, k + 1] + f[n - k, k]]]]
> > >
> > > number of partitions of n having no element greater than k.
> > > g[n_, 1] := 1; g[1, k_] := 1; g[n_, k_] := g[n, k] = If[k > n, g[n,
> > > k - 1], g[n, k - 1] + g[n - k, k]]
> > >
> >
> > thank you.
> > i did some blackbox changes to your function and other sequences popped up.
> >
> > changing:
> >
> > |n-k| to |n-2*k| appears to give A000009 number of partitions of n into distinct parts; number of partitions of n into odd parts.
> > |k+1| to |2*k+1| appears to give A000929 Dimension of n-th degree part of Steenrod algebra
> > |k+1| to |2*k| appears to give A018819 Binary partition function: number of partitions of n into powers of 2
> > |k+1| to |next_prime(k)| appears to give A034891 Number of different products of partitions of n; partitions of n into prime parts (1 included) (verified to 500).
> >
> strange. verfied A034891 to 1000 terms the last term being > 10^19.
> |k+1| seems a successor relation.
>
> |k+1| to |next_non_prime(k)| seems: A002095 Number of partitions of n into nonprime parts
> |k+1| to |3*k| seems partitons into powers of 3.
> |k+1| to |next_fibonacci(k)| seems A003107 Number of partitions of n into Fibonacci parts (with a single type of 1).
>
> does this stuff follow from theory?
>
>

strangely the period of "A034891 partitions of n into prime parts (1
included)" mod p appears to be p for the first 1000 primes.

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