# [seqfan] Re: Partitions and Dice

wouter meeussen wouter.meeussen at pandora.be
Sat Feb 21 14:06:12 CET 2009

```Frank wrote:

> This leads me to wonder: is t he number of partitions of n into k
> parts, each <= k, always equal to the number of partitions of n-1 into
> k-1 parts, each <= k+1?

yes, calculation seems to turn out that way:
checked to n=30. This produces "Frank's Triangle" :

<<DiscreteMath`Combinatorica`
partitionexact[n_,m_]:=TransposePartition/@(Prepend[#,m]&/@Partitions[n-m,m]
)

Table[Length at Select[partitionexact[n,k],Max[#]<=k&],{n,1,24},{k,n}]
1
0,1
0,1,1
0,1,1,1
0,0,2,1,1
0,0,2,2,1,1
0,0,2,3,2,1,1
0,0,1,4,3,2,1,1
0,0,1,4,5,3,2,1,1
0,0,0,5,6,5,3,2,1,1
0,0,0,4,8,7,5,3,2,1,1
0,0,0,4,9,10,7,5,3,2,1,1
0,0,0,3,11,12,11,7,5,3,2,1,1
0,0,0,2,11,16,14,11,7,5,3,2,1,1
0,0,0,1,12,19,19,15,11,7,5,3,2,1,1
0,0,0,1,11,23,24,21,15,11,7,5,3,2,1,1
0,0,0,0,11,25,31,27,22,15,11,7,5,3,2,1,1
0,0,0,0,9,29,37,36,29,22,15,11,7,5,3,2,1,1
0,0,0,0,8,30,46,45,39,30,22,15,11,7,5,3,2,1,1
0,0,0,0,6,32,52,58,50,41,30,22,15,11,7,5,3,2,1,1
0,0,0,0,5,32,61,70,66,53,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,3,32,68,86,82,71,55,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,2,30,76,101,104,90,74,56,42,30,22,15,11,7,5,3,2,1,1
0,0,0,0,1,29,81,120,127,116,95,76,56,42,30,22,15,11,7,5,3,2,1,1

row sums are Vladeta's A064174 "Number of partitions of n with nonnegative
rank".
Read backwards, approaches the partition numbers.
Always nice to see another "equal counts" versus "bijection" challenge.

Wouter.

```