a question

[Augustine Munagi]

Emeric,
You might get a hint from the following communication.

Discrete Mathematics, Volume 306, Issue 18, 28 September 2006, Pages 2234-2240.

Augustine


On 4/4/07, Emeric Deutsch <deutsch at duke.poly.edu> wrote:
> Dear Seqfans,
>
> The following is known:
> Given a partition p of n into k parts, [a_1^e_1,...,a_j^e_j]
> (e_1 + e_2 + ... + e_j = k), the number b(p) of Dyck paths of
> semilength n whose ascent lengths yield the partition p is
>        b(p)=n!/[(n-k+1)!*e_1!*e_2!*...*e_j!].
>
> Apparently, we have
>
>      SUM(b(p)*SUM(e_i*binom(a_i + 1, 2), i=1..j) = binom(2n+1,n-1),
>
> where the outer sum is over all partitions p of n.
>
> For example, for n = 4, the partitions 4, 31, 22, 211, 1111 yield
>      1*10 + 4*7 + 2*6 + 6*5 + 1*4 = 84.
>
> Has anybody seen this? Any idea for a proof? Any information?
> Many thanks.
>
> Emeric