a question
Augustine Munagi
aomunagi at gmail.com
Wed Apr 4 21:18:07 CEST 2007
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
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