# “MultiCatalan numbers”

Could anyone provide a reference for the following (sort of) generalization of Catalan numbers: the multinomial coefficient $$\binom{2k_1+3k_2+4k_3+...}{k_1+2k_2+3k_3+...,k_1,k_2,k_3,...}$$ is divisible by $k_1+2k_2+3k_3+...+1$.

Denoting the quotient by $C(k_1,k_2,k_3,...)$, one may call these the multiCatalan numbers. They definitely must appear somewhere in combinatorics, but I could not find any reference.

The reason I am sure these numbers must be known is that, for example: $(-1)^{k_1+k_2+...}C(k_1,k_2,...)$ is the coefficient at $x_1^{k_1}x_2^{k_2}\cdots$ of the composition inverse of the formal power series $t+x_1t^2+x_2t^3+...$;
$C(k_1,k_2,...)$ is the number of faces of the Stasheff polytope $S_{k_1+k_2+...}$ of shape $S_1^{k_1}\times S_2^{k_2}\times\cdots$ (here for convenience I have redenoted by $S_n$ the standard $K_{n+1}$; so $S_1$ is a point, $S_2$ a segment, $S_3$ a pentagon, etc.);
hence they also enumerate certain kinds of trees, etc., etc. ...

• Google OEIS A133437 and see the Loday reference. – Tom Copeland Oct 14 '13 at 10:13
• @TomCopeland Thanks a lot for that. It is certainly to the point and certainly provided lots of additional information to me. However I must say (so far) I could not find explicit appearance of the above numbers there. It is also true that just unwinding the particular case of the Lagrange inversion gives these numbers, so this gives a proof of the divisibility. But I also wanted to explicitly refer to a place where these numbers actually appear. – მამუკა ჯიბლაძე Oct 14 '13 at 10:48
• Some of the references cited in response to the earlier MO question, "Higher-dimensional Catalan numbers?," might help... – Joseph O'Rourke Oct 14 '13 at 11:20
• You read Section 6 of Loday's paper on inversion of power series and facets of associahedra, i.e. Stasheff polytopes? – Tom Copeland Oct 14 '13 at 13:45
• That's why I posed the question "Why do polytopes pop up ,..?" – Tom Copeland Oct 29 '13 at 1:52

$$K(n,k) = \frac{1}{k+1} \binom{n-3}{k} \binom{n+k-1}{k}\;.$$