Two papers that both address your question along similar lines are:

(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and

(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman

Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor.

Briefly, paper (1) defines (among other things) a poset $S(n,k), 0 \leq k \leq n$, with $|S(n,2)|$ equal to the $n^{th}$ Catalan number. The objects of this poset are defined in 3 ways (and shown to be equivalent): combinatorially (via cyclic polytopes), as "homotopies of homotopies" or as elements of "the free $(n-k)$-category on the $n$-simplex.

So if you like these definitions and think they are natural, $|S(n,k)|$ for $k >2$ gives you "higher dimensional" Catalan numbers.

Paper (2) does something nice, too, but I can't remember what.

Two papers that both address your question along similar lines are:

(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and

(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman

Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor.

Briefly, paper (1) defines (among other things) a poset $S(n,k), 0 \leq k \leq n$, with $|S(n,2)|$ equal to the $n^{th}$ Catalan number. The objects of this poset are defined in 3 ways (and shown to be equivalent): combinatorially (via cyclic polytopes), as "homotopies of homotopies" and as elements of "the free $(n-k)$-category on the $n$-simplex".

So if you like these definitions and think they are natural, $|S(n,k)|$ for $k >2$ gives you "higher dimensional" Catalan numbers.

Paper (2) does something nice, too, but I can't remember what.

Two papers that both address your question along similar lines are:

(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and

(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman

I think Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor.

Two papers that both address your question along similar lines are:

(1) Free n-category generated by a cube, oriented matroids, and higher Bruhat orders by Karpranov and Voevodsky and

(2) Arrangements of hyperplanes, higher braid groups and higher Bruhat orders by Manin and Schechtman

Maps between higher Bruhat orders and higher Stasheff-Tamari posets by H. Thomas serves as a follow-up and fills in in some of the details, but has a different flavor.

Briefly, paper (1) defines (among other things) a poset $S(n,k), 0 \leq k \leq n$, with $|S(n,2)|$ equal to the $n^{th}$ Catalan number. The objects of this poset are defined in 3 ways (and shown to be equivalent): combinatorially (via cyclic polytopes), as "homotopies of homotopies" or as elements of "the free $(n-k)$-category on the $n$-simplex.

So if you like these definitions and think they are natural, $|S(n,k)|$ for $k >2$ gives you "higher dimensional" Catalan numbers.

Paper (2) does something nice, too, but I can't remember what.