A092450 Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the product category [m]x[n], where [m] denotes the total order on m objects, viewed as a category.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 10, 5, 1, 1, 14, 68, 68, 14, 1, 1, 42, 544, 1396, 544, 42, 1, 1, 132, 4828, 37434, 37434, 4828, 132, 1, 1, 429, 46124, 1226228, 4073836, 1226228, 46124, 429, 1, 1, 1430, 465932, 47002628, 645463414, 645463414, 47002628

Offset: 0

Author

Hugh Robinson, Mar 24 2004

Keywords

• nonn
• tabl

Comments

Specifying a weak factorization system on a poset category is equivalent to specifying a set of morphisms that includes all identity morphisms and is closed under composition and pullback.

Examples

			T(2, 2) = 10:  the category has five nonidentity morphisms with relations ca = db = e.  a is a pullback of d and of e;  b is a pullback of c and of e.  So there are ten allowable sets of morphisms:  omitting identities for brevity, they are {}, {a}, {b}, {a,b}, {b,c}, {a,d}, {a,b,e}, {a,b,c,e}, {a,b,d,e}, {a,b,c,d,e}.
		

Links

• Hugh Robinson, Table of n, a(n) for n = 0..69
• Hugh Robinson, Haskell (ghc 7.4) program to generate the sequence

Crossrefs

Cf. A091378, A000108.

Formula

T(0, n) = T(n, 0) = 1. T(1, n) = T(n, 1) = C(n) the n-th Catalan number (A000108).

Extensions

More terms from Hugh Robinson, Oct 02 2011