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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.
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%I #17 Oct 17 2014 22:01:01

%S 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,

%T 1,1,132,4828,37434,37434,4828,132,1,1,429,46124,1226228,4073836,

%U 1226228,46124,429,1,1,1430,465932,47002628,645463414,645463414,47002628

%N 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.

%C 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.

%H Hugh Robinson, <a href="/A092450/b092450.txt">Table of n, a(n) for n = 0..69</a>

%H Hugh Robinson, <a href="/A092450/a092450.hs.txt">Haskell (ghc 7.4) program to generate the sequence</a>

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

%e 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}.

%Y Cf. A091378, A000108.

%K nonn,tabl

%O 0,8

%A _Hugh Robinson_, Mar 24 2004

%E More terms from _Hugh Robinson_, Oct 02 2011