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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.

%I A092450 #17 Oct 17 2014 22:01:01
%S A092450 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 A092450 1,1,132,4828,37434,37434,4828,132,1,1,429,46124,1226228,4073836,
%U A092450 1226228,46124,429,1,1,1430,465932,47002628,645463414,645463414,47002628
%N 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.
%C A092450 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 A092450 Hugh Robinson, <a href="/A092450/b092450.txt">Table of n, a(n) for n = 0..69</a>
%H A092450 Hugh Robinson, <a href="/A092450/a092450.hs.txt">Haskell (ghc 7.4) program to generate the sequence</a>
%F A092450 T(0, n) = T(n, 0) = 1. T(1, n) = T(n, 1) = C(n) the n-th Catalan number (A000108).
%e A092450 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 A092450 Cf. A091378, A000108.
%K A092450 nonn,tabl
%O A092450 0,8
%A A092450 _Hugh Robinson_, Mar 24 2004
%E A092450 More terms from _Hugh Robinson_, Oct 02 2011