cp's OEIS Frontend

A091378 Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the poset of order-preserving maps from [m] to [n+1] (where [m] denotes the total order on m objects), viewed as a category.

%I A091378 #13 Jun 29 2013 03:02:07
%S A091378 1,1,1,1,2,1,1,5,5,1,1,14,96,14,1,1,42,6560,6560,42,1,1,132,1738535,
%T A091378 771496766,1738535,132,1,1,429,2347585784
%N A091378 Triangle read by rows: T(m,n) = number of weak factorization systems (trivial Quillen model structures) on the poset of order-preserving maps from [m] to [n+1] (where [m] denotes the total order on m objects), viewed as a category.
%C A091378 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 A091378 Hugh Robinson, <a href="/A091378/a091378.hs.txt">Haskell (ghc 7.4) program to generate the sequence</a>
%F A091378 T(m, n) = T(n, m) because the corresponding categories are isomorphic. T(0, n) = T(n, 0) = 1. T(1, n) = T(n, 1) = C(n+1) the (n+1)st Catalan number (A000108).
%e A091378 T(1, 2) = 5:  the category is the total order on three objects:  it has three nonidentity morphisms a, b, c satisfying the relation ba = c.  Of the 8 possible sets of morphisms, {a, b} is not closed under composition and {c}, {b, c} are not closed under pullback since a is a pullback of c.  The other 5 sets generate weak factorization systems.
%e A091378 See A092450 for an example computing weak factorization systems on a category which is not a total order.
%Y A091378 Cf. A092450, A000108.
%K A091378 more,nonn,tabl
%O A091378 0,5
%A A091378 _Hugh Robinson_, Mar 01 2004
%E A091378 Corrected definition and more terms from _Hugh Robinson_, Oct 02 2011