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 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. 2
 1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 14, 96, 14, 1, 1, 42, 6560, 6560, 42, 1, 1, 132, 1738535, 771496766, 1738535, 132, 1, 1, 429, 2347585784 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 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. LINKS Hugh Robinson, Haskell (ghc 7.4) program to generate the sequence FORMULA 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). EXAMPLE 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. See A092450 for an example computing weak factorization systems on a category which is not a total order. CROSSREFS Cf. A092450, A000108. Sequence in context: A284731 A211400 A060854 * A156045 A119687 A086856 Adjacent sequences:  A091375 A091376 A091377 * A091379 A091380 A091381 KEYWORD more,nonn,tabl AUTHOR Hugh Robinson, Mar 01 2004 EXTENSIONS Corrected definition and more terms from Hugh Robinson, Oct 02 2011 STATUS approved

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Last modified October 17 12:01 EDT 2018. Contains 316279 sequences. (Running on oeis4.)