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A091378
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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.
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1
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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; internal format)
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OFFSET
| 0,5
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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.
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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).
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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.
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CROSSREFS
| Cf. A092450, A000108.
Sequence in context: A187617 A128612 A060854 * A156045 A119687 A086856
Adjacent sequences: A091375 A091376 A091377 * A091379 A091380 A091381
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KEYWORD
| more,nonn,tabl
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AUTHOR
| Hugh Robinson (hugh(AT)mit.edu), Mar 01 2004
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EXTENSIONS
| Corrected definition and more terms from Hugh Robinson (hughrobinson(AT)google.com), Oct 02 2011
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