|
|
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
|
|
|
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
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Corrected definition and more terms from Hugh Robinson, Oct 02 2011
|
|
STATUS
|
approved
|
|
|
|