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Permutations avoiding certain patterns: The case of length 4 and some generalizations
AbstractProving and disproving some earlier conjectures, we give a characterization of the numbers of permutations avoiding each pattern of length 4. Some implications for longer patterns are included
Some Triangulated Surfaces without Balanced Splitting
Let G be the graph of a triangulated surface of genus . A
cycle of G is splitting if it cuts into two components, neither of
which is homeomorphic to a disk. A splitting cycle has type k if the
corresponding components have genera k and g-k. It was conjectured that G
contains a splitting cycle (Barnette '1982). We confirm this conjecture for an
infinite family of triangulations by complete graphs but give counter-examples
to a stronger conjecture (Mohar and Thomassen '2001) claiming that G should
contain splitting cycles of every possible type.Comment: 15 pages, 7 figure
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