4,019 research outputs found
On the enumeration of tanglegrams and tangled chains
Tanglegrams are a special class of graphs appearing in applications
concerning cospeciation and coevolution in biology and computer science. They
are formed by identifying the leaves of two rooted binary trees. We give an
explicit formula to count the number of distinct binary rooted tanglegrams with
matched vertices, along with a simple asymptotic formula and an algorithm
for choosing a tanglegram uniformly at random. The enumeration formula is then
extended to count the number of tangled chains of binary trees of any length.
This includes a new formula for the number of binary trees with leaves. We
also give a conjecture for the expected number of cherries in a large randomly
chosen binary tree and an extension of this conjecture to other types of trees
Clusters, generating functions and asymptotics for consecutive patterns in permutations
We use the cluster method to enumerate permutations avoiding consecutive
patterns. We reprove and generalize in a unified way several known results and
obtain new ones, including some patterns of length 4 and 5, as well as some
infinite families of patterns of a given shape. By enumerating linear
extensions of certain posets, we find a differential equation satisfied by the
inverse of the exponential generating function counting occurrences of the
pattern. We prove that for a large class of patterns, this inverse is always an
entire function. We also complete the classification of consecutive patterns of
length up to 6 into equivalence classes, proving a conjecture of Nakamura.
Finally, we show that the monotone pattern asymptotically dominates (in the
sense that it is easiest to avoid) all non-overlapping patterns of the same
length, thus proving a conjecture of Elizalde and Noy for a positive fraction
of all patterns
Generating Permutations with Restricted Containers
We investigate a generalization of stacks that we call
-machines. We show how this viewpoint rapidly leads to functional
equations for the classes of permutations that -machines generate,
and how these systems of functional equations can frequently be solved by
either the kernel method or, much more easily, by guessing and checking.
General results about the rationality, algebraicity, and the existence of
Wilfian formulas for some classes generated by -machines are
given. We also draw attention to some relatively small permutation classes
which, although we can generate thousands of terms of their enumerations, seem
to not have D-finite generating functions
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