5,514 research outputs found
Enumeration of 3-letter patterns in compositions
Let A be any set of positive integers and n a positive integer. A composition
of n with parts in A is an ordered collection of one or more elements in A
whose sum is n. We derive generating functions for the number of compositions
of n with m parts in A that have r occurrences of 3-letter patterns formed by
two (adjacent) instances of levels, rises and drops. We also derive asymptotics
for the number of compositions of n that avoid a given pattern. Finally, we
obtain the generating function for the number of k-ary words of length m which
contain a prescribed number of occurrences of a given pattern as a special case
of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer
Conferenc
Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations
A composition of a nonnegative integer (n) is a sequence of positive integers
whose sum is (n). A composition is palindromic if it is unchanged when its
terms are read in reverse order. We provide a generating function for the
number of occurrences of arbitrary segmented partially ordered patterns among
compositions of (n) with a prescribed number of parts. These patterns
generalize the notions of rises, drops, and levels studied in the literature.
We also obtain results enumerating parts with given sizes and locations among
compositions and palindromic compositions with a given number of parts. Our
results are motivated by "encoding by restricted permutations," a relatively
undeveloped method that provides a language for describing many combinatorial
objects. We conclude with some examples demonstrating bijections between
restricted permutations and other objects.Comment: 12 pages, 1 figur
Polycation-Ï€ Interactions Are a Driving Force for Molecular Recognition by an Intrinsically Disordered Oncoprotein Family
Molecular recognition by intrinsically disordered proteins (IDPs) commonly involves specific localized contacts and target-induced disorder to order transitions. However, some IDPs remain disordered in the bound state, a phenomenon coined "fuzziness", often characterized by IDP polyvalency, sequence-insensitivity and a dynamic ensemble of disordered bound-state conformations. Besides the above general features, specific biophysical models for fuzzy interactions are mostly lacking. The transcriptional activation domain of the Ewing's Sarcoma oncoprotein family (EAD) is an IDP that exhibits many features of fuzziness, with multiple EAD aromatic side chains driving molecular recognition. Considering the prevalent role of cation-π interactions at various protein-protein interfaces, we hypothesized that EAD-target binding involves polycation- π contacts between a disordered EAD and basic residues on the target. Herein we evaluated the polycation-π hypothesis via functional and theoretical interrogation of EAD variants. The experimental effects of a range of EAD sequence variations, including aromatic number, aromatic density and charge perturbations, all support the cation-π model. Moreover, the activity trends observed are well captured by a coarse-grained EAD chain model and a corresponding analytical model based on interaction between EAD aromatics and surface cations of a generic globular target. EAD-target binding, in the context of pathological Ewing's Sarcoma oncoproteins, is thus seen to be driven by a balance between EAD conformational entropy and favorable EAD-target cation-π contacts. Such a highly versatile mode of molecular recognition offers a general conceptual framework for promiscuous target recognition by polyvalent IDPs. © 2013 Song et al
Sorting with a forklift
A fork stack is a generalised stack which allows pushes and pops of several
items at a time. We consider the problem of determining which input streams can
be sorted using a single forkstack, or dually, which permutations of a fixed
input stream can be produced using a single forkstack. An algorithm is given to
solve the sorting problem and the minimal unsortable sequences are found. The
results are extended to fork stacks where there are bounds on how many items
can be pushed and popped at one time. In this context we also establish how to
enumerate the collection of sortable sequences.Comment: 24 pages, 2 figure
Ascent Sequences Avoiding Pairs of Patterns
Ascent sequences were introduced by Bousquet-Melou et al. in connection with (2+2)-avoiding posets and their pattern avoidance properties were first considered by Duncan and SteingrÃmsson. In this paper, we consider ascent sequences of length n role= presentation style= display: inline; font-size: 11.2px; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; font-family: Verdana, Arial, Helvetica, sans-serif; position: relative; \u3enn avoiding two patterns of length 3, and we determine an exact enumeration for 16 different pairs of patterns. Methods include simple recurrences, bijections to other combinatorial objects (including Dyck paths and pattern-avoiding permutations), and generating trees. We also provide an analogue of the ErdÅ‘s-Szekeres Theorem to prove that any sufficiently long ascent sequence contains either many copies of the same number or a long increasing subsequence, with a precise bound
Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials
We say that a permutation is a Motzkin permutation if it avoids 132 and
there do not exist such that . We study the
distribution of several statistics in Motzkin permutations, including the
length of the longest increasing and decreasing subsequences and the number of
rises and descents. We also enumerate Motzkin permutations with additional
restrictions, and study the distribution of occurrences of fairly general
patterns in this class of permutations.Comment: 18 pages, 2 figure
The maximal spectral radius of a digraph with (m+1)^2 - s edges
It is known that the spectral radius of a digraph with k edges is \le
\sqrt{k}, and that this inequality is strict except when k is a perfect square.
For k=m^2 + \ell, \ell fixed, m large, Friedland showed that the optimal
digraph is obtained from the complete digraph on m vertices by adding one extra
vertex, and a corresponding loop, and then connecting it to the first \lfloor
\ell/2\rfloor vertices by pairs of directed edges (this is for odd \ell, for
even \ell we add one extra edge to the new vertex).
Using a combinatorial reciprocity theorem by Gessel, and a classification by
Backelin on the digraphs on s edges having a maximal number of walks of length
two, we obtain the following result: for fixed 0< s \neq 4, k=(m+1)^2 - s, m
large, the maximal spectral radius of a digraph with k edges is obtained by the
digraph which is constructed from the complete digraph on m+1 vertices by
removing the loop at the last vertex together with \lfloor s/2 \rfloor pairs of
directed edges that connect to the last vertex (if s is even, remove an extra
edge connecting to the last vertex).Comment: 11 pages, 9 eps figures. To be presented at the conference FPSAC03.
Submitted to Electronic Journal of Linear Algebra. Keywords: Spectral radius,
digraphs, 0-1 matrices, Perron-Frobenius theorem, number of walk
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