This grant supports a conference on Permutation Patterns in the Summer of 2011 at the California Polytechnic State University in San Luis Obispo, California. As in past conferences, we expect to attract many graduate students and junior researchers. In total, we expect around 70 participants and around 35 talks. The topic of the conference is permutation patterns and, more generally, the concept of pattern in mathematical objects such as partitions, words, and other structures such as genome sequences. A permutation of length n is a rearrangement of the integers 1 through n. If a permutation sigma can be found by erasing some of the integers in a permutation tau and renumbering the remaining integers in sequential order with 1 through k, then we say that sigma is involved in tau. The study of permutation patterns is concerned with understanding this involvement relation.

The definition of involvement can be rephrased in a variety of contexts such as geometric or model-theoretic. In computer science, sets of permutations which are closed under involvement are found in the study of sorting devices such as stacks, queues, and finite networks. The past decade or so has seen a confluence of ideas stemming from combinatorics and the computer science, creating new and developing avenues of research in mathematics, theoretical computer science, computability and complexity theory, and computational algebra. In addition to the continuing interest in the combinatorics and sorting mechanism problems, some major new directions include the structural theory of permutation patterns (constructions, decompositions, etc.), asymptotics, variations and generalizations such as the concept of generalized pattern avoidance, packing densities, algorithmic and decidability problems, and geometrical methods.

The study of permutation patterns has emerged as a significant branch of enumerative combinatorics over the past couple of decades. Hundreds of papers on permutation patterns have appeared relatively recently. The number of researchers in the area is increasing and the attendance in the annual conference on permutation patterns has steadily grown since its inception in 2003. Research is international across North America, Europe, Asia, and Australasia. This is a subject of global interest.

The subject of permutation patterns has a relatively low mathematical overhead. Some of the commonly used tools used to investigate permutation patterns include generating functions, algorithm design, experimental computation; tools which permeate combinatorics and computer science. Therefore, this subject is accessible to entry-level researchers. Our conference will provide opportunities for young researchers to learn about and begin to make contributions to the theory. The number of graduate students who have attended the last two meetings is an indication that research in this subject area is growing.

Project Report

** In 2011, partial support was given to female, underrepresented, and junior American participants in an international mathematics conference held at California Polytechnic State University in San Luis Obispo, California(Cal Poly) from June 20th, 2011 through June 24th, 2011. The topic was on the current status and recent progresses made in the study of ``permutation patterns'', a mathematical subject which has emerged as a significant branch of enumerative combinatorics over the past couple of decades. Hundreds of papers on permutation patterns have appeared relatively recently. The program included over 30 research talks and attracted a good number of participants. There were open problem sessions and ample opportunities for informal collaborations. The conference attracted a good number of students and junior researchers. Approximately 44% of attendees were undergraduate or graduate students whereas only 33% of conference attendees were tenured professors of mathematics. The grant provided some level of support every female mathematician coming from an institution within the United States except for two (who were supported from other means). Efforts were made to recruit female faculty members who otherwise may not have attended. Support was also offered to underrepresented minorities in the mathematical sciences coming from institutions in the United States. The Permutation Patterns 2011 Proceedings will be published as a special issue of the journal Pure Mathematics and Applications.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Tomek Bartoszynski
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California Polytechnic State University Foundation
San Luis Obispo
United States
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