CAMP Special Lecture

CAMP Special Lecture

This series of lectures is free and open to everyone.

If you would join this, in order to let us know, please register here before June 14.

    • Title. Alternating Permutations

  • Lecturer. Richard P. Stanley, MIT, USA

  • Date. June 29 (Mon.) - July 3 (Fri.), 2015

  • Time. 14:00 (June 29-30, July 1, July 3), 15:30 (July 2)

  • Venue. CAMP & NIMS

Detailed Time & Venue.

  • NIMS Seminar Room (3rd floor), 2:00 PM, June 29, 2015

  • NIMS Seminar Room (3rd floor), 2:00 PM, June 30, 2015

  • CAMP Seminar Room (M), 2:00 PM, July 1, 2015

  • CAMP Seminar Room (L), 3:30 PM, July 2, 2015

  • CAMP Seminar Room (L), 2:00 PM, July 3, 2015

Abstract. (for all lectures)

A permutation a1 a2 ... an of 1, 2, ..., n is alternating if a1 > a2 < a3 > a4 < ... . Alternating permutations have many fascinating properties related to combinatorics, geometry, representation theory, probability theory, and other areas. We will survey the highlights of this subject.

Lecture 1. (June 29, 14:00) Euler numbers and André's theorem

The number of alternating permutations of 1, 2, ..., n is denoted En and is called an Euler number. The first theorem on alternating permutations is the generating function

n≥0 En xn / n! = tan(x) + sec(x)

due to D. André in 1879. Some other combinatorial objects are also counted by the Euler numbers.

Lecture 2. (June 30, 14:00) Convex polytopes and alternating permutations

There are two interesting convex polytopes whose volume is En / n!. One of them is given by xi ≥ 0 (1 ≤ i ≤ n) and xi + xi+1 ≤ 1 (1 ≤ i ≤ n-1). They fit into a more general context of order polytopes and chain polytopes of posets.

Lecture 3. (July 1, 14:00) Longest alternating subsequences

What can one say about the length of the longest alternating subsequence of a random permutation? This theory is analogous but simpler than the better-known theory of the longest increasing subsequence of a random permutation. For example, if n>1 then the expected length of the longest alternating subsequence of a random permutation of 1, 2, ..., n is exactly (4n+1)/6.

Lecture 4. (July 2, 15:30) A symmetric group character related to alternating permutations

H. O. Foulkes defined a certain character χ of the symmetric group Sn whose values χ(w) are either 0 or ±Ek for some 1 ≤ k ≤ n (depending on w). This character allows us to enumerate certain classes of alternating permutations using representation theory. For instance, if p is prime then the number of alternating permutations in Sp that are also p-cycles is equal to

(Ep - (-1)(p-1)/2) / p.

No elementary proof is known.

Lecture 5. (July 3, 14:00) The cd-index of the symmetric group

The cd-index of Sn is a noncommutative polynomial in the indeterminates c and d that encodes a lot of combinatorial information. The number of terms in this polynomial is En, and there are interesting connections with alternating permutations.