CAMP Special Lecture

CAMP Special Lecture

This series of lectures is free and open to everyone.

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

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

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

∑_n≥0 E_n 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 E_n / n!. One of them is given by x_i ≥ 0 (1 ≤ i ≤ n) and x_i + x_i+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 S_n whose values χ(w) are either 0 or ±E_k 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 S_p that are also p-cycles is equal to

(E_p - (-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 S_n 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 E_n, and there are interesting connections with alternating permutations.

Participants