By C. Ward Henson, José Iovino, Alexander S. Kechris, Edward Odell, Catherine Finet, Christian Michaux

This quantity offers articles from 4 striking researchers who paintings on the cusp of study and good judgment. The emphasis is on energetic examine subject matters; many effects are provided that experience now not been released ahead of and open difficulties are formulated. significant attempt has been made by way of the authors to make their articles obtainable to mathematicians new to the realm

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How many sequences (X1 , X2 , . . , Xk ) are there of subsets of the set [n] = {1, 2, . . , n} such that X1 ∩ X2 ∩ · · · ∩ Xk = ∅? Let f (k, n) be this number. If we were not particularly inspired we could perhaps argue as follows. Suppose X1 ∩ X2 ∩ · · · ∩ Xk−1 = T , where #T = i. If Yj = Xj − T , then Y1 ∩ · · · ∩ Yk−1 = ∅ and Yj ⊆ [n] − T . Hence, there are f (k − 1, n − i) sequences (X1 , . . , Xk−1 ) such that X1 ∩ X2 ∩ · · · ∩ Xk−1 = T . (n − i)! i-element subsets T of [n]. Hence, n f (k, n) = i=0 Let Fk (x) = n≥0 f (k, n)x n /n!.

An ), the following simple algorithm may be used to deﬁne (w, f ). First, write down the number n and regard it as starting a cycle C1 of w. Let f (C1 ) = an + 1. Assuming n, n − 1, . . , n − i + 1 have been inserted into the disjoint cycle notation for w, we now have two possibilities: i. 0 ≤ an−i ≤ t − 1. Then start a new cycle Cj with the element n − i to the left of the previously inserted elements, and set f (Cj ) = an−i + 1. ii. an−i = t + k where 0 ≤ k ≤ i − 1. Then insert n − i into an old cycle so that it is not the leftmost element of any cycle, and so that it appears to the right of k + 1 of the numbers previously inserted.

A. The map Sn → Sn deﬁned above is a bijection. b. If w ∈ Sn has k cycles, then w has k left-to-right maxima. If w ∈ SS where #S = n, then let ci = ci (w) be the number of cycles of w of length i. Note that n = ici . Deﬁne the type of w, denoted type(w), to be the sequence (c1 , . . , cn ). The total number of cycles of w is denoted c(w), so c(w) = c1 (w) + · · · + cn (w). 2 Proposition. The number of permutations w ∈ SS of type (c1 , . . 2c2 c2 ! · · · ncn cn !. 24 What Is Enumerative Combinatorics?