David Callan

Note: Pattern avoidance in flattened partitions

To flatten a set partition (with apologies to Mathematica^(R)) means to form a permutation by erasing the dividers between its blocks. Of course, the result depends on how the blocks are listed. For the usual listing-increasing entries in each block and blocks arranged in increasing order of their first entries-we count the partitions of [n] whose flattening avoids a single 3-letter pattern. Five counting sequences arise: a null sequence, the powers of 2, the Fibonacci numbers, the Catalan numbers, and the binomial transform of the Catalan numbers.

On permutations avoiding 1243, 2134, and another 4-letter pattern

Abstract We enumerate permutations avoiding 1324, 2143, and a third 4-letter pattern τ a step toward the goal of enumerating avoiders for all triples of 4-letter patterns. The enumeration is already known for all but five patterns τ, which are treated in this paper.

Enumeration of small Wilf classes avoiding 1324 and two other 4-letter patterns

Abstract Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

Enumeration of 3- and 4-Wilf classes of four 4-letter patterns

Let Sn be the symmetric group of all permutations of n letters. We show that there are precisely 27 (respectively, 15) Wilf classes consisting of exactly 3 (respectively, 4) symmetry classes of subsets of four 4-letter patterns.

Enumeration of 2-Wilf Classes of Four 4-letter Patterns

Let Sn be the symmetric group of all permutations of n letters. We show that there are precisely 64 Wilf classes consisting of exactly 2 symmetry classes of subsets of four 4-letter patterns.

The Maximum Associativeness of Division: 11091

11091 [2004, 534]. Proposed by David Callan, University of Wisconsin, Madison, WI. Let x0, xi, x2, ... ,xn be indeterminates. It is well known that the number of ways to insert nonredundant parentheses into the repeated quotient jco -+xx -fx2 -f-?xn to make it a meaningful expression is the nth Catalan number, C?. The resulting fractions are not all the same, because division is not associative, but neither are they all different. Which fraction occurs most frequently, and how often does it occur? (The Catalan sequence begins with 1, 1, 2, 5, 14, 42, 132, and 429.)

Dyck Paths Skipping Every Fourth Point: 11013

Solution by Marc Renault, Shippensburg University, Shippensburg, PA. It is well known that the number of Dyck n-paths is the Catalan number C, and that the Catalan numbers satisfy the recurrence relation Cn i= 1 Ci Cn--i. Let A = {(4k, 0): 1 < k < n 1}, let d2n be the number of Dyck (2n)-paths that avoid A, and let d be the number of Dyck (2n)-paths with at least one point in A. Clearly C2n = d2n + di,. A Dyck (2n)-path whose first point in A is (4k, 0) consists of a Dyck (2k)-path that avoids {(4j, 0): 1 < j i k 1} followed by a Dyck (2n 2k)-path. Thus, the number of Dyck (2n)-paths whose first point in A is (4k, 0) is d2 C2n-2k, and in total d, = =E l d2kC2n-2k. Noting that d2 = 2 = 2C1, we proceed by induction. With d2k = 2C2k-1 for 1 _ k < n 1, we have

Another Binomial Coefficient Identity: 10878

We claim that setting u = and u = yields, respectively, f (N 1) 0. For the first claim, note that the first two terms inside sum to less than -(n2 + n)-1, and the rest is less than u4 C 0 u2, which produces a negative total when n > 5. For the second claim, when n > 8 the first two terms suffice (that is, u3/3 > 1/n), while for 5 5 the local minima occur at [(n + N/3n + 1 1)/21 and at [(n /3n I + 1)/2].

Math Bite: Enumerating Certain Sparse Matrices

Remark The n X n matrices satisfying (i) and (ii) above, but with all line-sums equal to r, are enumerated using generating functions in the recently-published Enumerative Comhinatorics, Vol. 2, by Richard Stanley (Problem 5.62). The generating function yields the surprisingly simple formula n!2 in the special case r = 3. The picture above gives a combinatorial explanation; it shows that such a matrix can be uniquely represented as P + 2Q, where P and Q range over all arbitrary n X n permutation matrices.