Julian West

Generating trees and the Catalan and Schro¨der numbers

Abstract A permutation π ϵ Sn avoids the subpattern τ iff π has no subsequence having all the same pairwise comparisons as τ, and we write π ϵ Sn(τ). We present a new bijective proof of the well-known result that /vbSn(123)/vb = /vbSn(132)/vb = cn, the nth Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that /vbSn(3142,2413)/vb = sn -1, the Schroder number, and characterize the deque-sortable permutations of Knuth, also counted by sn - 1.

Generating trees and forbidden subsequences

Abstract We discuss an enumerative technique called generating trees which was introduced in the study of Baxter permutations. We apply the technique to some other classes of permutations with forbidden subsequences. We rederive some known results, e.g. |Sn(132,231)| = 2n and |Sn(123,132,213)| = Fn, and add several new ones: Sn(123,3241), Sn(123,3214), Sn(123,2143). Finally, we argue for the broader use of generating trees in combinatorial enumeration.

Forbidden subsequences and Chebyshev polynomials

Abstract In (West, Discrete Math. 157 (1996) 363–374) it was shown using transfer matrices that the number | S n (123; 3214)| of permutations avoiding the patterns 123 and 3214 is the Fibonacci number F 2 n (as are also | S n (213; 1234)| and | S n (213; 4123)|). We now find the transfer matrix for | S n (123; r , r − 1,…,2, 1, r − 1)|, | S n (213; 1,2,…, r , r + 1)|, and | S n (213; r + 1, 1, 2,…, r )|, determine its characteristic polynomial in terms of the Chebyshev polynomials, and go on to determine the generating function as a quotient of modified Chebyshev polynomials. This leads to an asymptotic result for each r which collapses to the exact results 2 n when r = 2 and F 2 n when r = 3 and to the Catalan number c n as r → ∞. We observe that our generating function also enumerates certain lattice paths, plane trees, and directed animals, giving hope that these areas of combinatorics can be applied to enumerating permutations with excluded subsequences.

A New Class of Wilf-Equivalent Permutations

For about 10 years, the classification up to Wilf equivalence of permutation patterns was thought completed up to length 6. In this paper, we establish a new class of Wilf-equivalent permutation patterns, namely, (n − 1, n − 2, n, τ) ∼ (n − 2, n, n − 1, τ) for any τ∈Sn−3. In particular, at level n = 6, this result includes the only missing equivalence (546213) ∼ (465213), and for n = 7 it completes the classification of permutation patterns by settling all remaining cases in S7.

Raney Paths and a Combinatorial Relationship between Rooted Nonseparable Planar Maps and Two-Stack-Sortable Permutations

An encoding of the set of two-stack-sortable permutations (TSS) in terms of lattice paths and ordered lists of strings is obtained. These lattice paths are called Raney paths. The encoding yields combinatorial decompositions for two complementary subsets of TSS, which are the analogues of previously known decompositions for the set of nonseparable rooted planar maps (NS). This provides a combinatorial relationship between TSS and NS, and, hence, a bijection is determined between these sets that is different, simpler, and more refined than the previously known bijection.

Explicit enumeration of 321, hexagon-avoiding permutations

Abstract The 321 , hexagon-avoiding ( 321-hex ) permutations were introduced and studied by Billey and Warrington (J. Alg. Comb. 13 (2001) 111–136). as a class of elements of S n whose Kazhdan–Lusztig polynomials and the singular loci of whose Schubert varieties have certain fairly simple and explicit descriptions. This paper provides a 7-term linear recurrence relation leading to an explicit enumeration of the 321-hex permutations. A complete description of the corresponding generating tree is obtained as a by-product of enumeration techniques used in the paper, including Schensteds 321-subsequences decomposition, a 5-parameter generating function and the symmetries of the octagonal patterns avoided by the 321-hex permutations.

The Permutations 123 p 4 … p m and 321 p 4 … p m are Wilf-Equivalent

Abstract. Write p1, p2…pm for the permutation matrix δpi, j. Let Sn (M) be the set of n×n permutation matrices which do not contain the m×m permutation matrix M as a submatrix. In [7] Simion and Schmidt show bijectively that |Sn (123) |=|Sn (213) |. In [9] this was generalised to a bijection between Sn (12 p3…pm) and Sn (21 p3…pm). In the present paper we obtain a bijection between Sn (123 p4…pm) and Sn (321 p4…pm).

The Dyck pattern poset

We introduce the notion of pattern in the context of lattice paths, and investigate it in the specific case of Dyck paths. Similarly to the case of permutations, the pattern-containment relation defines a poset structure on the set of all Dyck paths, which we call the Dyck pattern poset. Given a Dyck path P, we determine a formula for the number of Dyck paths covered by P, as well as for the number of Dyck paths covering P. We then address some typical pattern-avoidance issues, enumerating some classes of pattern-avoiding Dyck paths. We also compute the generating function of Dyck paths avoiding any single pattern in a recursive fashion, from which we deduce the exact enumeration of such a class of paths. Finally, we describe the asymptotic behavior of the sequence counting Dyck paths avoiding a generic pattern, we prove that the Dyck pattern poset is a well-ordering and we propose a list of open problems.

A Robinson-Schensted Algorithm for a Class of Partial Orders

LetPbe a finite partial order which does not contain an induced subposet isomorphic with 3+1, and letGbe the incomparability graph ofP. Gasharov has shown that the chromatic symmetric functionXGhas nonnegative coefficients when expanded in terms of Schur functions; his proof uses the dual Jacobi?Trudi identity and a sign-reversing involution to interpret these coefficients as numbers ofP-tableau. This suggests the possibility of a direct bijective proof of this result, generalizing the Robinson?Schensted correspondence. We provide such an algorithm here under the additional hypothesis thatPdoes not contain an induced subposet isomorphic with {xa

Complementary Algorithms for Tableaux

We study four operations defined on pairs of tableaux. Algorithms for the first three involve the familiar procedures of jeu de taquin, row insertion, and column insertion. The fourth operation, hopscotch, is new, although specialised versions have appeared previously. Like the other three operations, this new operation may be computed with a set of local rules in a growth diagram, and it preserves the Knuth equivalence class. Each of these four operations gives rise to an a priori distinct theory of dual equivalence. We show that these four theories coincide. The four operations are linked via the involutive tableau operations of complementation and conjugation.