WIDTH-k GENERALIZATIONS OF CLASSICAL PERMUTATION STATISTICS ROBERT DAVIS 7 Abstract. Weintroducenewnaturalgeneralizationsofthe classicaldescentandinversion 1 statisticsforpermutations,calledwidth-k descents andwidth-k inversions. Thesevariations 0 inducegeneralizationsoftheexcedanceandmajorstatistics,providingaframeworkinwhich 2 the most well-known equidistributivity results for classical statistics are paralleled. We n explore additional relationships among the statistics providing specific formulas in certain a special cases. Moreover, we explore the behavior of these width-k statistics in the context J of pattern avoidance. 7 1 ] O 1. Introduction C h. Let S denote the set of permutations σ = a ···a of [n], and let S = S ∪S ∪.... A n 1 n 1 2 at function st : Sn → N is called a statistic, and the systematic study of permutation statistics m is generally accepted to have begun with MacMahon [6]. In particular, four of the most [ well-known statistics are the descent, inversion, major, and excedance statistics, defined 1 respectively by v 8 desσ = |{i ∈ [n−1] | a > a }| i i+1 8 invσ = |{(i,j) ∈ [n]2 | i < j and a > a }| 7 i j 4 majσ = i 0 . 1 i∈Desσ X 0 excσ = |{i ∈ [n] | a > i}| i 7 1 where Desσ = {i ∈ [n−1] | a > a }. : i i+1 v Given any statistic st, one may form the generating function i X Fst(q) = qstσ. r n a σX∈Sn A famous result due to MacMahon [6] states that Fdes(q) = Fexc(q), and that both equal n n are to the Eulerian polynomial A (q). The Eulerian polynomials themselves may be defined n via the identity A (q) (1+j)nqj = n . (1−q)n+1 j≥0 X Anotherwell-knownresult, duetoMacMahon[5]andRodrigues[7],statesFinv(q) = Fmaj(q) = n n [n] ! Foata famously provided combinatorial proofs of the equidistributivity of each; see [4] q for a treatment of these. Thus any statistic st for which Fst(q) = A (q) is called Eulerian, n n Department of Mathematics, Michigan State University, East Lansing, MI 48824-1027, USA. E-mail: davisr@math.msu.edu. 1 2 ROBERTDAVIS and if Fst(q) = [n] ! then st is called Mahonian. These four statistics have many general- n q izations; in this article, we discuss new variations, induced from a simple generalization of des. For each of the following definitions, we assume n ∈ Z , k ∈ [n−1], ∅ 6= K ⊆ [n−1], >0 and σ = a a ...a ∈ S . We define a width-k descent of σ to be an index i ⊆ [n−k] for 1 2 n n which a > a . Thus the width-1 descents are the usual descents of a permutation. Denote i i+k the set of all width-k descents of σ by Des (σ) = {i ∈ [n−k] | a > a }, k i i+k and set des (σ) = |Des (σ)|. k k If one is interested in descents of σ of various widths, first let K ⊆ [n−1] denote the set of widths under consideration. Now, we define Des (σ) to be the multiset Des (σ), K k∈K k and des (σ) = |Des (σ)|. K K S Now, define a width-k inversion of σ to be a pair (i,j) for which a > a and j −i = mk i j for some positive integer m. Denote the set of width-k inversions by Inv (σ) = {(i,j) ∈ [n]2 | a > a and j −k = mk,k ∈ Z} k i j and set inv (σ) = |Inv (σ)|. k k Again, one may beinterested in width-k inversions for multiple values of k, so for K ⊆ [n−1] define Inv (σ) = Inv (σ). Additionally, let inv (σ) = |Inv (σ)|. K k∈K k K K Example 1.1. If σ = 4136572, then S Des (σ) = {1,4,5} and Inv (σ) = {(1,3),(1,7),(3,7),(4,7),(5,7)}. {2,3} {2,3} Thus, des (σ) = 3, inv (σ) = 5. {2,3} {2,3} As we will see in the next section, the above definitions motivate us to generalize exc and maj in such a way that well-known known relationships among des,inv,maj,exc are paralleled. However, it is not very convenient to work with exc and maj directly. So, this K K paper will focus much more on des and inv . K K The main focus of this paper is to explore these new statistics and their relationships among each other. While they are well-behaved for special cases of K ⊆ [n− 1], formulas for more general cases have been more elusive. Section 3 continues this exploration by considering the same statistics for classes of permutations avoiding a variety of patterns. 2. Main Results We begin with simple expressions for des (σ) and inv (σ) for a fixed σ ∈ S . First, we K K n point out that if K ⊆ [n−1], j ∈ K and ij ∈ K for some positive integer i, then for any σ ∈ S , inv (σ) = inv (σ). This occurs because for any k ∈ [n−1], inv (σ) counts all n K K\{ij} k descents whose widths are multiples of k. Thus ij is already accounted for in inv (σ). K\{j} This follows quickly from the definitions of inv and des . k k Proposition 2.1. For any nonempty K ⊆ [n−1], inv (σ) = (−1)|K′|+1inv (σ), K lcm(K′) ∅(K′⊆K X where we set inv (σ) = 0 if lcm(K′) ≥ n. lcm(K′) WIDTH-k GENERALIZATIONS OF CLASSICAL PERMUTATION STATISTICS 3 Proof. We first consider the case where K = {k}. The elements of Inv (σ) are pairs of the k form (i,i+jk) for some positive integer j. Such an element exists if and only if there is a width-(jk) descent of σ at i. Thus, inv (σ) simply counts the number of width-jk descents k of σ for all possible j. This leads to the equality inv (σ) = des (σ). k jk j≥1 X The formula for general K then follows from inclusion-exclusion: by adding the number of all width-k inversions for all k ∈ K, we are twice counting any instances of a width-lcm(k ,k ) 1 2 inversion since such an inversion is also of widths k and k . If K contains three distinct 1 2 elements k ,k ,k , then lcm(k ,k ,k ) would have been added three times (once for each 1 2 3 1 2 3 inv (σ)) and subtracted three times (once for each inv (σ), i < j), so it must be added ki {ki,kj} again for the sum. Extending this argument to range over larger subsets of K results in the (cid:3) claimed formula. Example 2.2. Let us returnto σ = 4136572. Wesaw fromExample 1.1that inv (σ) = 5, {2,3} where four inversions have width 2 and two have width 3. But since the inversion (1,7) has width both 2 and 3, it must also have width lcm(2,3) = 6. So, it contributes two summands of 1 and one summand of −1. We come now to a function which helps demonstrate the interactions among the width-k statistics. Let n and k be positive integers for which n = dk + r for some d,r ∈ Z with 0 ≤ r < k. To each σ = a ...a ∈ S we may then associate the set of disjoint substrings 1 n n β (σ) = {β1 (σ),...,βk (σ)} where n,k n,k n,k a a a ...a if i ≤ r βi (σ) = i i+k i+2k i+dk n,k a a a ...a if r < i < k ( i i+k i+2k i+(d−1)k Now, define φ : S → Sr ×Sk−r n d+1 d by setting φ(σ) = (stdβ1 (σ),...,stdβk (σ)), where std is the standardization map, that is, n,k n,k the permutation obtained by replacing the smallest element of σ with 1, the second-smallest element with 2, etc. Note in particular that each stdβi (σ) is a permutation of [d +1] or n,k [d]. The first of the identities in the following proposition was originally established in [8], though with slightly different notation. We provide an alternate proof and extend their result to width-k inversions. Theorem 2.3. Let n and k be positive integers such that n = dk+r, where 0 ≤ r < k, and let A (q) denotethe ith Eulerian polynomial. Also let M denote themultinomial coefficient i n,k n M = n,k (d+1)r,dk−r (cid:18) (cid:19) where ij indicates i repeated j times. We then have the identities Fdesk(q) = M Ar (q)Ak−r(q) n n,k d+1 d Finvk(q) = M [d+1]r![d]k−r! n n,k q q 4 ROBERTDAVIS Proof. Let k ∈ [n − 1] and consider φ defined above. Note that φ is an M -to-one func- n,k tion since, given (σ ,...,σ ) ∈ Sr × Sk−r, there are M ways to partition [n] into the 1 k d+1 d n,k subsequences βi (σ) such that stdβi (σ) = σ for all i. Also note that n,k n,k i k des (σ) = des(stdβi (σ)). k n,k i=1 X Thus, Finvk(q) = qdeskσ n σX∈Sn = M qdesσ1...qdesσk n,k  (σ1,...,σk)X∈Srd+1×Skd−r  = M Ar (q)Ak−r(q).  n,k d+1 d This proves the first identity. The second identity follows completely analogously, with the main modification being that an element of Inv (σ) corresponds to a usual inversion in some unique stdβj (σ). (cid:3) k n,k Given des and inv , one must wonder what the corresponding generalizations of exc and K K maj are whose relationships with des and inv parallel that of the classical statistics. To K K do this, we define the multiset k Exc (σ) = {{j ∈ [n−1] | ⌈j/k⌉ ∈ Exc(stdβi (σ))}} K n,k k∈Ki=1 [ ] and set exc (σ) = |Exc (σ)|, and also set K K i maj (σ) = . K k kX∈Ki∈DXesk(σ)(cid:24) (cid:25) We could equivalently state that k k maj(stdβi (σ)) and exc (σ) = exc(stdβi (σ)). n,k K n,k i=1 i=1 X X These are exactly the definitions needed in order to obtain identities that parallel Fdes(q) = n Fexc(q) and Finv(q) = Fmaj(q), as we will soon see. n n n One important distinct to make between exc and exc is the following. If σ = a a ...a K 1 2 n and τ = b b ...b , then even if a = b for some i, one cannot say i ∈ Exc (σ) if and only 1 2 n i i K if i ∈ Exc (τ). For example, if σ = 4136572, then 1 ∈ Exc (σ), but if τ = 4153627 then K 2 1 ∈/ Exc (τ). 2 Example 2.4. Again let σ = 4136572. We then have exc (σ) = |{1,3,1,2}| = 4 {2,3} and 1 5 4 maj (σ) = + + = 6. {2,3} 2 2 3 (cid:24) (cid:25) (cid:24) (cid:25) (cid:24) (cid:25) WIDTH-k GENERALIZATIONS OF CLASSICAL PERMUTATION STATISTICS 5 k G G (q) G (q) 6,k 8,k 9,k 1 6A (q) 8A (q) 9A (q) 5 7 9 2 180A (q)2 1120A (q)2 9q−1A (q) 2 3 9 3 6! 8q−2A (q) 45360A (q)3 7 2 4 180q−2A (q)2 8! 9q−3A (q) 2 9 5 6q−4A (q) 8q−4A (q) 9q−4A (q) 5 7 9 6 1120q−4A (q)2 45360q−3A (q)3 3 2 7 8q−6A (q) 9q−6A (q) 7 9 8 9q−7A (q) 9 Table 1. The polynomials G (q) for n = 6,8,9 and 1 ≤ k ≤ n. n,k By constructing a nearly identical argument as in the proof Theorem 2.3, and using the facts that Fdes(q) = Fexc(q) and Finv(q) = Fmaj(q), we have the following corollary. n n n n Corollary 2.5. The identities Fndesk(q) = Fnexck(q) and Fninvk(q) = Fnmajk(q) hold. Now that we have established the analogous parallels between the four classical statistics and their width-k counterparts, we wish to explore what other structure is present. A simple proposition relates des and inv when k is large. k k Corollary 2.6. For all k ≥ n/2, Fdesk(q) = Finvk(q). n n Proof. Since k ≥ n/2, the sets βi (σ) contain at most two elements. So, width-k descents n,k and width-k inversions of σ are identical. (cid:3) We now show that interesting behavior occurs when considering the function G (q) = qdesk(σ)−desn−k(σ). n,k σX∈Sn According to computational data, the following conjecture holds for all n ≤ 9 and 1 ≤ k < n for which gcd(k,n) = 1. Conjecture 2.7. If gcd(k,n) = 1, then G (q) = nq1−kA (q). n,k n−1 Several illustrative polynomials are given in Table 2. This does not hold more generally, and it would be interesting to determine if a general formula exists. Question 2.8. For which values of n,k does there exist a closed formula for G (q), and n,k what is the formula? 3. Pattern Avoidance We say that a permutation σ ∈ S contains the pattern π ∈ S if there exists a sub- n m sequence σ′ of σ such that std(σ′) = π. If no such subsequence exists, then we say that σ avoids the pattern π. If Π ⊆ S, then we say σ avoids Π if σ avoids every element of Π. The set of all permutations of S avoiding Π is denoted Av (Π). In a mild abuse of notation, if n n Π = {π}, we will write Av (π). n 6 ROBERTDAVIS In this section, we consider the functions Fst(Π;q) = qstσ, n σ∈AXvn(Π) which specializes to Fst(q) if Π = ∅. In most instances, Fdesk will be the main focus, but n n FdesK and Finvk will also make appearances. n n An important concept within pattern avoidance is that of Wilf equivalence. Two sets Π,Π′ ⊂ S are said to be Wilf equivalent if |Av (Π)| = |Av (Π′)| for all n. In this case, n n we write Π ≡ Π′ to denote this Wilf equivalence, which is indeed an equivalence relation. For example, it is known [3, 6] that whenever π,π′ ∈ S , then |Av (π)| = |Av (π′)| = C , 3 n n n where 1 2n C = n n+1 n (cid:18) (cid:19) is the nth Catalan number. Proving whether Π ≡ Π′ is often quite difficult, and their Wilf equivalence does not imply that Fst(Π;q) = Fst(Π′;q). However, in some instances, the problems of establishing these n n identities have straightforward solutions by applying basic transformations on the elements of the avoidance classes. Given π = a ...a ∈ S , let πr denote its reversal and πc denote 1 n m its complement, respectively defined by πr = a a ...a and πc = (m+1−a )(m+1−a )...(m+1−a ). m m−1 1 1 2 m Similarly, given Π ⊆ S, we let Πr = {πc | π ∈ Π} and Πc = {πr | π ∈ Π} be the reversal and complement of Π, respectively. Our results of this section begin with a multivariate generalization of Fst(Π;q), and with n showing how its specializations describe relationships among Π,Πr, and Πc. Definition 3.1. Fix Π ⊆ S. Define n−1 T (Π;t ,...,t ) = tdesk(σ). n 1 n−1 k σ∈AXvn(Π)Yk=1 Several conclusions quickly follow. This function specializes to FdesK(Π;q) by setting n t = q for i ∈ K and t = 1 for all i ∈/ K. We can also recover FinvK(Π;q) from T by setting i i n n t = q whenever i ∈ [n−1]∩kZ for some k ∈ K, and setting t = 1 otherwise. Additionally, i i when Π = {π}, a nice duality appears, providing a mild generalization of Lemma 2.1 from [1]. Proposition 3.2. For any Π ⊆ S, let Π′ denote either Πr or Πc. We then have T (Π′;t ,...,t ) = tn−1tn−2···t T (Π;t−1,...,t−1 ). n 1 n−1 1 2 n−1 n 1 n−1 Consequently, T ((Πr)c;t ,...,t ) = T (Π;t ,...,t ). n 1 n−1 n 1 n−1 Proof. It is enough to prove the claim when the set of patterns being avoided is {π} for some π ∈ S , since the full result follows by applying the argument to all elements of Π m simultaneously. WIDTH-k GENERALIZATIONS OF CLASSICAL PERMUTATION STATISTICS 7 First consider when π′ = πc and σ ∈ Av (π). Because σ ∈ Av (π) if and only if σc ∈ n n Av (πc), we have that for each k, i ∈ Des (σ) if and only if i ∈/ Des (σc). This implies n k k Des (σc) = [n−k]\Des (σ), hence des (σc) = n−k −des (σ). So, k k k k n−1 T (πc;t ,...,t ) = tdesk(σ) n 1 n−1 k σ∈AXvn(πc)kY=1 n−1 = tdesk(σc) k σ∈AXvn(π)kY=1 n−1 = tn−k−desk(σ) k σ∈AXvn(π)kY=1 = tn−1tn−2···t T (π;t−1,...,t−1 ). 1 2 n−1 n 1 n−1 Proving that the result holds for π′ = πr follows similarly. The second identity in the proposition statement holds by applying the first identity twice: first for Πr and then for Πc. (cid:3) The above identities significantly reduce the amount of work needed to study FdesK(Π;q) n for all Π ⊆ S . For the remainder of this paper, we systematically approach Π for |Π| ≤ 2. n 3.1. Avoiding singletons. By Proposition 3.2, we immediately get Fdesk(123;q) = qn−kFdesk(321;q−1) n n and Fdesk(132;q) = Fdesk(213;q) = qn−kFdesk(231;q−1) = qn−kFdesk(312;q−1). n n n n So, studying Fdesk(π;q) for π ∈ S reduces to studying the function for a choice of one n 3 pattern from {123,321} and one from the remaining patterns. For some choices of Π, the permutations in Av (Π) are especially highly structured, which leads to similar arguments n throughout the rest of this paper. Webeginwithπ = 312. Noticethatifa ...a ∈ Av (312)anda = 1,thenstd(a ...a ) ∈ 1 n n i 1 i−1 Av (312), std(a ...a ) ∈ Av (312), and i−1 i+1 n n−i max{a ,...,a } < min{a ,...,a }. 1 i−1 i+1 n Proposition 3.3. For all n, k n Fdesk(312;q) = C Fdesk(312;q)+ qFdesk(312;q)Fdesk(312;q) n i−1 n−i i−1 n−i i=1 i=k+1 X X where C is the ith Catalan number. i Proof. First consider when σ = a ...a ∈ Av (312) and a = 1 for some i ≤ k. By the 1 n n i discussion preceding thisproposition, j ∈/ Des (σ)foranyj ≤ i. So, noneoftheC possible k i−1 permutations that make up std(a ...a ) contribute to des (σ). The only contributions to 1 i−1 k des (σ) come from std(a ...a ) ∈ Av (312;q). The overall contribution to Fdesk(312;q) k i+1 n n−i n is the first summand of the identity. Now suppose a = 1 for some i > k. Each choice of a ...a , contributes to des (σ) as i 1 i k usual, but there will be an additional width-k descent produced at i − k. The elements a ...a contribute to des (σ) as before. The overall contribution to Fdesk(312;q) is the i+1 n k n 8 ROBERTDAVIS first summand of the identity. Since we have considered all possible indices i for which we could have a = 1, we add the two cases together and are done. (cid:3) i Note that when we set q = 1, the above recursion specializes to the well-known recursion n for Catalan numbers C = C C . n+1 i=0 i n−i Wecanusetheprevious propositioninconjunction withProposition3.2to produceformu- laeforFdesk(Π;q)andFinvk(ΠP;q) whenever Πisa single element ofS other than123or 321. n n 3 The only nontrivial work required, then, is to compute the degrees of the two polynomials. Corollary 3.4. For all n, n−k n−i degFdesk(312;q) = n−k and degFinvk(312;q) = . n n k i=1 (cid:22) (cid:23) X Proof. The degrees of the above polynomials are given by identifying a permutation in Av (312) with the most possible descents. This is satisfied by n(n − 1)...21 ∈ Av (312) n n which has n−k descents of width k. Determining the number of width-k inversions in this (cid:3) permutation is done similarly. Although 321 is Wilf equivalent to 312, it is not so obvious how to construct a recurrence relation for Fdesk(321;q). To help describe the elements of Av (321), first let σ = a ...a ∈ n n 1 n S and call a a left-right maximum if a > a for all j < i. Thus σ ∈ Av (321) if and i i j n only if its set of non-left-right maxima form an increasing subsequence of σ. Indeed, if the non-left-right maxima did contain a descent, then there would be some left-right maximum preceding both elements, which violates the condition that σ avoid 321. Despite such a description, using it to reveal Fdesk(321;q) has thus far been unsuccessful. So, we leave the n following as an open question. Question 3.5. Is there a closed formula or simple recursion for Fdesk(321;q) (equivalently, n for Fdesk(123;q))? n 3.2. Avoiding doubletons. At this point, we begin studying the functions Fdesk(Π;q) n when avoiding doubletons from S . Recall that, by the Erd˝os-Szekeres theorem [2], there 3 are no permutations in S for n ≥ 5 that avoid both 123 and 321. Thus, we will not consider n this pair. Additionally, the functions Finvk(Π;q) are quite unwieldy, so we will not consider n these either. For our first nontrivial example, we begin with {123,132}. Permutations a ...a ∈ 1 n Av (123,132) have the following structure: for any i = 1,...,n, if a = n, then the substring n i a a ...a is decreasing and consists of the elements from the interval [n − i + 1,n − 1]. 1 2 i−1 Additionally, the substring a ...a is an element of Av (123,132). This structure makes i+1 n n−i it easy to show that there are 2n−1 elements of Av (123,132) [9]. n Proposition 3.6. For all n and 1 ≤ k ≤ n−1, k Fdesk(123,132;q) = qmin(i,n−k)Fdesk(123,132;q) n n−i i=1 X n−k + qmin(i−1,n−k−1)Fdesk(123,132;q) n−i i=k+1 X +2n−max(k+1,n−k+1)qn−k−1. WIDTH-k GENERALIZATIONS OF CLASSICAL PERMUTATION STATISTICS 9 Proof. Let σ = a ...a ∈ Av (123,132;q). If a = n for i ≤ k, then it is clear from the 1 n n i preceding description of the elements in Av (123,132) that all of 1,...,min(i,n − k) are n elements of Des (σ). If j > n − k for some j, then j ∈/ Des (σ) since a does not exist. k k j+k The remaining elements of Des (σ) arise as a width-k descent of a ...a , which accounts k i+1 n for the factor of Fdesk(123,132;q) in the first summand. n−i Now, if k +1 ≤ n−k and a = n for i = k +1,...,n−k, then every element of 1,...,i i except i−k is anelement ofDes (σ). If k+1 > n−k, thenthesecond summand is empty and k nothing is lost by continuing to the case of a = n for i ≥ max(k+1,n−k+1). Again, the i remaining elements of Des (σ) are the width-k descents of a ...a , hence the additional k i+1 n factor of Fdesk(123,132;q). This accounts for the second summand. n−i Finally, let m = max(k +1,n−k + 1). If a = n for i ≥ m, then all of 1,...,n−k are i descents except i−k, and these are the only possible width-k descents. In particular, none of i+1,i+2,...,n can be the index for a width-k descent. This leads to the sum n n−1 qn−k−1|Av (123,132)| = 1+ 2n−i−1 qn−k−1 = 2n−mqn−k−1, n−i ! i=m i=m X X (cid:3) which accounts for the third summand. Next, we consider {123,312}. We proceed similarly as before but with some minor differ- ences, reflecting the new structure we encounter. If σ = a ...a ∈ Av (123,312) and a = 1 1 n n i for some i < n, then σ is of the form σ = i(i−1)...21n(n−1)...(i+2)(i+1), since neither subsequence a ...a anda ...a may contain anascent. If a = 1, though, 1 i−1 i+1 n n then std(a ...a ) ∈ Av (123,312). 1 n−1 n−1 Proposition 3.7. For all n and 1 ≤ k < n, k n−1 Fdesk(123,312;q) = qmax(0,n−k−i) + qmax(n−2k,i−k) n i=1 i=k+1 X X +qFdesk(123,312;q) n−1 Proof. Suppose a = 1 for some i ≤ k. Using the description of elements in Av (123,312), i n we know there are n−k −i width-k descents. Since there is only one such permutation for each i, we simply add all of the qn−k−i, so long as n−k −i ≥ 0. If this inequality does not hold, then there are no width-k descents in this range. This accounts for the first summand. The second summand is computed very similarly to that of the second summand in Proposition 3.6. The final summand is a direct result of noting that when a = 1, then n (a −1)...(a −1) may be any element of Av (123,312;q), which accounts for the fac- 1 n−1 n−1 tor Fdesk(123,312;q). For each of these choices, we know n−k ∈ Des (σ), hence the factor n−1 k of q. Adding the sums results in the identity claimed. (cid:3) Now we consider {132,231}. Note that if a ...a ∈ Av (132,231), then a 6= n for any 1 n n i 1 < i < n. If a = n, then std(a ...a ) ∈ Av (132,231), and similarly if a = n. Once 1 2 n n−1 n again, this allows us to quickly compute that |Av (132,231)| = 2n−1. n 10 ROBERTDAVIS Proposition 3.8. Let K = {k ,...,k } be a nonempty subset of [n−1] whose elements are 1 l listed in increasing order. We then have l+1 FdesK(132,231;q) = (1+qi−1)ki−ki−1 n i=1 Y where k = 1 and k = n. 0 l+1 Proof. It follows from the description of elements in Av (132,231) that there are two sum- n mands in a recurrence for FdesK(132,231;q): one corresponding to a = n and one corre- n 1 sponding to a = n. When a = n, then there are |K| = l copies of 1 ∈ Des (σ); if a = n, n 1 K n thena makes no contribution toDes (σ). Thus, by deleting nfromσ, we get therecurrence n K FdesK(132,231;q) = (1+ql)FdesK(132,231;q). n n−1 Repeating this, a factor of 1+ql appears until we get to FdesK(132,231;q) = (1+ql)n−klFdesK(132,231;q). n k l At this point, note that FdesK(132,231;q) = FdesK\{kl}(132,231;q). k k l l (cid:3) Repeating the previous argument ends up with the identity claimed. Next, consider {132,213}. If σ = a ...a ∈ Av (132,213), then if a = n for any i, then 1 n n i a ...a must be increasing in order to avoid 213. Moreover, 1 i−1 max(a ,...,a ) < a i+1 n 1 in order to avoid 132, and std(a ...a ) ∈ Av (132,213). i+1 n n−i Proposition 3.9. For all n and 1 ≤ k < n, k Fdesk(132,213;q) = qmin(i,n−k)Fdesk(132,213;q) n n−i i=1 X n−k + qmin(k,n−i)Fdesk(132,213;q) n−i i=k+1 X n + qn−iFdesk(132,213;q). n−i i=max(k+1,n−k+1) X Proof. The proof of this is entirely analogous to the proof of Proposition 3.6. (cid:3) Next, we consider {132,312}. For each i = 1,...,n−1, either a = max{a ,...,a }+1 i+1 1 i or a = min{a ,...,a }−1. This doubleton often results in especially pleasant formulas, i+1 1 i and our results are no exception. Proposition 3.10. Let K = {k ,...,k } be a nonempty subset of [n− 1] whose elements 1 l are listed in increasing order. We then have l+1 FdesK(132,312;q) = (1+qi−1)ki−ki−1 n i=1 Y where k = 1 and k = n. 0 l+1

Width-$k$ Generalizations of Classical Permutation Statistics

Similar Width-$k$ Generalizations of Classical Permutation Statistics