Generalized Pattern-Matching Conditions for 𝐶_𝑘≀𝑆_𝑛

Abstract

We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product 𝐶_𝑘≀𝑆_𝑛 of the cyclic group 𝐶_𝑘 and the symmetric group 𝑆_𝑛. In particular, we derive the generating functions for the number of matches that occur in elements of 𝐶_𝑘≀𝑆_𝑛 for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of 𝐶_𝑘≀𝑆_𝑛. Our research leads to connections to many known objects/structures yet to be explained combinatorially.

1. Introduction

The goal of this paper is to study pattern-matching conditions on the wreath product 𝐶_𝑘≀𝑆_𝑛 of the cyclic group 𝐶_𝑘 and the symmetric group 𝑆_𝑛. 𝐶_𝑘≀𝑆_𝑛 is the group of 𝑘^𝑛𝑛! signed permutations where we allow 𝑘 signs of the form 1=𝜔⁰,𝜔,𝜔²,…,𝜔^𝑘−1 for some primitive 𝑘th root of unity 𝜔. We can think of the elements 𝐶_𝑘≀𝑆_𝑛 as pairs 𝛾=(𝜎,𝜖) where 𝜎=𝜎₁⋯𝜎_𝑛∈𝑆_𝑛 and 𝜖=𝜖₁⋯𝜖_𝑛∈{1,𝜔,…,𝜔^𝑘−1}^𝑛. For ease of notation, if 𝜖=(𝜔^𝑤1,𝜔^𝑤2,…,𝜔^𝑤𝑛) where 𝑤_𝑖∈{0,1,…,𝑘−1} for 𝑖=1,2,…,𝑛, then we simply write 𝛾=(𝜎,𝑤) where 𝑤=𝑤₁𝑤₂⋯𝑤_𝑛∈{0,1,…,𝑘−1}^𝑛.

Given a sequence 𝜎=𝜎₁⋯𝜎_𝑛 of distinct integers, let red(𝜎) be the permutation found by replacing the 𝑖th largest integer that appears in 𝜎 by 𝑖. For example, if 𝜎=2754, then red(𝜎)=1432. Given a permutation 𝜏 in the symmetric group 𝑆_𝑗, we say a permutation 𝜎=𝜎₁⋯𝜎_𝑛∈𝑆_𝑛 has a 𝜏-match starting at position 𝑖 provided red(𝜎_𝑖⋯𝜎_{𝑖+𝑗−1})=𝜏. Let 𝜏-mch(𝜎) be the number of 𝜏-matches in the permutation 𝜎. Similarly, we say that 𝜏 occurs in 𝜎 if there exist 1≤𝑖₁<⋯<𝑖_𝑗≤𝑛 such that red (𝜎_𝑖1⋯𝜎_𝑖𝑗)=𝜏. We say that 𝜎   avoids 𝜏 if there are no occurrences of 𝜏 in 𝜎.

We can define similar notions for words over a finite alphabet [𝑘]={0,1,…,𝑘−1}. Given a word 𝑤=𝑤₁⋯𝑤_𝑛∈[𝑘]^𝑛, let red(𝑤) be the word found by replacing the 𝑖th largest integer that appears in 𝑤 by 𝑖−1. For example, if 𝑤=27247, then red(𝑤)=02012. Given a word 𝑢∈[𝑘]^𝑗 such that red(𝑢)=𝑢, we say a word 𝑤∈[𝑘]^𝑛 has a 𝑢-match starting at position 𝑖 provided red(𝑤_𝑖⋯𝑤_{𝑖+𝑗−1})=𝑢. Let 𝑢-mch(𝑤) be the number of 𝑢-matches in the word 𝑤. Similarly, we say that 𝑢 occurs in a word 𝑤 if there exist 1≤𝑖₁<⋯<𝑖_𝑗≤𝑛 such that red(𝑤_𝑖1⋯𝑤_𝑖𝑗)=𝑢. We say that 𝑤   avoids 𝑢 if there are no occurrences of 𝑢 in 𝑤.

There are a number of papers on pattern matching and pattern avoidance in 𝐶_𝑘≀𝑆_𝑛 [1–4]. For example, the following pattern matching condition was studied in [2–4].

Definition 1. Let (𝜏,𝑢)∈𝐶_𝑘≀𝑆_𝑗, Υ be a subset of 𝐶_𝑘≀𝑆_𝑗 and (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛. (1)One says that (𝜎,𝑤) has an exact occurrence of (𝜏,𝑢) (resp., Υ) if there are 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑗≤𝑛 such that (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),𝑤_𝑖1⋯𝑤_𝑖𝑗)=(𝜏,𝑢) (resp., (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),𝑤_𝑖1⋯𝑤_𝑖𝑗)∈Υ).(2)One says that (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛  avoids an exact occurrence of  (𝜏,𝑢) (resp., Υ) if there are no exact occurrences of (𝜏,𝑢) (resp., Υ) in (𝜎,𝑤).(3)One says that there is an exact  (𝜏,𝑢)-match in  (𝜎,𝑤)   starting at position  𝑖 (resp., exact  Υ-match in  (𝜎,𝑤)  starting at position  𝑖) if red𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1},𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1}=(𝜏,𝑢)resp.,red𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1},𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1}∈Υ. (1)

That is, an exact occurrence or an exact match of (𝜏,𝑢)∈𝐶_𝑘≀𝑆_𝑗 in an element (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 is just an ordinary occurrence or match of 𝜏 in 𝜎 where the corresponding signs agree exactly. For example, Mansour [3] proved via recursion that for any (𝜏,𝑢)∈𝐶_𝑘≀𝑆₂, the number of elements in 𝐶_𝑘≀𝑆_𝑛 which avoid exact occurrences of (𝜏,𝑢) is ∑^𝑛_𝑗=0𝑗!(𝑘−1)^𝑗𝑛_𝑗². This generalized a result of Simion [5] who proved the same result for the hyperoctahedral group 𝐶₂≀𝑆_𝑛. Similarly, Mansour and West [4] determined the number of permutations in 𝐶₂≀𝑆_𝑛 that avoid all possible exact occurrences of 2 or 3 element sets of patterns of elements of 𝐶₂≀𝑆₂. For example, let 𝐾¹_𝑛 be the number of (𝜎,𝜖)∈𝐶₂≀𝑆_𝑛 that avoid all exact occurrences of the patterns in the set {(12,00),(12,01),(21,10)}, let 𝐾²_𝑛 be the number of (𝜎,𝜖)∈𝐶₂≀𝑆_𝑛 that avoid all exact occurrences of the patterns in the set {(12,01),(12,10),(21,01)}, and let 𝐾³_𝑛 be the number of (𝜎,𝜖)∈𝐶₂≀𝑆_𝑛 that avoid all exact occurrences of the patterns in the set {(12,00),(12,01),(21,00)}. Then Mansour and West [4] proved that 𝐾¹_𝑛=𝐹_2𝑛+1,𝐾²_𝑛=𝑛!^𝑛_𝑗=0𝑛𝑗⁻¹,𝐾³_𝑛=𝑛!+𝑛!^𝑛_𝑗=11𝑗, (2) where 𝐹_𝑛 is the 𝑛th Fibonacci number.

An alternative matching condition arises when we drop the requirement of the exact matching of signs and replace it by the condition that the two sequences of signs match in the sense of words described above. That is, we will consider the following matching conditions.

Definition 2. Let (𝜏,𝑢)∈𝐶_𝑘≀𝑆_𝑗 where red(𝑢)=𝑢, Υ be a subset of 𝐶_𝑘≀𝑆_𝑗 where for all (𝜏,𝑢)∈Υ, red(𝑢)=𝑢, and (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛. (1)One says that (𝜎,𝑤) has an occurrence of (𝜏,𝑢) (resp., Υ) if there are 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑗≤𝑛 such that (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),red(𝑤_𝑖1⋯𝑤_𝑖𝑗))=(𝜏,𝑢) (resp., (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),red(𝑤_𝑖1⋯𝑤_𝑖𝑗))∈Υ).(2)We say that (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛avoids   (𝜏,𝑢) (resp., Υ) if there are no occurrences of (𝜏,𝑢) (resp., Υ) in (𝜎,𝑤).(3)One says that there is a (𝜏,𝑢)-match in (𝜎,𝑤)starting at position 𝑖 (resp., Υ-matchin (𝜎,𝑤)starting at position 𝑖) if  (red(𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1}),𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1})=(𝜏,𝑢) (resp., (red(𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1}), red(𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1})) ∈Υ).

For example, suppose that (𝜏,𝑢)=(12,00) and (𝜎,𝑤)=(1324,1222). Then there are no exact occurrences or exact matches of (𝜏,𝑢) in (𝜎,𝑤). However, there are two occurrences of (𝜏,𝑢), one in positions 2 and 4 and one in positions 3 and 4. Thus, there are two occurrences of (𝜏,𝑢) in (𝜎,𝑤), and there is a (𝜏,𝑢)-match in (𝜎,𝑤) starting at position 3.

Finally, we will consider a more general matching condition which generalizes both occurrences and matches and exact occurrences and exact matches in 𝐶_𝑘≀𝑆_𝑛.

Definition 3. Let (𝜏,𝑢)∈𝐶_𝑘≀𝑆_𝑗, let Υ be a subset of 𝐶_𝑘≀𝑆_𝑗, and let 𝐴^→=(𝐴₁,…,𝐴_𝑗) be a sequence of subsets of [𝑘], and (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛. (1)One says that (𝜎,𝑤) has an occurrence of (𝜏,𝑢,𝐴^→) (resp., (Υ,𝐴^→)) if there are 1≤𝑖₁<𝑖₂<⋯<𝑖_𝑗≤𝑛 such that (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),red(𝑤_𝑖1⋯𝑤_𝑖𝑗))=(𝜏,red(𝑢)) (resp., (red(𝜎_𝑖1⋯𝜎_𝑖𝑗),red(𝑤_𝑖1⋯𝑤_𝑖𝑗)) is equal to (𝜏,red(𝑢)) for some (𝜏,𝑢) in Υ) and 𝑤_𝑖𝑠∈𝐴_𝑠 for 𝑠=1,2,…,𝑗.(2)One says that (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛avoids   (𝜏,𝑢) (resp., Υ) if there are no occurrences of (𝜏,𝑢) (resp., Υ) in (𝜎,𝑤).(3)We say that there is a (𝜏,𝑢)-match in  (𝜎,𝑤)  starting at position  𝑖 (resp., Υ-match in  (𝜎,𝑤)  starting at position  𝑖) if (red(𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1}),   red(𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1}))=(𝜏,red(𝑢))  (resp.,(red(𝜎_𝑖𝜎_𝑖+1⋯𝜎_{𝑖+𝑗−1}), red(𝑤_𝑖𝑤_𝑖+1⋯𝑤_{𝑖+𝑗−1})) is equal to (𝜏,red(𝑢)) for some (𝜏,𝑢) in Υ) and 𝑤_𝑖+𝑠∈𝐴_𝑠+1 for 𝑠=0,1,…,𝑗−1.

Thus, a (𝜏,𝑢,𝐴^→)-occurrence or (𝜏,𝑢,𝐴^→)-match where 𝑢=𝑢₁⋯𝑢_𝑗 and 𝐴^→=(𝐴₁,…,𝐴_𝑗) is such that 𝐴_𝑖={𝑢_𝑖} for 𝑖=1,…,𝑗 is just an exact occurrence or exact match of (𝜏,𝑢). Similarly, a (𝜏,𝑢,𝐴^→)-occurrence or (𝜏,𝑢,𝐴^→)-match where 𝑢=𝑢₁⋯𝑢_𝑗, red(𝑢)=𝑢, and 𝐴^→=(𝐴₁,…,𝐴_𝑗) is such that 𝐴_𝑖=[𝑘] for 𝑖=1,…,𝑗 is just an occurrence or match of (𝜏,𝑢).

Suppose we are given (𝜏,𝑢)∈𝐶_𝑘≀𝑆_𝑗, Υ⊆𝐶_𝑘≀𝑆_𝑗, 𝐴^→=(𝐴₁,…,𝐴_𝑗) where 𝐴_𝑖⊆[𝑘] for 𝑖=1,…,𝑗, and (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛. We let (𝜏,𝑢)-mch((𝜎,𝑤)) (resp., (𝜏,𝑢)-Emch((𝜎,𝑤))) denote the number of (𝜏,𝑢)-matches (resp., exact (𝜏,𝑢)-matches) in (𝜎,𝑤). We let Υ-mch((𝜎,𝑤)) (resp., Υ-Emch((𝜎,𝑤))) denote the number of Υ-matches (resp., exact Υ-matches) in (𝜎,𝑤). We let (𝜏,𝑢,𝐴^→)-mch((𝜎,𝑤)) be the number of (𝜏,𝑢,𝐴^→)-matches in (𝜎,𝑤) and let (Υ,𝐴^→)-mch((𝜎,𝑤)) be the number of (Υ,𝐴^→)-matches in (𝜎,𝑤).

The main result of this paper is to derive a generating function for the distribution of (𝜏,𝑢,𝐴^→)-matches where (𝜏,𝑢) is any element of 𝐶_𝑘≀𝑆₂. To state our main result, we first need some notation. We define the 𝑝,𝑞-analogues of 𝑛, 𝑛!, (^𝑛_𝑘), and 𝑛𝑎₁,…,𝑎_𝑚 by [𝑛]_𝑝,𝑞=𝑝^𝑛−𝑞^𝑛𝑝−𝑞=𝑝^𝑛−1+𝑝^𝑛−2𝑞+⋯+𝑝𝑞^𝑛−2+𝑞^𝑛−1,[𝑛]_𝑝,𝑞!=[𝑛]_𝑝,𝑞[𝑛−1]_𝑝,𝑞⋯[2]_𝑝,𝑞[1]_𝑝,𝑞,𝑛𝑘_𝑝,𝑞=[𝑛]_𝑝,𝑞![𝑘]_𝑝,𝑞![𝑛−𝑘]_𝑝,𝑞!,𝑛𝑎₁,…,𝑎_{𝑚𝑝,𝑞}=[𝑛]_𝑝,𝑞!𝑎_1𝑝,𝑞!⋯𝑎_{𝑚𝑝,𝑞}!, (3)

respectively. We define the 𝑞-analogues of 𝑛, 𝑛!, (^𝑛_𝑘), and 𝑛𝑎₁,…,𝑎_𝑚 by [𝑛]_1,𝑞, [𝑛]_1,𝑞!, [^𝑛_𝑘]_1,𝑞, and 𝑛𝑎₁,…,𝑎_𝑚1,𝑞, respectively.

Next suppose that Υ⊆𝐶_𝑘≀𝑆₂ and 𝐴^→=(𝐴₁,𝐴₂) where 𝐴₁,𝐴₂⊆[𝑘]. Then we will say that (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 is a maximum packing for (Υ,𝐴^→) if (𝜎,𝑤) has (Υ,𝐴^→)-matches starting at positions 1,2,…,𝑛−1. We let ℳ𝒫_(Υ,^→_𝐴),𝑛 denote the set of (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 which are maximum packings for (Υ,𝐴^→). Given any word 𝑤=𝑤₁⋯𝑤_𝑛∈[𝑘]^𝑛, we let 𝑧(𝑤)=∏^𝑛_𝑖=1𝑧_𝑤𝑖. For any 𝜎=𝜎₁⋯𝜎_𝑛∈𝑆_𝑛, we let inv(𝜎) (resp., coinv(𝜎)) equal the number of pairs (𝑖,𝑗) such that 1≤𝑖<𝑗≤𝑛 and 𝜎_𝑖>𝜎_𝑗 (resp., 𝜎_𝑖<𝜎_𝑗). We then define mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1=_{(𝜎,𝑤)∈ℳ𝒫→𝐴(Υ,),𝑛}𝑝^coinv(𝜎)𝑞^inv(𝜎)𝑧(𝑤). (4) We shall also be interested in the specializations mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;1,…,1=_{()𝜎,𝑤∈ℳ𝒫→𝐴(Υ,),𝑛}𝑝^coinv(𝜎)𝑞^inv(𝜎),mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;1,𝑟,…,𝑟^𝑘−1=_{(𝜎,𝑤)∈ℳ𝒫→𝐴(Υ,),𝑛}𝑝^coinv(𝜎)𝑞^inv(𝜎)𝑟^‖𝑤‖, (5) where for any word 𝑤=𝑤₁⋯𝑤_𝑛∈[𝑘]^𝑛, ||𝑤||=𝑤₁+⋯+𝑤_𝑛. In the special case, where Υ={(𝜏,𝑢)} where (𝜏,𝑢)∈𝐶_𝑘≀𝑆₂, we will denote mp_(Υ,^→_𝐴),𝑛(𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1) as mp_(𝜏,𝑢,^→_𝐴),𝑛(𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1). For example, if 𝜏=12, 𝑢=01, and 𝐴^→=([𝑘],[𝑘]), then (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 is a maximum packing for (𝜏,𝑢,𝐴^→) if and only if 𝜎=12⋯𝑛 is the identity permutation and 𝑤=𝑤₁⋯𝑤_𝑛 is such that 0≤𝑤₁<⋯<𝑤_𝑛≤𝑘−1. Thus, mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1=𝑝^{𝑛2𝑘−1}_𝑖=0(1+𝑧_𝑖𝑡)_𝑡𝑛, (6) where for any power series 𝑓(𝑡)=∑_𝑛≥0𝑓_𝑛𝑡^𝑛, we write 𝑓(𝑡)∣_𝑡𝑛=𝑓_𝑛 for the coefficient of 𝑡^𝑛 in 𝑓(𝑡). Then it is easy to prove that mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;1,…,1=𝑝^𝑛2𝑘𝑛,mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;1,𝑟,…,𝑟^𝑘−1=𝑝^𝑛2𝑟^𝑛2𝑘𝑛_𝑟. (7) If 𝜏=12, 𝑢=00, and 𝐴^→=([𝑘],[𝑘]), then (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 is a maximum packing for (𝜏,𝑢,𝐴^→) if and only if 𝜎=12⋯𝑛 is the identity permutation and 𝑤=𝑖^𝑛 for some 0≤𝑖≤𝑘−1. Thus, mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1=𝑝^{𝑛2𝑘−1}_𝑖=0𝑧^𝑛_𝑖. (8) It then follows that mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;1,…,1=𝑘𝑝^𝑛2,mp_(𝜏,𝑢,^→_𝐴),𝑛𝑝,𝑞;1,𝑟,…,𝑟^𝑘−1=[𝑘]_𝑟𝑛𝑝^𝑛2. (9) Similarly if Υ={(12,00),(12,01)} and 𝐴^→=([𝑘],[𝑘]), then (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 is a maximum packing for (Υ,𝐴^→) if and only if 𝜎=12⋯𝑛 is the identity permutation and 𝑤=𝑤₁⋯𝑤_𝑛 is such that 0≤𝑤₁≤⋯≤𝑤_𝑛≤𝑘−1. Thus, mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1=𝑝^{𝑛2𝑘−1}_𝑖=011−𝑧_𝑖𝑡_𝑡𝑛. (10) We shall then prove that mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;1,…,1=𝑝^𝑛2𝑛,𝑛+𝑘−1mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;1,𝑟,…,𝑟^𝑘−1=𝑝^𝑛2𝑛𝑛+𝑘−1_𝑟. (11)

The main result of this paper is to prove the following.

Theorem 4. For any Υ⊆𝐶_𝑘≀𝑆₂ and 𝐴^→=(𝐴₁,𝐴₂) where 𝐴₁,𝐴₂⊆[𝑘], _𝑛≥0𝑡^𝑛[𝑛]_𝑝,𝑞!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑝^coinv(𝜎)𝑞^inv(𝜎)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))}𝑧(𝑤)=1×1−𝑧₀+⋯+𝑧_𝑘−1𝑡+_𝑛≥2𝑡^𝑛[𝑛]_𝑝,𝑞!(𝑥−1)^𝑛−1mp_(Υ,^→_𝐴),𝑛×𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1⁻¹. (12)

We note that (12) can be specialized to give natural analogues for generating functions of rises and descents in 𝐶_𝑘≀𝑆_𝑛 where we compare pairsusing the product order. That is, instead of thinking of an element of 𝐶_𝑘≀𝑆_𝑛 as a pair (𝜎₁⋯𝜎_𝑛,𝑤₁⋯𝑤_𝑛), we can think of it as a sequence of pairs (𝜎₁,𝑤₁)(𝜎₂,𝑤₂)⋯(𝜎_𝑛,𝑤_𝑛). We then define a partial order on such pairs by the usual product order. That is, (𝑖₁,𝑗₁)≤(𝑖₂,𝑗₂) if and only if 𝑖₁≤𝑖₂ and 𝑗₁≤𝑗₂. Then we define the following sets and statistics for elements (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛: Des((𝜎,𝑤))=𝑖∶𝜎_𝑖>𝜎_𝑖+1&𝑤_𝑖≥𝑤_𝑖+1,des((𝜎,𝑤))=|Des((𝜎,𝑤))|,Ris((𝜎,𝑤))=𝑖∶𝜎_𝑖<𝜎_𝑖+1&𝑤_𝑖≤𝑤_𝑖+1,ris((𝜎,𝑤))=|Ris,𝑊((𝜎,𝑤))|Des((𝜎,𝑤))=𝑖∶𝜎_𝑖>𝜎_𝑖+1&𝑤_𝑖=𝑤_𝑖+1,𝑤des((𝜎,𝑤))=|𝑊Des,𝑊((𝜎,𝑤))|Ris((𝜎,𝑤))=𝑖∶𝜎_𝑖<𝜎_𝑖+1&𝑤_𝑖=𝑤_𝑖+1,𝑤ris((𝜎,𝑤))=|𝑊Ris,𝑆((𝜎,𝑤))|Des((𝜎,𝑤))=𝑖∶𝜎_𝑖>𝜎_𝑖+1&𝑤_𝑖>𝑤_𝑖+1,𝑠des((𝜎,𝑤))=|𝑆Des,𝑆((𝜎,𝑤))|Ris((𝜎,𝑤))=𝑖∶𝜎_𝑖<𝜎_𝑖+1&𝑤_𝑖<𝑤_𝑖+1,𝑠ris((𝜎,𝑤))=|𝑆Ris((𝜎,𝑤))|. (13) We shall refer to Des((𝜎,𝑤)) as the descent set of (𝜎,𝑤), 𝑊Des((𝜎,𝑤)) as the weak descent set of (𝜎,𝑤), and 𝑆Des((𝜎,𝑤)) as the strict descent set of (𝜎,𝑤). Similarly, we will refer to Ris((𝜎,𝑤)) as the rise set of (𝜎,𝑤), 𝑊Ris((𝜎,𝑤)) as the weak rise set of (𝜎,𝑤), and 𝑆Ris((𝜎,𝑤)) as the strict rise set of (𝜎,𝑤). It is easy to see that  𝑖∈𝑊Des((𝜎,𝑤)) if and only if there is a (21,00)-match starting at position 𝑖, 𝑖∈𝑆Des((𝜎,𝑤)) if and only if there is a (21,10)-match starting at position 𝑖, and 𝑖∈Des((𝜎,𝑤)) if and only if there is a Υ-match starting at position 𝑖 where Υ={(21,00),(21,10)}.

Similarly,  𝑖∈𝑊Ris((𝜎,𝑤)) if and only if there is a (12,00)-match starting at position 𝑖, 𝑖∈𝑆Ris((𝜎,𝑤)) if and only if there is a (12,01)-match starting at position 𝑖, and 𝑖∈Ris((𝜎,𝑤)) if and only if there is a Υ-match starting at position 𝑖 where Υ={(12,00),(12,01)}.

If 𝜎=𝜎₁⋯𝜎_𝑛∈𝑆_𝑛, then we define the reverse of 𝜎, 𝜎^𝑟 by 𝜎^𝑟=𝜎_𝑛⋯𝜎₁. Similarly, if 𝑤=𝑤₁⋯𝑤_𝑛∈[𝑘]^𝑛, then we define 𝑤^𝑟=𝑤_𝑛⋯𝑤₁. It is easy to see that ris((𝜎,𝑤))=des𝜎^𝑟,𝑤^𝑟,𝑤ris((𝜎,𝑤))=𝑤des𝜎^𝑟,𝑤^𝑟,𝑠ris((𝜎,𝑤))=𝑠des𝜎^𝑟,𝑤^𝑟. (14) Thus, we need to find the distributions for only one of the corresponding pairs. Substituting 𝑧₀=𝑧₁=⋯=𝑧_𝑘−1=1 into (12) in these special cases will yield the following generating functions for rises, strict rises, and weak rises in 𝐶_𝑘≀𝑆_𝑛.

Theorem 5. Consider the following: _𝑛≥0𝑡^𝑛𝑛!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑥^{ris((𝜎,𝑤))}=1−𝑥1−𝑥+∑_𝑛≥1((𝑥−1)𝑡)^𝑛/𝑛!_𝑛^{𝑛+𝑘−1}_𝑛≥0,𝑡^𝑛𝑛!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑥^{wris((𝜎,𝑤))}=1−𝑥1−𝑥+𝑘𝑒^{(𝑥−1)𝑡},−1_𝑛≥0𝑡^𝑛𝑛!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑥^{sris((𝜎,𝑤))}=1−𝑥1−𝑥+∑_𝑛≥1((𝑥−1)𝑡)^𝑛/𝑛!^𝑘_𝑛. (15)(16)

Other distribution results for (𝜏,𝑢)-matches follow from these results. For example, if 𝜎=𝜎₁⋯𝜎_𝑛∈𝑆_𝑛, then we define the complement of 𝜎, 𝜎^𝑐 by 𝜎^𝑐=𝑛+1−𝜎₁⋯𝑛+1−𝜎_𝑛. (17) If 𝑤=𝑤₁⋯𝑤_𝑛∈[𝑘]^𝑛, then we define the complement of 𝑤, 𝑤^𝑐 by 𝑤^𝑐=𝑘−1−𝑤₁⋯𝑘−1−𝑤_𝑛. (18) We can then consider maps 𝜙_𝑎,𝑏∶𝐶_𝑘≀𝑆_𝑛→𝐶_𝑘≀𝑆_𝑛 where 𝜙_𝑎,𝑏((𝜎,𝑤))=(𝜎^𝑎,𝑤^𝑏) for 𝑎,𝑏∈{𝑟,𝑐}. Such maps will easily allow us to establish that the distribution of (𝜏,𝑢)-matches is the same for various classes of (𝜏,𝑢)’s. For example, one can use such maps to show that the distributions of (12,01)-matches, (21,01)-matches, (12,10)-matches, and (21,10)-matches are all the same.

Another interesting case is when we let Υ={(12,01),(12,10)}. In this case, we have a Υ-match in (𝜎,𝑤) starting at 𝑖 if and only if 𝜎_𝑖<𝜎_𝑖+1 and 𝑤_𝑖≠𝑤_𝑖+1. In that case, (12) can be specialized to prove the following.

Theorem 6. Consider the following: _𝑛≥0𝑡^𝑛𝑛!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑥^{Υ-mch((𝜎,𝑤))}=(𝑘−1)(1−𝑥)(𝑘−1)(1−𝑥)+𝑘𝑒^{(𝑘−1)(𝑥−1)𝑡}.−1 (19)

Substituting 𝑧_𝑖=𝑟^𝑖 for 𝑖=0,…,𝑘−1 into (12) will yield the following generating function: 𝐷^Υ_𝑘𝑥,𝑝,𝑞,𝑟,𝑡=_𝑛≥0𝑡^𝑛[𝑛]_𝑝,𝑞!_{(𝜎,𝑤)∈𝐶𝑘≀𝑆𝑛}𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑟^‖𝑤‖𝑥^{Υ-mch((𝜎,𝑤))}. (20) Let Υ_𝐫=),(,Υ{(12,0012,01)}_𝐰=,Υ{(12,00)}_𝐬=,Υ{(12,01)}_𝐝={(12,01),(12,10)}. (21) Thus, Υ_𝐫-matches correspond to rises, Υ_𝐰-matches correspond to weak rises, and Υ_𝐬-matches correspond to strict rises. We shall find 𝐷^Υ𝐚_𝑘(𝑥,𝑝,𝑞,𝑟,𝑡) for 𝐚∈{𝐫,𝐰,𝐬} and find 𝐷^Υ𝐝_𝑘(𝑥,𝑝,𝑞,1,𝑡). For example, we will show that the following generating function for the distribution of inversions, coinversions, and rises over 𝐶_𝑘≀𝑆_𝑛 is an immediate consequence of (12).

Lemma 7. Consider the following: 𝐷^Υ𝐫_𝑘𝑥,𝑝,𝑞,𝑟,𝑡=_𝑛≥0𝑡^𝑛[𝑛]_𝑝,𝑞!_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑟^‖𝑤‖𝑥^{ris((𝜎,𝑤))}=1−𝑥1−𝑥+∑_𝑛≥1𝑝^𝑛2((𝑥−1)𝑡)^𝑛/[𝑛]_𝑝,𝑞!_𝑛^{𝑛+𝑘−1}_𝑟, (22) which reduces to (15) when one sets 𝑝=𝑞=𝑟=1.

We shall prove (12) by applying a ring homomorphism, defined on the ring Λ of symmetric functions over infinitely many variables 𝑥₁,𝑥₂,…, to a simple symmetric function identity. There has been a long line of research, [6–15], which shows that a large number of generating functions for permutation statistics can be obtained by applying homomorphisms defined on the ring of symmetric functions Λ over infinitely many variables 𝑥₁,𝑥₂,… to simple symmetric function identities. For example, the 𝑛th elementary symmetric function, 𝑒_𝑛, and the 𝑛th homogeneous symmetric function, ℎ_𝑛, are defined by the generating functions 𝐸(𝑡)=_𝑛≥0𝑒_𝑛𝑡^𝑛=_𝑖1+𝑥_𝑖𝑡,𝐻(𝑡)=_𝑛≥0ℎ_𝑛𝑡^𝑛=_𝑖11−𝑥_𝑖𝑡. (23) We let 𝑃(𝑡)=∑_𝑛≥0𝑝_𝑛𝑡^𝑛 where 𝑝_𝑛=∑_𝑖𝑥^𝑛_𝑖 is the 𝑛th power symmetric function. A partition of 𝑛 is a sequence 𝜇=(𝜇₁,…,𝜇_𝑘) such that 0<𝜇₁≤⋯≤𝜇_𝑘 and 𝜇₁+⋯+𝜇_𝑘=𝑛. We write 𝜇⊢𝑛 if 𝜇 is a partition of 𝑛, and we let ℓ(𝜇) denote the number of parts of 𝜇. If 𝜇⊢𝑛, we set ℎ_𝜇=∏^ℓ(𝜇)_𝑖=1ℎ_𝜇𝑖, 𝑒_𝜇=∏^ℓ(𝜇)_𝑖=1𝑒_𝜇𝑖, and 𝑝_𝜇=∏^ℓ(𝜇)_𝑖=1𝑝_𝜇𝑖. Let Λ_𝑛 denote the space of homogeneous symmetric functions of degree 𝑛 over infinitely many variables 𝑥₁,𝑥₂,… so that Λ=⨁_𝑛≥0Λ_𝑛. It is well known that {𝑒_𝜆∶𝜆⊢𝑛}, {ℎ_𝜆∶𝜆⊢𝑛}, and {𝑝_𝜆∶𝜆⊢𝑛} are all bases of Λ_𝑛. It follows that {𝑒₀,𝑒₁,…} is an algebraically independent set of generators for Λ and, hence, we can define a ring homomorphism 𝜉∶Λ→𝑅 where 𝑅 is a ring by simply specifying 𝜉(𝑒_𝑛) for all 𝑛≥0.

Now it is well known that 𝐻(𝑡)=1𝐸(),−𝑡𝑃(𝑡)=∑_𝑛≥1(−1)^𝑛−1𝑛𝑒_𝑛𝑡^𝑛𝐸().−𝑡 (24)(25) A surprisingly large number of results on generating functions for various permutation statistics in the literature and large number of new generating functions can be derived by applying homomorphisms on Λ to simple identities such as (24) and (25). We shall show that (12) can be proved by applying appropriate ring homomorphisms to identity (24).

The outline of this paper is as follows. In Section 2, we will provide the necessary background in symmetric functions that we will need to derive our generating functions. In Section 3, we will prove (12) and give a number of cases where we can compute mp_(Υ,^→_𝐴)(𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1) or its specializations which will prove all of the formulas in the special cases described above. We shall also show that if 𝑁𝑀_(Υ,^→_𝐴),𝑛 denotes the number of (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛 such that (𝜎,𝑤) has no (Υ,𝐴^→)-matches, then the sequence (𝑁𝑀_(Υ,^→_𝐴),𝑛)_𝑛≥0 appears in the OIES [16] in several special cases. In Section 4, we will study 𝑁𝑀_(Υ,^→_𝐴),𝑛 as a function of 𝑘, the size of the underlying cyclic group 𝐶_𝑘. We will show that in several cases, 𝑁𝑀_(Υ,^→_𝐴),𝑛 is a polynomial in 𝑘 whose coefficients have interesting combinatorial properties. Finally, in Section 5, we will discuss some related results and directions for future research.

2. Symmetric Functions

In this section, we give the necessary background on symmetric functions needed for our proofs of the generating functions.

Let Λ denote the ring of symmetric functions over infinitely many variables 𝑥₁,𝑥₂,… with coefficients in the field of complex numbers ℂ. The 𝑛th elementary symmetric function 𝑒_𝑛 in the variables 𝑥₁,𝑥₂,… is defined by 𝐸(𝑡)=_𝑛≥0𝑒_𝑛𝑡^𝑛=_𝑖1+𝑥_𝑖𝑡, (26) and the 𝑛th homogeneous symmetric function ℎ_𝑛 in the variables 𝑥₁,𝑥₂,… is defined by 𝐻(𝑡)=_𝑛≥0ℎ_𝑛𝑡^𝑛=_𝑖11−𝑥_𝑖𝑡. (27) Thus, 𝐻(𝑡)=1𝐸().−𝑡 (28) Let 𝜆=(𝜆₁,…,𝜆_ℓ) be an integer partition, that is, 𝜆 is a finite sequence of weakly increasing positive integers. Let ℓ(𝜆)=𝑙 denote the number of parts of 𝜆. If the sum of these integers is 𝑛, we say that 𝜆 is a partition of 𝑛 and write 𝜆⊢𝑛. For any partition 𝜆=(𝜆₁,…,𝜆_ℓ), let 𝑒_𝜆=𝑒_𝜆1⋯𝑒_𝜆ℓ. The well known fundamental theorem of symmetric functions says that {𝑒_𝜆∶𝜆isapartition} is a basis for Λ or that {𝑒₀,𝑒₁,…} is an algebraically independent set of generators for Λ. Similarly, if we define ℎ_𝜆=ℎ_𝜆1⋯ℎ_𝜆ℓ, then {ℎ_𝜆∶𝜆isapartition} is also a basis for Λ. Since {𝑒₀,𝑒₁,…} is an algebraically independent set of generators for Λ, we can specify a ring homomorphism 𝜃 on Λ by simply defining 𝜃(𝑒_𝑛) for all 𝑛≥0.

Since the elementary symmetric functions 𝑒_𝜆 and the homogeneous symmetric functions ℎ_𝜆 are both bases for Λ, it makes sense to talk about the coefficient of the homogeneous symmetric functions when written in terms of the elementary symmetric function basis. These coefficients have been shown to equal the sizes of certain sets of combinatorial objects up to a sign. A brick tabloid of shape (𝑛), and type 𝜆=(𝜆₁,…,𝜆_𝑘) is a filling of a row of 𝑛 squares of cells with bricks of lengths 𝜆₁,…,𝜆_𝑘 such that bricks do not overlap. One brick tabloid of shape (12) and type (1,1,2,3,5) is displayed in Figure 1.

Figure 1 

A brick tabloid of shape (12) and type (1,1,2,3,5).

Let ℬ_𝜆,𝑛 denote the set of all 𝜆-brick tabloids of shape (𝑛) and let 𝐵_𝜆,𝑛=|ℬ_𝜆,𝑛|. Through simple recursions stemming from (28), Eğecioğlu and Remmel proved in [17] that ℎ_𝑛=_𝜆⊢𝑛(−1)^{𝑛−ℓ(𝜆)}𝐵_𝜆,𝑛𝑒_𝜆. (29)

We end this section with two lemmas that will be needed in later sections. Both of the lemmas follow from simple codings of a basic result of Carlitz [18] that 𝑛𝑘_𝑞=_{𝑟∈ℛ(1𝑘0𝑛−𝑘)}𝑞^inv(𝑟), (30) where ℛ(1^𝑘0^𝑛−𝑘) is the number of rearrangements of 𝑘 1's and 𝑛−𝑘 0's. We start with a lemma from [19]. Fix a brick tabloid 𝑇=(𝑏₁,…,𝑏_ℓ(𝜇))∈ℬ_𝜇,𝑛. Let IF(𝑇) denote the set of all fillings of the cells of 𝑇=(𝑏₁,…,𝑏_ℓ(𝜇)) with the numbers 1,…,𝑛 so that the numbers increase within each brick reading from left to right. We then think of each such filling as a permutation of 𝑆_𝑛 by reading the numbers from left to right in each row. For example, Figure 2 pictures an element of IF(3,6,3) whose corresponding permutation is 461215781011239.

Figure 2 

An element of IF(3,6,3).

Then the following lemma which is proved in [19] gives a combinatorial interpretation to 𝑝^{∑ℓ(𝜇)𝑖=1𝑏𝑖2𝑛}_{𝑏1,…,𝑏ℓ(𝜇)𝑝,𝑞}.

Lemma 8. If 𝑇=(𝑏₁,…,𝑏_ℓ(𝜇)) is a brick tabloid in ℬ_𝜇,𝑛, then 𝑝^{∑ℓ(𝜇)𝑖=1𝑏𝑖2}𝑛𝑏₁,…,𝑏_{ℓ(𝜇)𝑝,𝑞}=_{𝜎∈IF(𝑇)}𝑞^inv(𝜎)𝑝^coinv(𝜎). (31)

Another well known combinatorial interpretation for 𝑛+𝑘−1𝑘−1𝑞 ([13]) is that it is equal to the sum of the sizes of the partitions that are contained in an 𝑛×(𝑘−1) rectangle. Thus one has the following lemma.

Lemma 9. Consider the following: _{0≤𝑎1≤⋯≤𝑎𝑛≤𝑘−1}𝑞^{𝑎1+⋯+𝑎𝑛}=𝑛𝑛+𝑘−1_𝑞. (32)

3. Generating Functions

The main goal of this section is to prove (12). That is, we will prove the following theorem.

Theorem 10. Let Υ⊆𝐶_𝑘≀𝑆₂ and 𝐴^→=(𝐴₁,𝐴₂) where 𝐴₁,𝐴₂⊆[𝑘]. For all 𝑘≥2, _𝑛≥0𝑡^𝑛[𝑛]_𝑝,𝑞!_{(𝜎,𝑤)∈𝐶𝑘≀𝑆𝑛}𝑝^coinv(𝜎)𝑞^inv(𝜎)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))}𝑧(𝑤)=1×1−𝑧₀+⋯+𝑧_𝑘−1𝑡+_𝑛≥2𝑡^𝑛[𝑛]_𝑝,𝑞!(𝑥−1)^𝑛−1mp_(Υ,^→_𝐴),𝑛×𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1⁻¹. (33)

Proof. Define a ring homomorphism Γ∶Λ→ℚ(𝑝,𝑞,𝑧₀,…,𝑧_𝑘−1,𝑥) by setting Γ𝑒₀Γ𝑒=1,₁=𝑧₀+⋯+𝑧_𝑘−1,Γ𝑒_𝑛=(−1)^𝑛−1(𝑥−1)^𝑛−11[𝑛]_𝑝,𝑞!mp_(Υ,^→_𝐴),𝑛×𝑝,𝑞;𝑧₀,…𝑧_𝑘−1for𝑛≥1. (34) Then we claim that [𝑛]_𝑝,𝑞!Γℎ_𝑛=_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑧(𝑤)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))}, (35) for all 𝑛≥1. That is, [𝑛]_𝑝,𝑞!Γℎ_𝑛=[𝑛]_𝑝,𝑞!_𝜇⊢𝑛(−1)^{𝑛−ℓ(𝜇)}𝐵_𝜇,(𝑛)Γ𝑒_𝜇=[𝑛]_𝑝,𝑞!_𝜇⊢𝑛(−1)^{𝑛−ℓ(𝜇)}×_{(𝑏1,…,𝑏ℓ(𝜇))∈ℬ𝜇,𝑛}^ℓ(𝜇)_𝑗=1(−1)^𝑏𝑗−1(𝑥−1)^𝑏𝑗−1×𝑧₀+⋯+𝑧_𝑘−1𝜒𝑏_𝑗+1=1𝑏_{𝑗𝑝,𝑞}!mp_(Υ,^→_{𝐴),𝑏𝑗}𝑝,𝑞;𝑧₀,𝑧₁,…,𝑧_𝑘−1×𝜒𝑏_𝑗>1=_{𝜇⊢𝑛𝑏1,…,𝑏ℓ(𝜇)∈ℬ𝜇,𝑛}𝑝^{∑ℓ(𝜇)𝑗=1𝑏𝑗2}𝑛𝑏₁,…,𝑏_{ℓ(𝜇)𝑝,𝑞}×^ℓ(𝜇)_𝑗=1𝑧₀+⋯+𝑧_𝑘−1𝜒𝑏_𝑗+𝑝=1_−𝑏𝑗2𝑏_{𝑗𝑝,𝑞}!mp_(Υ,^→_{𝐴),𝑏𝑗}𝑝,𝑞;𝑧₀,𝑧₁,…,𝑧_𝑘−1×𝜒𝑏_𝑗>1. (36)
Next we want to give a combinatorial interpretation to (36). By Lemma 8, for each brick tabloid 𝑇=(𝑏₁,…,𝑏_ℓ(𝜇)), we can interpret 𝑝^{∑ℓ(𝜇)𝑗=1𝑏𝑗2𝑛}_{𝑏1,…,𝑏ℓ(𝜇)𝑝,𝑞} as the sum of the weights of all fillings of 𝑇 with a permutation 𝛼∈𝑆_𝑛 such that 𝛼 is increasing in each brick, and we weight 𝛼 with 𝑞^inv(𝛼)𝑝^coinv(𝛼). If 𝑏_𝑗=1 and covers cell 𝑟, then we will interpret the factor 𝑧₀+⋯+𝑧_𝑘−1 as picking an element 𝑤_𝑟∈[𝑘] to put on top of 𝛼_𝑟. Next suppose that 𝑏_𝑗>1 and covers cells 𝑟,𝑟+1,…,𝑟+𝑠. Let 𝛼_𝑟<𝛼_𝑟+1<⋯<𝛼_𝑟+𝑠 be the elements of 𝛼 in cells 𝑟,𝑟+1,…,𝑟+𝑠, respectively. Then we interpret 𝑝^−𝑛2mp_(Υ,^→_{𝐴),𝑏𝑗}(𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1) as the numbers of ways of picking a maximum packing (𝛽^(𝑗),𝑣^(𝑗))∈𝐶_𝑘≀𝑆_𝑏𝑗 for (Υ,𝐴^→) where we weight (𝛽^(𝑗),𝑣^(𝑗)) by 𝑝^−𝑏𝑗2𝑝^{coinv(𝛽(𝑗))}𝑞^{inv(𝛽(𝑗))}𝑧(𝑣^(𝑗)). We then reorder by the proper contribution after reordering which is coinv(𝛽^(𝑗)) to the coinversion count of the resulting permutation and inv(𝛽^(𝑗)) to the inversion count of the resulting permutation. Finally, we interpret ∏^ℓ(𝜇)_𝑗=1(𝑥−1)^𝑏𝑗−1 as all ways of picking a label of the cells of each brick except the final cell with either an 𝑥 or a −1. For completeness, we label the final cell of each brick with 1. For example, suppose that Υ={(21,00),(21,01)}, 𝑘=3, and 𝐴^→=([3],[3]). Thus (𝜎,𝑤) is a maximum packing for (Υ,𝐴^→) only if 𝜎 is strictly decreasing and 𝑤 is a weakly increasing word over the alphabet {0,1,2}. Then at the top of Figure 3, we have pictured the brick tabloid 𝑇=(3,1,5,2) along with a permutation 𝛼 which is increasing within bricks. Below that, we have picked our choices of (𝛽^(𝑗),𝑣^(𝑗)) for the bricks 𝑏_𝑗 for 𝑗=1,3,4 and choice of 1 for brick 𝑏₂ which is of length 1. These choices result in the filled brick tabloid pictured at the bottom of Figure 3. We have also specified a labeling of the cells with either 𝑥, −1, or 1 so that the last cell of each brick is labeled with 1 and the remaining cells are labeled with either 𝑥 of −1. We shall call all such objects created in this way filled labeled brick tabloids, and let ℱ_𝑛 denote the set of all filled labeled brick tabloids that arise in this way. Thus, a 𝐶∈ℱ_𝑛 consists of a brick tabloid 𝑇=(𝑏₁,…,𝑏_𝑡), an element (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛, and a labeling 𝐿 of the cells of 𝑇 with elements from {𝑥,1,−1} such that (1)if 𝑏_𝑗=1 and covers cell 𝑟, then 𝑤_𝑟 is an arbitrary element of [𝑘], (2)if 𝑏_𝑗>1 and covers cells 𝑟,𝑟+1,…,𝑟+𝑠, then (red(𝜎_𝑟𝜎_𝑟+1⋯𝜎_𝑟+𝑠),𝑤_𝑟𝑤_𝑟+1⋯𝑤_𝑟+𝑠) is an element of ℳ𝒫_(Υ,^→_{𝐴),𝑏𝑗}, (3)the final cell of each brick is labeled with 1,(4)each cell which is not a final cell of a brick is labeled with 𝑥 or −1.
We then define the weight 𝑤(𝐶) of 𝐶 to be 𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑧(𝑤) times the product of all the 𝑥 labels in 𝐿 and the sign sgn(𝐶) of 𝐶 to be the product of all the −1 labels in 𝐿. For example, if 𝐶=(𝑇,(𝜎,𝑤),𝐿) is the filled labeled brick tabloid pictured at the bottom of Figure 3, then 𝑤(𝐶)=𝑞²⁸𝑝²⁷𝑧⁵₀𝑧⁴₁𝑧²₂𝑥⁴ and sgn(𝐶)=−1. It follows that [𝑛]_𝑝,𝑞!Γℎ_𝑛=_{𝐶∈ℱ𝑛}sgn(𝐶)𝑤(𝐶). (37)
Next we define a weight-preserving sign-reversing involution 𝐼∶ℱ_𝑛→ℱ_𝑛. To define 𝐼(𝐶), we scan the cells of 𝐶=(𝑇,(𝜎,𝑤),𝐿) from left to right looking for the leftmost cell 𝑡 such that either (i) 𝑡 is labeled with −1 or (ii) 𝑡 is at the end of a brick 𝑏_𝑗, and the brick 𝑏_𝑗+1 immediately following 𝑏_𝑗 has the property that if 𝑏_𝑗 together with 𝑏_𝑗+1 cover cells 𝑟,𝑟+1,…,𝑟+𝑠, then (red(𝜎_𝑟𝜎_𝑟+1⋯𝜎_𝑟+𝑠),𝑤_𝑟𝑤_𝑟+1⋯,𝑤_𝑟+𝑠) is a maximum packing for (Υ,𝐴^→). In case (i), 𝐼(𝐶)=(𝑇^′,(𝜎^′,𝑤^′),𝐿^′) where 𝑇^′ is the result of replacing the brick 𝑏 in 𝑇 containing 𝑡 by two bricks 𝑏^∗ and 𝑏^∗∗ where 𝑏^∗ contains the cell 𝑡 plus all the cells in 𝑏 to the left of 𝑡 and 𝑏^∗∗ contains all the cells of 𝑏 to the right of 𝑡, 𝜎=𝜎^′, 𝑤=𝑤^′, and 𝐿^′ is the labeling that results from 𝐿 by changing the label of cell 𝑡 from −1 to 1. In case (ii), 𝐼(𝐶)=(𝑇^′,(𝜎^′,𝑤^′),𝐿^′) where 𝑇^′ is the result of replacing the bricks 𝑏_𝑗 and 𝑏_𝑗+1 in 𝑇 by a single brick 𝑏, 𝜎=𝜎^′, 𝑤=𝑤^′, and 𝐿^′ is the labeling that results from 𝐿 by changing the label of cell 𝑡 from 1 to −1. If neither case (i) or case (ii) applies, then we let 𝐼(𝐶)=𝐶. For example, if 𝐶 is the element of ℱ₁₂ pictured in Figure 3, then 𝐼(𝐶) is pictured in Figure 4.
It is easy to see that 𝐼 is a weight-preserving sign-reversing involution and hence 𝐼 shows that [𝑛]_𝑝,𝑞!Γℎ_𝑛=_{𝐶∈ℱ𝑛(𝐶),𝐼=𝐶}sgn(𝐶)𝑤(𝐶). (38)
Thus, we must examine the fixed points 𝐶=(𝑇,𝜎,𝑤,𝐿) of 𝐼. First there can be no −1 labels in 𝐿 so that sgn(𝐶)=1. Moreover, if 𝑏_𝑗 and 𝑏_𝑗+1 are two consecutive bricks in 𝑇 and 𝑡 is the last cell of 𝑏_𝑗, then it cannot be the case that there is (Υ,𝐴^→)-match starting at position 𝑡 in (𝜎,𝑤) since otherwise we could combine 𝑏_𝑗 and 𝑏_𝑗+1. For any such fixed point, we associate an element (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛. For example, a fixed point of 𝐼 is pictured in Figure 5 where 𝜎=5431111097682 and 𝑤=00110011202.
It follows that if cell 𝑡 is at the end of a brick, then 𝑡 is not the start of a (Υ,𝐴^→)-match. However, if 𝑣 is a cell which is not at the end of a brick, then our definitions force 𝑣 to be the start of a (Υ,𝐴^→)-match. Since each such cell 𝑣 must be labeled with an 𝑥, it follows that sgn(𝐶)𝑤(𝐶)=𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑧(𝑤)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))}. Vice versa, if (𝜎,𝑤)∈𝐶_𝑘≀𝑆_𝑛, then we can create a fixed point 𝐶=(𝑇,(𝜎,𝑤),𝐿) by having the bricks in 𝑇 end at cells of the form 𝑡 where 𝑡 is not the start of a (Υ,𝐴^→)-match, labeling each cell 𝑡 which is the start of a (Υ,𝐴^→)-match with 𝑥, and labeling the remaining cells with 1. Thus, we have shown that [𝑛]_𝑝,𝑞!Γℎ_𝑛=_{()𝜎,𝑤∈𝐶𝑘≀𝑆𝑛}𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑧(𝑤)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))} (39) as desired.
Applying Γ to the identity 𝐻(𝑡)=(𝐸(−𝑡))⁻¹, we get _𝑛≥0Γℎ_𝑛𝑡^𝑛=_𝑛≥0𝑡^𝑛[𝑛]_𝑝,𝑞!×_{(𝜎,𝑤)∈𝐶𝑘≀𝑆𝑛}𝑞^inv(𝜎)𝑝^coinv(𝜎)𝑧(𝑤)𝑥^{(Υ,→𝐴)-mch((𝜎,𝑤))}=11+∑_𝑛≥1(−𝑡)^𝑛Γ𝑒_𝑛=1×1+𝑧₀+⋯+𝑧_𝑘−1(−𝑡)+_𝑛≥1(−1)^𝑛𝑡^𝑛×(−1)^𝑛−1(𝑥−1)^𝑛−1[𝑛]_𝑝,𝑞!×mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1⁻¹=1×1−𝑧₀+⋯+𝑧_𝑘−1𝑡+_𝑛≥2(𝑥−1)^𝑛−1𝑡^𝑛/[𝑛]_𝑝,𝑞!×mp_(Υ,^→_𝐴),𝑛𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1⁻¹, (40) which proves (33).

Figure 3 

A composite object 𝐶∈ℱ12.

Figure 4 

𝐼(𝐶) for 𝐶 in Figure 3.

Figure 5 

A fixed point of 𝐼.

To be able to use Theorem 10, we need to be able to compute mp_(Υ,^→_𝐴),𝑛(𝑝,𝑞;𝑧₀,…,𝑧_𝑘−1). In fact, this is easy to do in most cases. Thus, we end this section by giving several such examples. We will examine the cases when (i)Υ=Υ_𝐬={(12,01)} and 𝐴^→=([𝑘],[𝑘]), (ii)Υ=Υ_𝐫={(12,00),(12,01)} and 𝐴^→=([𝑘],[𝑘]), (iii)Υ=Υ_𝐰={(12,00)} and 𝐴^→=([𝑘],[𝑘]), (iv)Υ=Υ_𝐝={(12,01),(12,10)} and 𝐴^→=([𝑘],[𝑘]), (v)Υ={(12,𝑎𝑏)} and 𝐴^→=({𝑎},{𝑏}), (vi)Υ