Research Article | Open Access
Generalized Pattern-Matching Conditions for
We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of 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 . Our research leads to connections to many known objects/structures yet to be explained combinatorially.
The goal of this paper is to study pattern-matching conditions on the wreath product of the cyclic group and the symmetric group . is the group of signed permutations where we allow signs of the form for some primitive th root of unity . We can think of the elements as pairs where and . For ease of notation, if where for , then we simply write where .
Given a sequence of distinct integers, let red be the permutation found by replacing the th largest integer that appears in by . For example, if , then . Given a permutation in the symmetric group , we say a permutation has a -match starting at position provided . Let -mch be the number of -matches in the permutation . Similarly, we say that occurs in if there exist such that red . We say that avoids if there are no occurrences of in .
We can define similar notions for words over a finite alphabet . Given a word , let red be the word found by replacing the largest integer that appears in by . For example, if , then red. Given a word such that red, we say a word has a -match starting at position provided . Let -mch be the number of -matches in the word . Similarly, we say that occurs in a word if there exist such that red. We say that avoids if there are no occurrences of in .
Definition 1. Let , be a subset of and . (1)One says that has an exact occurrence of (resp., ) if there are such that (resp., ).(2)One says that avoids an exact occurrence of (resp., ) if there are no exact occurrences of (resp., ) in .(3)One says that there is an exact -match in starting at position (resp., exact -match in starting at position ) if
That is, an exact occurrence or an exact match of in an element is just an ordinary occurrence or match of in where the corresponding signs agree exactly. For example, Mansour  proved via recursion that for any , the number of elements in which avoid exact occurrences of is . This generalized a result of Simion  who proved the same result for the hyperoctahedral group . Similarly, Mansour and West  determined the number of permutations in that avoid all possible exact occurrences of 2 or 3 element sets of patterns of elements of . For example, let be the number of that avoid all exact occurrences of the patterns in the set , let be the number of that avoid all exact occurrences of the patterns in the set , and let be the number of that avoid all exact occurrences of the patterns in the set . Then Mansour and West  proved that where is the th 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 where red, be a subset of where for all , red, and . (1)One says that has an occurrence of (resp., ) if there are such that (resp., ).(2)We say that avoids (resp., ) if there are no occurrences of (resp., ) in .(3)One says that there is a -match in starting at position (resp., -matchin starting at position ) if (resp., , ).
For example, suppose that and . Then there are no exact occurrences or exact matches of in . However, there are two occurrences of , one in positions 2 and 4 and one in positions 3 and 4. Thus, there are two occurrences of in , and there is a -match in starting at position .
Finally, we will consider a more general matching condition which generalizes both occurrences and matches and exact occurrences and exact matches in .
Definition 3. Let , let be a subset of , and let be a sequence of subsets of , and . (1)One says that has an occurrence of (resp., ) if there are such that (resp., is equal to for some in ) and for .(2)One says that avoids (resp., ) if there are no occurrences of (resp., ) in .(3)We say that there is a -match in starting at position (resp., -match in starting at position ) if , , is equal to for some in ) and for .
Thus, a -occurrence or -match where and is such that for is just an exact occurrence or exact match of . Similarly, a -occurrence or -match where , , and is such that for is just an occurrence or match of .
Suppose we are given , , where for , and . We let -mch (resp., -Emch) denote the number of -matches (resp., exact -matches) in . We let -mch (resp., -Emch) denote the number of -matches (resp., exact -matches) in . We let -mch be the number of -matches in and let -mch be the number of -matches in .
The main result of this paper is to derive a generating function for the distribution of -matches where is any element of . To state our main result, we first need some notation. We define the -analogues of , , , and by
respectively. We define the -analogues of , , , and by , , , and , respectively.
Next suppose that and where . Then we will say that is a maximum packing for if has -matches starting at positions . We let denote the set of which are maximum packings for . Given any word , we let . For any , we let (resp., ) equal the number of pairs such that and (resp., ). We then define We shall also be interested in the specializations where for any word , . In the special case, where where , we will denote as . For example, if , , and , then is a maximum packing for if and only if is the identity permutation and is such that . Thus, where for any power series , we write for the coefficient of in . Then it is easy to prove that If , , and , then is a maximum packing for if and only if is the identity permutation and for some . Thus, It then follows that Similarly if and , then is a maximum packing for if and only if is the identity permutation and is such that . Thus, We shall then prove that
The main result of this paper is to prove the following.
Theorem 4. For any and where ,
We note that (12) can be specialized to give natural analogues for generating functions of rises and descents in where we compare pairsusing the product order. That is, instead of thinking of an element of as a pair , we can think of it as a sequence of pairs . We then define a partial order on such pairs by the usual product order. That is, if and only if and . Then we define the following sets and statistics for elements : We shall refer to as the descent set of , as the weak descent set of , and as the strict descent set of . Similarly, we will refer to as the rise set of , as the weak rise set of , and as the strict rise set of . It is easy to see that if and only if there is a -match starting at position , if and only if there is a -match starting at position , and if and only if there is a -match starting at position where .
Similarly, if and only if there is a -match starting at position , if and only if there is a -match starting at position , and if and only if there is a -match starting at position where .
If , then we define the reverse of , by . Similarly, if , then we define . It is easy to see that Thus, we need to find the distributions for only one of the corresponding pairs. Substituting into (12) in these special cases will yield the following generating functions for rises, strict rises, and weak rises in .
Theorem 5. Consider the following:
Other distribution results for -matches follow from these results. For example, if , then we define the complement of , by If , then we define the complement of , by We can then consider maps where for . Such maps will easily allow us to establish that the distribution of -matches is the same for various classes of ’s. For example, one can use such maps to show that the distributions of -matches, -matches, -matches, and -matches are all the same.
Another interesting case is when we let . In this case, we have a -match in starting at if and only if and . In that case, (12) can be specialized to prove the following.
Theorem 6. Consider the following:
Substituting for into (12) will yield the following generating function: Let Thus, -matches correspond to rises, -matches correspond to weak rises, and -matches correspond to strict rises. We shall find for and find . For example, we will show that the following generating function for the distribution of inversions, coinversions, and rises over is an immediate consequence of (12).
Lemma 7. Consider the following: which reduces to (15) when one sets .
We shall prove (12) by applying a ring homomorphism, defined on the ring of symmetric functions over infinitely many variables , 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 over infinitely many variables to simple symmetric function identities. For example, the th elementary symmetric function, , and the th homogeneous symmetric function, , are defined by the generating functions We let where is the th power symmetric function. A partition of is a sequence such that and . We write if is a partition of , and we let denote the number of parts of . If , we set , , and . Let denote the space of homogeneous symmetric functions of degree over infinitely many variables so that . It is well known that , , and are all bases of . It follows that is an algebraically independent set of generators for and, hence, we can define a ring homomorphism where is a ring by simply specifying for all .
Now it is well known that 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 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 or its specializations which will prove all of the formulas in the special cases described above. We shall also show that if denotes the number of such that has no -matches, then the sequence appears in the OIES  in several special cases. In Section 4, we will study as a function of , the size of the underlying cyclic group . We will show that in several cases, is a polynomial in 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 denote the ring of symmetric functions over infinitely many variables with coefficients in the field of complex numbers . The th elementary symmetric function in the variables is defined by and the th homogeneous symmetric function in the variables is defined by Thus, Let be an integer partition, that is, is a finite sequence of weakly increasing positive integers. Let denote the number of parts of . If the sum of these integers is , we say that is a partition of and write . For any partition , let . The well known fundamental theorem of symmetric functions says that is a basis for or that is an algebraically independent set of generators for . Similarly, if we define , then is also a basis for . Since is an algebraically independent set of generators for , we can specify a ring homomorphism on by simply defining for all .
Since the elementary symmetric functions and the homogeneous symmetric functions are both bases for , 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 , and type is a filling of a row of squares of cells with bricks of lengths such that bricks do not overlap. One brick tabloid of shape and type is displayed in Figure 1.
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  that where is the number of rearrangements of 1's and 0's. We start with a lemma from . Fix a brick tabloid . Let denote the set of all fillings of the cells of with the numbers so that the numbers increase within each brick reading from left to right. We then think of each such filling as a permutation of by reading the numbers from left to right in each row. For example, Figure 2 pictures an element of whose corresponding permutation is .
Then the following lemma which is proved in  gives a combinatorial interpretation to .
Lemma 8. If is a brick tabloid in , then
Another well known combinatorial interpretation for () is that it is equal to the sum of the sizes of the partitions that are contained in an rectangle. Thus one has the following lemma.
Lemma 9. Consider the following:
3. Generating Functions
The main goal of this section is to prove (12). That is, we will prove the following theorem.
Theorem 10. Let and where . For all ,
Proof. Define a ring homomorphism by setting
Then we claim that
for all . That is,
Next we want to give a combinatorial interpretation to (36). By Lemma 8, for each brick tabloid , we can interpret as the sum of the weights of all fillings of with a permutation such that is increasing in each brick, and we weight with . If and covers cell , then we will interpret the factor as picking an element to put on top of . Next suppose that and covers cells . Let be the elements of in cells , respectively. Then we interpret as the numbers of ways of picking a maximum packing for where we weight by . We then reorder by the proper contribution after reordering which is to the coinversion count of the resulting permutation and to the inversion count of the resulting permutation. Finally, we interpret as all ways of picking a label of the cells of each brick except the final cell with either an or a . For completeness, we label the final cell of each brick with . For example, suppose that , , and . Thus is a maximum packing for only if is strictly decreasing and is a weakly increasing word over the alphabet . Then at the top of Figure 3, we have pictured the brick tabloid along with a permutation which is increasing within bricks. Below that, we have picked our choices of for the bricks for and choice of 1 for brick 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 , , or so that the last cell of each brick is labeled with 1 and the remaining cells are labeled with either of . We shall call all such objects created in this way filled labeled brick tabloids, and let denote the set of all filled labeled brick tabloids that arise in this way. Thus, a consists of a brick tabloid , an element , and a labeling of the cells of with elements from such that (1)if and covers cell , then is an arbitrary element of , (2)if and covers cells , then is an element of , (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 or .
We then define the weight of to be times the product of all the labels in and the sign of to be the product of all the labels in . For example, if is the filled labeled brick tabloid pictured at the bottom of Figure 3, then and . It follows that
Next we define a weight-preserving sign-reversing involution . To define , we scan the cells of from left to right looking for the leftmost cell such that either (i) is labeled with or (ii) is at the end of a brick , and the brick immediately following has the property that if together with cover cells , then is a maximum packing for . In case (i), where is the result of replacing the brick in containing by two bricks and where contains the cell plus all the cells in to the left of and contains all the cells of to the right of , , , and is the labeling that results from by changing the label of cell from to . In case (ii), where is the result of replacing the bricks and in by a single brick , , , and is the labeling that results from by changing the label of cell from to . If neither case (i) or case (ii) applies, then we let . For example, if is the element of pictured in Figure 3, then is pictured in Figure 4.
It is easy to see that is a weight-preserving sign-reversing involution and hence shows that
Thus, we must examine the fixed points of . First there can be no labels in so that . Moreover, if and are two consecutive bricks in and is the last cell of , then it cannot be the case that there is -match starting at position in since otherwise we could combine and . For any such fixed point, we associate an element . For example, a fixed point of is pictured in Figure 5 where and .
It follows that if cell is at the end of a brick, then is not the start of a -match. However, if is a cell which is not at the end of a brick, then our definitions force to be the start of a -match. Since each such cell must be labeled with an , it follows that . Vice versa, if , then we can create a fixed point by having the bricks in end at cells of the form where is not the start of a -match, labeling each cell which is the start of a -match with , and labeling the remaining cells with . Thus, we have shown that as desired.
Applying to the identity , we get