A lecture hall theorem for -falling partitions
For an integer , a partition is called -falling, a notion introduced by Keith, if the least nonnegative residues mod of ’s form a nonincreasing sequence. We extend a bijection originally due to the third author to deduce a lecture hall theorem for such -falling partitions. A special case of this result gives rise to a finite version of Pak-Postnikov’s -generalization of Euler’s theorem. Our work is partially motivated by a recent extension of Euler’s theorem for all moduli, due to Keith and Xiong. We note that their result actually can be refined with one more parameter.
Key words and phrases:Partitions; Stockhofe-Keith map; lecture hall partitions; -falling partitions.
2010 Mathematics Subject Classification:05A17, 11P83
A partition of a positive integer is a nonincreasing sequence of positive integers such that . The ’s are called the parts of , and is called the weight of , usually denoted as . For convenience, we often allow parts of size zero and append as many zeros as needed.
Being widely perceived as the genesis of the theory of partition, Euler’s theorem asserts that the set of partitions of into odd parts and the set of partitions of into distinct parts are equinumerous. Equivalently,
Among numerous generalizations and refinements of Euler’s theorem [1, 2, 5, 6, 7, 8, 9, 10, 12, 14, 17, 16, 15, 18, 20, 22, 24], the one that arguably attracted the most attention is the following finite version named the Lecture Hall Theorem, first discovered by Bousquet-Mélou and Eriksson.
If is a partition of length with some parts possibly zero such that
then is called a lecture hall partition of length . Let be the set of lecture hall partitions of length .
Theorem 1.1 (Theorem 1.1, ).
It can be easily checked that any partition into distinct parts less than or equal to satisfies the inequality condition in (1.1). That is, for any , which shows that (1.2) indeed yields Euler’s theorem when .
In 1883, Glaisher  found a purely bijective proof of Euler’s theorem and was able to extend it to the equinumerous relationship between partitions with parts repeated less than times and partitions into non-multiples of for any . That is,
Recently, Xiong and Keith  obtained a substantial refinement of Glaisher’s result with respect to certain partition statistics, which we define next.
Throughout this paper, we will assume that . For any partition , let
We define its -alternating sum type to be the -tuple and its -alternating sum . We note that the -alternating sum type of does not put any restriction on .
We define its -length type to be the -tuple and its -length . Note that the -length type of is independent of the parts in that are multiples of .
Let us define the following two subsets of partitions:
: the set of partitions in which each non-zero part can be repeated at most times;
: the set of partitions in which each non-zero part is not divisible by , called -regular partitions.
Theorem 1.2 (Theorem 2.1, ).
The natural desire to find certain “lecture hall version” for the result of Xiong and Keith motivated us to take on this investigation. While the version with full generality matching their result is yet to be found, we do obtain a lecture hall theorem for -falling partitions.
A partition is called -falling, which was introduced by Keith in , if the least nonnegative residues mod of ’s form a nonincreasing sequence. We denote the set of -falling and -regular partitions (-falling regular partitions for short) as . For , let
and be a subset of with certain ratio conditions between parts. Due to the complexity of the conditions, the definition of is postponed to section 3. A partition in is called an -falling lecture hall partition.
We now state the main result of this paper.
Theorem 1.3 (-falling lecture hall theorem).
For and ,
Another result of this paper is a refinement of Theorem 1.2. Let us consider the residue sequence of a partition. Namely, for , we take for each part the least non-negative residue modulo and denote the resulting sequence as . Recall the permutation statistic ascent:
for any word , which is consisted of totally ordered letters. We extend this statistic to partitions via their residue sequences and let .
We have the following refinement of Theorem 1.2.
To make this paper self-contained, in the next section we first recall the Stockhofe-Keith map and then prove Theorem 1.4. In section 3, we define -falling lecture hall partitions and prove Theorem 1.3, one special case of which gives rise to a lecture hall theorem (see Theorem 3.1) for Pak-Postnikov’s -generalization  of Euler’s theorem. We conclude in the final section with some outlook for future work.
2. Preliminaries and a proof of Theorem 1.4
In this section, we first recall further definitions and notions involving partitions for later use. After that, we will recap the Stockhofe-Keith map and prove Theorem 1.4.
Given two (infinite) sequences and , we define the usual linear combination as
for any two nonnegative integers and .
For a partition , its conjugate partition is a partition resulting from choosing as the number of parts of that are not less than [3, Definition 1.8].
The following lemma (see for instance [23, Lemma 1]) follows via the conjugation of partitions.
The conjugation map is a weight-preserving bijection such that
1) This immediately follows via the conjugation of partitions, so we omit the details.
2) Again, by conjugation, we see that . Also, by the definition,
Thus . ∎
Using conjugation, we can derive an interesting set of partitions that are equinumerous to , namely -flat partitions:
: the set of partitions in which the differences between consecutive parts are at most , called -flat partitions.
The two sets and are clearly in one-to-one correspondence via conjugation.
2.1. Stockhofe-Keith map
Given any partition , we define its base -flat partition, denoted as , as follows. Whenever there are two consecutive parts and with , we subtract from each of the parts . We repeat this until we reach a partition in , which is taken to be .
Suppose we are given a partition . We now describe step-by-step how to get a partition via the aforementioned Stockhofe-Keith map .
Insert each part in , from the largest one to the smallest one, into according to the following insertion method. Note that after each insertion, we always arrive at a new -flat partition. In particular, the final partition we get, say , is in as well.
Conjugate to get .
For example, let us take and . We use -modular Ferrers graphs  to illustrate the process of deriving . See Figure 1 below. For the readers’ convenience, we have colored the inserted cells red for step 2.
Theorem 2.3 (Theorem 3.1, ).
-regular partitions of any given -length type are in bijection with -flat partitions of the same -length type.
2.2. Proof of Theorem 1.4
In view of the proof of Theorem 1.2 in , all it remains is to examine the map with respect to the extra parameter . Suppose , and the largest part of is , then we have , according to the definition of -flat partitions. Next, during step 2, we insert columns of into , and each insertion will give rise to a new part in that is divisible by , therefore we see . The above discussion gives us
as desired. ∎
For , let . Then, the extra parameter tracked by gives us
which has previously been derived by Keith [13, Theorem 6] as well (he used instead of our ). Moreover, this refinement is reminiscent of Sylvester’s bijection for proving Euler’s theorem, in which case and we always have , see for example Theorem 1 (item 4) in .
3. A lecture hall theorem for -falling partitions
We will first handle the case with a single residue class. Let us fix . For , let
where is the number of nonzero parts in , and we make the convention that for a fraction , forces .
For any ,
The above result can be viewed as a finite (or “lecture hall”) version of the following Pak-Postnikov’s -generalization [16, Theorem 1] of Euler’s partition theorem since and .
For and , the number of partitions of into parts congruent to modulo , equals the number of partitions of with exactly parts of maximal size, (if any) second by size parts, (if any) third by size parts, etc.
Proof of Theorem 3.1.
Since a partition has all its parts congruent to modulo , with the largest part , we see that and we have
It remains to prove the first equality. We achieve this by constructing a bijection from to that is weight-preserving and sends to . This bijection is based on the third author’s original idea from , which was later generalized to deal with the -sequence version in . We modify it to suit the current settings.
Define as follows. For , let be the sequence obtained from the empty sequence by recursively inserting the parts of in nonincreasing order according to the following insertion procedure. We define .
The effect of this insertion is that we use up a complete part , so the weight of the sequence and its -alternating sum are increased by and , respectively. Also, it can be checked easily that the returned sequence satisfies the condition for . We omit the details.
The map is indeed invertible since the parts of were inserted in nonincreasing order, i.e., from the largest to smallest. If the parts are not inserted in this order, is not necessarily invertible. The inverse of , namely , can be described similarly in this algorithmic fashion. For a given partition with , define to be the sequence obtained from the empty sequence by adding nondecreasing parts one at a time that are derived from peeling off partially or entirely certain parts of according to the following deletion procedure.
The effect of this deletion is that the weight of the sequence and its -alternating sum are decreased by and , respectively. Also, it should be noted that this deletion process must stop after a finite number of steps. Since belongs to , there must be such that
Let be the largest such . Then, for any and
which shows such must pass (Test D).
To finish the proof, we make the following claims about and without giving the proofs, since all of them are essentially the same as those found in , which is the case when and .
Each insertion outputs a new , and in particular, is well-defined.
Each deletion outputs a new , and in particular, is well-defined.
The deletion procedure reverses the insertion procedure, consequently is the inverse of .
Before we move on, we provide an example for the insertion procedure.
Let , , , and . We insert into as follows. Note that
So we get
In Figure 2, we illustrate the process of getting by applying to using -modular Ferrers’ graphs. Newly inserted cells after each step are colored red.
As we will see, the bijection plays a crucial role in our proof of Theorem 1.3. We need a few more definitions.
For a partition , we define two local statistics “first bigger” and “last bigger”. For each , where is the number of nonzero parts in , suppose
Then we let .
Note that since , such and must exist and , so and are well-defined.
Fix a positive integer . For a partition , we call it an -falling lecture hall partition of order , if , and the following two conditions hold.
For , .
We denote the set of all -falling lecture hall partitions of order as .
Partitions in satisfying condition (1) above are said to be of -alternating type in .
Recall the definition of -falling regular partitions. A partition is -falling regular if the parts are not multiples of and their positive residues are nonincreasing.
For a chosen vector , let
3.1. Proof of Theorem 1.3
For some and a vector , we consider two embeddings,
such that and . To be precise, for a given partition , both and change the residue of each part of mod to be uniformly the predetermined value . In terms of the corresponding -modular Ferrers’ graph, the two maps keep all the cells labelled , but relabel all the remaining cells as . So in general, neither of these two maps preserves the weight of the partition, but they do keep the number of cells in their -modular Ferrers’ graphs the same.
Moreover, the given vector and the -falling condition uniquely determine the preimage of any partition in . Similarly, the condition (1) in Definition 3.5 together with dictate the preimage of any partition in . This entitles us to define a bijection
where is the -length type of the partition it acts on.
It has been proved in Theorem 3.1 that is a bijection satisfying , and the discussion above shows that both and are invertible. Consequently, we see that is indeed a bijection such that for any , and we complete the proof. ∎