Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities
aa r X i v : . [ m a t h . C O ] J un EULER’S PARTITION THEOREM FOR ALL MODULI ANDNEW COMPANIONS TOROGERS-RAMANUJAN-ANDREWS-GORDON IDENTITIES
XINHUA XIONG, WILLIAM J. KEITH
Abstract.
We generalise Euler’s partition theorem involving odd parts anddifferent parts for all moduli and provide new companions to Rogers-Ramanujan-Andrews-Gordon identities related to this theorem. Introduction
In the theory of partitions, Euler’s partition theorem involving odd parts anddifferent parts is one of the famous theorems. It claims that the number of partitionsof n into odd parts is equal to the number of partitions n into different parts. Byconstructing a bijection, Sylvester [17] not only proved Euler’s theorem, but alsoprovided a refinement of it which can be stated as, “the number of partitions of n into odd parts with exactly k different parts is equal to the number of partitionsof n into different parts such that exactly k sequences of consecutive integers occurin each partition.” Bessenrodt [9] proved that Sylvester’s bijection implies thatthe number of partitions of n into different parts with the alternating sum Σ isequal to the number of partitions of n with Σ odd parts. Here for a partition λ = ( λ , λ , . . . , λ k ), the alternating sum is defined by(1) Σ = λ − λ + λ − λ + · · · + ( − k +1 λ k . In [13], Kim and Yee gave a different description of Sylvester’s bijection whichprovides a simpler proof of the refinement of Euler’s theorem due to Bessenrodt.There are several other refinements and variants of Euler’s theorem. See [1, 4, 6,7, 8, 14, 18, 19].We can think of Euler’s theorem as a theorem on partitions involving modulustwo by interpreting odd parts as parts ≡ . The first nontrivial generali-sation of Euler’s theorem for all moduli in this sense is the following theorem dueto Pak-Postnikov.
Theorem 1 (Pak-Postnikov [15]) . The number of partitions of n with type ( c, m − c, c, m − c, . . . ) is equal to the number of partitions of n with all parts ≡ c (mod m ) . By the type ( c, m − c, c, m − c, . . . ) for a partition λ , it means that λ has thelength divisible by m by allowing zero as parts and has c ≥ m − c second largest parts, etc. So it has the form: λ = λ = λ = · · · = λ c > λ c +1 = λ c +2 = · · · = λ m > λ m +1 = · · · = λ m + c > . . . Mathematics Subject Classification.
Key words and phrases. partitions.
The second author’s doctoral thesis [12] has a chapter devoted to various identitiesof this nature, such as for m - falling or m - rising partitions, those in which leastpositive residues of each part modulo m form a nonincreasing or nondecreasingsequence.In this paper, we prove a theorem which generalises Euler’s partition theoremmentioned above for all moduli, and simultaneously generalises Pak-Postnikov’stheorem. We also show that this theorem provides new companions to Rogers-Ramanujan-Andrews-Gordon identities.In Section 2 we give all definitions necessary to state our theorem. In Section 3we prove the theorem by a bijection originally due to Stockhofe in [16] (an Englishtranslation of the original German can be found as an appendix to [12]), slightlyextended and heavily specialized for the present purpose. The original map was ageneral bijection on all partitions; we will show that the properties required holdwhen specialized to the sets of interest for the theorem. All concepts and claimsnecessary will be defined and proved here to keep the paper self-contained.2. Statement of theorem
In order to state our theorem, we introduce some notations and terminologies. • For a partition λ = ( λ , λ , λ , . . . , λ r ) of n , we denote n by | λ | . We sometimeswrite a partition as the form λ + λ + λ + · · · + λ r or the form λ ≥ λ ≥ λ ≥ · · · ≥ λ r . • Let m ≥ λ ≥ λ ≥ λ ≥ · · · ≥ λ km ( k ≥ m , we define its alternatingsum type to be an ( m − , Σ , . . . , Σ m − , Σ m − )given by Σ = k X i =1 λ ( i − m +1 − λ ( i − m +2 , Σ = k X i =1 λ ( i − m +2 − λ ( i − m +3 , · · · Σ m − = k X i =1 λ ( i − m + m − − λ im . For example, if m = 3, then the partition 6 + 5 + 4 + 3 + 2 + 1 has the alternatingsum type (6 − − , − −
1) = (2 , , , . . . , , Σ , , . . . , c th position. If m = 2, the alternating sum type is exactly thealternating sum is given by (1). • When we speak of alternating sum types for a partition of length k , we allowthe last few parts to be zero so that any partition has length ⌈ k/m ⌉ m . For exampleif m = 3, the partition 5 + 4 + 3 + 3 has length 6 by viewing it as 5 + 4 + 3 + 3 + 0 + 0and it has two basic units: 5 + 4 + 3 and 3 + 0 + 0 . • For a partition λ with all parts m ), known as an m -regular partition,we define its length type to be the ( m − l , l , l , . . . , l m − , l m − ),where for 1 ≤ i ≤ m − l i is the number of parts of λ which are congruent to i ULER’S PARTITION THEOREM FOR ALL MODULI AND NEW COMPANIONS TO ROGERS-RAMANUJAN-ANDREWS-GORDON IDENTITIES3 modulo m . For example, for a partition λ with all parts congruent to c , its lengthtype is (0 , , . . . , , l, . . . , l is the number of parts of λ and lies at the c th position. • Given a partition λ = ( λ , . . . , λ k ), its conjugate λ ′ is ( { λ i ≥ } , { λ i ≥ } , . . . ).Now we can state our theorem. Theorem 2.
Let m ≥ . Let P be the set of partitions in which each part can berepeated at most m − times. (This implies that their alternating sum types cannotbe (0 , , . . . , .) Let Q be the set of partitions with no parts ≡ m ) . Thenwe have the partition identity: X λ ∈ P z Σ ( λ )1 z Σ ( λ )2 . . . z Σ m − ( λ ) m − q | λ | = X µ ∈ Q z l ( µ )1 z l ( µ )2 . . . z l m − ( µ ) m − q | µ | . Equivalently, the number of partitions of n with the alternating sum type (Σ , Σ , . . . Σ m − ) is equal to the number of partitions of n with Σ parts congruent to modulo m , Σ parts congruent to modulo m , . . . , Σ m − parts congruent m − modulo m . If we let z = z = . . . z m − = z , we get the result that the number of partitionsof n with parts repeated at most m − + Σ + · · · + Σ m − is equal to the number of partitions of n with no parts congruent to0 modulo m and Σ + Σ + · · · + Σ m − parts, which is a refinement of Glaisher’stheorem: Theorem 3 (Glaisher [14]) . The number of partitions of n with parts repeated atmost m − times is equal to the number of partitions of n with no parts is congruentto modulo m . When the alternating sum type is pure type, this theorem reduces to Theorem1.1 due to Pak-Postnikov. When m is 2, this theorem reduces to the refinement ofEuler’s theorem due to Bessenrodt, Kim and Yee.We give n = 10 and m = 3 , P , their alternating sum types and the numbers on the left, and the correspondingparts for partitions in Q on the right. We only list all partitions with mixed types. XINHUA XIONG, WILLIAM J. KEITH
Partitions in P (Σ , Σ ) ♯ (1 ,
2) 4 (3 ,
1) 4 (cid:26) (cid:27) (2 ,
3) 2 (cid:26) (cid:27) (4 ,
2) 2 (cid:26) (cid:27) (6 ,
1) 2 (cid:8) (cid:9) (1 ,
5) 1 (cid:8) (cid:9) (3 ,
4) 1 (cid:8) (cid:9) (5 ,
3) 1 (cid:8) (cid:9) (7 ,
2) 1 (cid:8)
10 + 1 (cid:9) (9 ,
1) 1 Partitions in Q ( l , l ) ♯ (1 ,
2) 4 (3 ,
1) 4 (cid:26) (cid:27) (2 ,
3) 2 (cid:26) (cid:27) (4 ,
2) 2 (cid:26) (cid:27) (6 ,
1) 2 (cid:8) (cid:9) (1 ,
5) 1 (cid:8) (cid:9) (3 ,
4) 1 (cid:8) (cid:9) (5 ,
3) 1 (cid:8) (cid:9) (7 ,
2) 1 (cid:8) (cid:9) (9 ,
1) 1Partitions in P (Σ , Σ , Σ ) ♯ (1 , ,
1) 3 (cid:26) (cid:27) (2 , ,
0) 2 (cid:26) (cid:27) (3 , ,
1) 2 (cid:26) (cid:27) (4 , ,
0) 2 Partitions in Q ( l , l , l ) ♯ (1 , ,
1) 3 (cid:26) (cid:27) (2 , ,
0) 2 (cid:26) (cid:27) (3 , ,
1) 2 (cid:26) (cid:27) (4 , ,
0) 2
ULER’S PARTITION THEOREM FOR ALL MODULI AND NEW COMPANIONS TO ROGERS-RAMANUJAN-ANDREWS-GORDON IDENTITIES5 (cid:8) (cid:9) (0 , , (cid:8) (cid:9) (1 , ,
3) 1 (cid:8) (cid:9) (1 , ,
1) 1 (cid:8) (cid:9) (2 , ,
2) 1 (cid:8) (cid:9) (2 , ,
0) 1 (cid:8) (cid:9) (3 , ,
1) 1 (cid:8) (cid:9) (4 , ,
2) 1 (cid:8) (cid:9) (4 , ,
0) 1 (cid:8) (cid:9) (5 , ,
1) 1 (cid:8) (cid:9) (6 , ,
0) 1 (cid:8) (cid:9) (7 , ,
1) 1 (cid:8) (cid:9) (8 , ,
0) 1 (cid:8) (cid:9) (0 , ,
2) 1 (cid:8) (cid:9) (1 , ,
3) 1 (cid:8) (cid:9) (1 , ,
1) 1 (cid:8) (cid:9) (2 , ,
2) 1 (cid:8) (cid:9) (2 , ,
0) 1 (cid:8) (cid:9) (3 , ,
1) 1 (cid:8) (cid:9) (4 , ,
2) 1 (cid:8) (cid:9) (4 , ,
0) 1 (cid:8) (cid:9) (5 , ,
1) 1 (cid:8) (cid:9) (6 , ,
0) 1 (cid:8) (cid:9) (7 , ,
1) 1 (cid:8) (cid:9) (8 , ,
0) 1 Proof of the main theorem
We begin with a simple lemma.
Lemma 1.
The conjugates λ ′ of partitions λ with alternating sum type ( s , . . . , s m − ) are precisely those partitions of length type ( s , . . . , s m − ) .Proof. Suppose λ km + i − λ km + i +1 = c contributes a nonzero amount to s i . Thenin the conjugate partition, c parts of size km + i appear. The converse also holds.Call m - flat a partition in which all differences between consecutive parts arestrictly less than m and the smallest part is less than m . These are clearly theconjugates of partitions in P . We will prove by bijection that Theorem 4.
There is a bijection between m -regular partitions of any given lengthtype ( ℓ , . . . , ℓ m − ) and m -flat partitions of the same length type.Proof. In fact, the bijection even preserves the sequential order of the nonzeroresidues modulo m ; our map will consist of rearranging units of size m .It is useful to define two operations analogous to scalar multiplication and vectoraddition on partitions. For convenience, assume that all partitions are equippedwith infinite tails consisting solely of zeros. The scalar multiple of a partition λ = ( λ , λ , . . . ) by the positive integer m is the partition mλ = ( mλ , mλ , . . . ).Given two partitions λ = ( λ , λ , . . . ) and µ = ( µ , µ , . . . ), we can define their(infinite-dimensional) vector sum λ + µ = ( λ + µ , λ + µ , . . . ). (Partition additionin the literature sometimes means taking the nonincreasing sequence of the multisetunion of all parts of both partitions; we will not require this operation.)Begin with an m -flat partition. We will remove multiples of m to constructa partition π , via a sequence of intermediate partitions λ (0) , λ (1) , λ (2) , etc. Aswe do so, we will use the removed multiples of m to construct a second partition mσ = ( mσ , mσ , . . . ).Initialize σ = (), the empty partition. Step 1.
First, construct λ (0) by removing from λ any parts divisible by m forwhich, after removal, the partition λ (0) is still m -flat. These will be parts such that XINHUA XIONG, WILLIAM J. KEITH • λ i = k i m = λ i +1 , i.e. all but the last of a repeated part divisible by m ; • λ = k m if the largest part is divisible by m ; or • parts λ i = k i m , i >
1, such that λ i − = k i m + j , λ i +1 = ( k i − m + j ,with 0 < j < j < m .Thus the remaining parts divisible by m in λ (0) are all distinct, not the largest(or smallest) parts, and any remaining part λ i = k i m lies between λ i − = k i m + j and λ i +1 = ( k i − m + j with 0 < j ≤ j < m .For each part λ i = k i m removed, add k i m to mσ as a part.For the previous step, the order of removal did not matter, although of course mσ is arranged in nonincreasing order. In the next step, we work from the largestpart divisible by m to the smallest. Step 2.
Begin with λ (0) and set j = 0.(1) If λ ( j ) has no parts divisible by m , stop.(2) If λ i = k i m is the largest part in λ ( j ) divisible by m , remove λ i from λ ( j ) .Renumber following parts.(3) In addition, reduce by m all parts λ through λ i − . The remaining partitionis now λ ( j +1) . Increment j .(4) Add m ( k i + i −
1) to mσ as a part.(5) Repeat.The following lemma concerning parts removed in Step 2 will be useful when wewish to prove that this process is reversible. Lemma 2.
Parts added to σ in Step 2 are always at least the size of those removedin Step 1, and are added in nondecreasing order of size. The largest possible sizeof a part added to mσ in Step 2 is the number of parts in λ not divisible by m .Proof. If λ is an m -flat partition and λ i = k i m + j , j m ), with λ i + c = ( k i − m + j the next smaller part which is nonzero modulo m , thisnecessarily requires 0 < j < j < m . In this case refer to λ i as a descent of λ .If a part k i m appears between λ i and λ i + c defining a descent, we speak of k i m asappearing within the descent.Parts divisible by m are not the largest part of λ ( j ) and do not appear withindescents of λ ( j ) : these were removed in Step 1.Suppose part λ i = k i m appears in λ ( j ) , so that when removed we will add part m ( k i + i −
1) to mσ . The next part, if any, which will be removed is λ i + s − = k i + s − m , s ≥
2, after the renumbering. (That is, it was λ i + s before renumbering.)We have k i + s − < k i , decreasing by exactly 1 for each descent passed as we readfrom (after renumbering) λ i to λ i + s − , plus 1 immediately, in essence thinking ofthe passage from k i m to λ i as a descent. The total decrease is at most s −
1, since λ i + s − and λ i + s − cannot be descents (parts k i m in Step 2 do not appear withindescents).On the other hand, the number of parts added due to subtraction from previousparts always increases by s −
1: one for each part passed regardless of whether it isa descent or not, less 1 because 1 fewer part exists prior to λ i + s − after removal of λ i . Thus we have k i + s − + ( i + s − − ≥ k i + i − i = 0 for apotential largest part. The first k i + i − i and increased by i exactly. ULER’S PARTITION THEOREM FOR ALL MODULI AND NEW COMPANIONS TO ROGERS-RAMANUJAN-ANDREWS-GORDON IDENTITIES7
Finally, the largest a part removed in Step 2 can be is if we add as much aspossible, with steps across descents being irrelevant; that is, the largest possiblepart that could be removed is a part λ i = m which is the next-to-last part, followedby a single part not divisible by m . Since all previous parts divisible by m wouldhave been removed at this step, clearly in this case 1 + i − λ not divisible by m . By the previous clauses, this is the largest removal.Thus all claims of the lemma hold. Step 3.
After Step 2, we now have some λ ( j ) which is simultaneously m -regularand m -flat, and mσ consisting of parts divisible by m . Set π = λ ( j ) . Our finalpartition is π + m ( σ ′ ) . Since by our lemma the largest part of σ was less than orequal to the number of parts in λ not divisible by m , its conjugate has at most thisnumber of parts, so we only add multiples of m to such parts in π . The resultingpartition has all parts not divisible by m .Since no step in this construction alters the residue modulo m of a part notdivisible by m , it is an easy lemma that Lemma 3.
The length type of the parts of λ not divisible by m , read as a partition,is the same as the length type of the partition π + m ( σ ′ ) . We now briefly show that the map is reversible. Starting with a partition µ intoparts not divisible by m , we will construct a sequence of partitions π ( j ) which beginwith the m -flat, m -regular portion of µ and have parts from σ inserted. Step 3 Reverse.
It is easy to break a partition µ into a flat plart plus acollection of parts divisible by m , as π + m ( σ ′ ). Whenever µ i − µ i +1 ≥ m (includingfor the smallest part: treat the next part as 0), add 1 to σ ′ for parts 1 through i .Subtract m from all parts µ through µ i . Repeat. When done with all possibleremovals, conjugate σ ′ to obtain σ . Step 2 Reverse.
Observe that if we wish to insert a part mσ into π , we mustdetermine whether it is to be inserted in the reverse of Step 2 or Step 1. Step2 insertions occur when part mσ is larger than k for ( π ( j ) ) = k m + j . Theposition where such a part can be inserted is unique, since by the proof of Lemma2 there can be only one i such that σ = k i + i − mσ would notbe appearing within a descent. Passing a column that is not a descent changes theamount to be added; passing a column that is a descent does not, but is not a placewhere parts are added in Step 2. Step 1 Reverse.
This step is easy since the order in which parts are insertedwill not matter. Once parts are small enough that they can be inserted into π ( j ) while retaining flatness, insert all at once. A part of size k i m will go precisely aftera part of size k i m if one already exists, or within the descent at λ i = k i m + j ifthe next part is not divisible by m .The result is our desired m -flat partition.An example of the bijection may be illustrative. Let our modulus be m = 5.Let our starting partition be (9 , , , , , , , , , , , , , , , , , , , , , , , , λ =(22 , , , , , , , , XINHUA XIONG, WILLIAM J. KEITH − −
13 = 6). So far σ · ,
5) = (3 , · , , , , , , σ , obtaining (25 , ,
5) so far, and are left with the followingpartition, (17 , , , , , σ and finishing with σ · , , ,
5) = (5 , , , ·
5, and π = (12 , , , , σ , obtaining σ ′ = (4 , , , , π :2 5 5 5 5 5 54 5 5 5 53 5 5 5 51 5 5 52 5 5Our final partition is (32 , , , , , , , ULER’S PARTITION THEOREM FOR ALL MODULI AND NEW COMPANIONS TO ROGERS-RAMANUJAN-ANDREWS-GORDON IDENTITIES9
If we were to reverse our map, we would observe that σ ′ has five parts of size atleast 2 since the smallest part of µ is 2 + 5 + 5, and so forth obtain σ ; observingthat the largest part of σ is 5, we determine that we should set i = 3, since setting i = 4 is too large (adding part 5 and following it by 1 + 5 + 5 would not result in aflat partition), whereas i = 2 would not result in a partition at all (15 preceded by14). The other insertions are likewise unique.4. New companions to Rogers-Ramanujan-Andrews-Gordon identities
Besides Euler’s partition theorem involving odd parts and different parts, Rogers-Ramanujan-Andrews-Gordon identities are another famous partition theorem; see[2, 3, 5, 10]. Recall that the first Rogers-Ramanujan identity (partition version) saysthat the number of partitions of n with the condition that the difference betweenany two parts is at least 2 (called Rogers-Ramanujan partitions) is equal to thenumber of partitions of n such that each part is congruent to 1 or 4 modulo 5.From our viewpoint, partitions with each part congruent to 1 or 4 modulo 5 areexactly partitions belonging to Q with length types ( l , , , l ), ( l , l ) = (0 , Theorem 5.
The number of partitions of n where the difference between any twoparts is at least is equal to the number of partitions of n with parts repeated atmost times and alternating sum types (Σ , , , Σ ) , where (Σ , Σ ) = (0 , . We give an example to illustrate this theorem. Let n = 11, the partitions of 11with the condition that the difference is at least 2 are11 ,
10 + 1 , , , , , . And the partitions of 11 with alternating sum types (Σ , , , Σ ) are3 + 2 + 2 + 2 + 2 (1 , , , , , , , , , , , , , , , , , , , , , , , ,
11 (11 , , , . We list the alternating sum type following each partition. We have a similar com-panion on the second Rogers-Ramanujan identity:
Theorem 6.
The number of partitions of n where the difference between any twoparts is at least and is not a part is equal to the number of partitions of n with parts repeated at most times and alternating sum type (0 , Σ , Σ , and (Σ , Σ ) = (0 , . We still use n = 11 to illustrate this theorem. The partitions of 11 with thecondition that the difference is at least 2 and 1 is not a part are11 , , , . And the partitions of 11 with alternating sum types (0 , Σ , Σ ,
0) are3 + 3 + 3 + 1 + 1 (0 , , , , , , , , , , ,
0) 5 + 5 + 1 (0 , , , . We list the corresponding alternating sum type following each partition.For Andrews-Gordon’s identities, we have
Theorem 7.
Let d ≥ , ≤ i ≤ d . Suppose the conjecture is true, then thenumber of partitions λ + λ + λ + · · · + λ r of n such that no more than i − of the parts are and pairs of consecutive integers appear at most d − times isequal to the number of partitions of n with parts repeated at most d times andalternating sum type (Σ , Σ , . . . , Σ d − , Σ d ) = (0 , , . . . , satisfying that both Σ i and Σ d +1 − i are zero. Acknowledgements
The first author would like to thank Professor Peter Paule and Professor Chris-tian Krattenthaler for their comments on an earlier version of this paper during theStrobl meeting. The first author was supported by the Austria Science Foundation(FWF) grant SFB F50-06 (Special Research Program “Algorithmic and Enumer-ative Combinatorics”) and partially supported by the National Natural ScienceFoundation of China (11101238).