Stanley--Elder--Fine theorems for colored partitions
aa r X i v : . [ m a t h . C O ] M a r STANLEY–ELDER–FINE THEOREMS FOR COLOREDPARTITIONS
HARTOSH SINGH BAL AND GAURAV BHATNAGAR
Abstract.
We give a new proof of a partition theorem popularly known asElder’s theorem, but which is also credited to Stanley and Fine. We extendthe theorem to the context of colored partitions (or prefabs). More specifically,we give analogous results for b -colored partitions, where each part occurs in b colors; for b -colored partitions with odd parts (or distinct parts); for partitionswhere the part k comes in k colors; and, overpartitions. Introduction
The purpose of this paper is to extend a charming theorem in the theory ofpartitions which appeared in Stanley [9, Ch 1, Ex. 80], but is usually attributed toElder, and more recently, has been found in the work of Fine; see Gilbert [8] for acomprehensive history. Our extensions appears in the context of colored partitions(or prefabs). As a consequence, we give analogous results for partitions where thepart k occurs in b colors, and partitions where the part k appears in k colors. Wealso consider colored partitions with odd or distinct parts and overpartitions. Inthe process, we extend results of Andrews and Merca [2] and Gilbert [8].We recall some of the terminology from the theory of integer partitions. Apartition of n is a way of writing n as an unordered sum of numbers. It is representedas a sequence of non-increasing, positive integers λ : λ ≥ λ ≥ λ ≥ . . . , with n = λ + λ + λ + · · · . The symbol λ ⊢ n is used to say that λ is a partition of n ; we also say λ has weight n . If λ ⊢ n , we have n = X k kf k where the non-negative integers f k = f k ( λ ) denote the frequency of k in λ , that is,the number of times k comes in λ . For example, the partition has frequencies: f = 4 , f = 1 , f = 2 , f = 1 .One of the quantities in the Stanley–Elder–Fine theorem is F k ( n ) := X λ ⊢ n f k ( λ ) , Date : March 5, 2021.2010
Mathematics Subject Classification.
Primary: 11P81; Secondary: 05A17.
Key words and phrases.
Integer partitions, Stanley’s theorem, Elder’s theorem, colored parti-tions, prefabs, partitions with k colors of k . the total number of k ’s appearing in all the partitions of n . The other is G k ( n ) = X λ ⊢ n g k ( λ ) , where g k ( λ ) := the number of parts which appear at-least k times in λ .The Stanley–Elder–Fine theorem says that for all n , F k ( n ) = G k ( n ) , (1.1)for k = 1 , , . . . , n .There are several proofs of this result, many of a combinatorial nature (see [8]for references). Here is another, rather simple, combinatorial proof of (1.1). Wefirst show F k ( n ) = p ( n − k ) + p ( n − k ) + p ( n − k ) + · · · , (1.2)for k = 1 , , . . . . (We take p ( m ) = 0 for m < .) Observe that for k = 1 , , . . . , n , F k ( n ) = p ( n − k ) + F k ( n − k ) , because adding k to each partition of n − k yields a partition of n , and vice-versa,on deletion of k from any partition containing k as a part, we obtain a partition of n − k . This gives (1.2) by iteration.Next, consider G k ( n ) . If we add · · · + 1 ( k -times) to any partition of n − k , we obtain a partition of n where appears as a part at-least k times; if weadd · · · + 2 ( k -times) to any partition of n − k , we obtain a partition of n where appears as a part at-least k times; and so on. The process is reversible.Thus G k ( n ) = p ( n − k ) + p ( n − k ) + p ( n − k ) + · · · (1.3)for k = 1 , , , . . . ; and F k ( n ) = G k ( n ) for all k = 1 , , . . . , n .The “counting by rows = counting by columns” quality of the Stanley–Elder–Finetheorem is reflected in this proof.The objective of this paper is to extend this proof to colored partitions generatedby ∞ Y k =1 − q k ) b k , where b k is a sequence of non-negative numbers. These are called prefabs by Wilf[10, §3.14], but we prefer the imagery of partitions with colored parts. Each part k comes in b k colors. They reduce to ordinary partitions when b k = 1 for all k . Weare able to extend (1.1) to the cases b k = b (a positive number) and when b k = k for all k . In addition, we consider partition objects from generating functions thatare products of such products; in particular, we consider overpartitions.This paper is organized as follows. In §2, we prove an analogue of (1.2) forcolored partitions. In §3 we consider b -colored partitions, where each part has b colors. Next, in §4, we consider b -colored partitions with odd or distinct parts.In §5 we consider partitions where the part k comes in k colors. The number ofsuch partitions is the same as the number of plane partitions of n . Next, in §6 weconsider overpartitions. We conclude in §7 by giving credit where credit is due. TANLEY–ELDER–FINE THEOREMS FOR COLORED PARTITIONS 3 The frequency function for colored partitions
The objective of this section is to obtain a key relation for the frequency functionfor colored partitions. We use notation from [5] to represent colored partitions.Let u k represent a set containing b k copies of k . The elements of u k are repre-sented as k , k , . . . . The elements of u k can be regarded as k with different colors.Consider two sets u j and u k , containing, respectively, b j and b k colors. We now usethe symbol u j + u k to denote the set of partitions u j + u k := { j a + k b : j a ∈ u j , k b ∈ u k } . Here u j represents u j + u j . This definition is extended by induction to finite sums P a i u i where a i ≥ . By a colored partition of weight n , we mean an element π where π ∈ X i a i u i , with | π | = P ni =1 ia i = n. For a partition π , let f k c ( π ) be the number of parts equal to k c in π , i.e., thefrequency of k c in π . Then the frequency of k in π is f k ( π ) = X k c ∈ x k f k c ( π ) . We denote the sum of the frequencies of all partitions of size n by F k ( n ) .For a partition π , let f k c ( π ) be the number of parts equal to k c in π , i.e., thefrequency of k c in π . Then the frequency of k in π is f k ( π ) = X k c ∈ x k f k c ( π ) . We denote the sum of the frequencies of all partitions of size n by F k ( n ) . Theorem 2.1.
Let h ( n ) represent the number of colored partitions of size n where k comes in b k colors. Let F k ( n ) be the frequency of k in all the partitions of n .Then we have the recurrence relation F k ( n ) = b k ( h ( n − k ) + h ( n − k ) + h ( n − k ) + · · · ) (2.1) Remark.
Here b k is a sequence of non-negative integers. A more general version ofthis theorem, where the b k are complex numbers, is proved in [5]. Proof.
We prove F k ( n − k ) + b k h ( n − k ) . (2.2)For X = P j c j u j , let | X | denote the number of partitions in X , and let F k ( X ) denote the number of k ’s in X . Consider a set of partitions of weight n representedby X = c k u k + Y , where Y = P j = k c j x j . Here c k > . Now the number ofpartitions in X are (cid:18) c k + | u k | − | u k | − (cid:19) | Y | = (cid:18) c k + b k − b k − (cid:19) | Y | . We relate X with partitions obtained by deleting one k .The contribution to F k ( n ) from X is F k ( X ) = c k (cid:18) c k + b k − b k − (cid:19) | Y | . H. S. BAL AND G. BHATNAGAR
Using the elementary identity ( n + 1) (cid:18) n + kk − (cid:19) = k (cid:18) n + k − k − (cid:19) + n (cid:18) n + k − k − (cid:19) we find that F k ( X ) = ( c k − (cid:18) c k − b k − b k − (cid:19) | Y | + b k (cid:18) c k − b k − b k − (cid:19) | Y | . Now the first of the two terms on the right is F k ( X − u k ) , the number of k ’s in X − u k . Thus on summing over all X that contains u k , we obtain F k ( n − k ) . Inthe second term, the quantity (cid:18) c k − b k − b k − (cid:19) | Y | is the number of partitions in X − u k . The weight of each partition is n − k . Summingover all the partitions X of weight n which contain u k , we obtain h ( n − k ) . Thisshows (2.2). (cid:3) Partitions with the same number of colors for each part
We consider b -colored partitions where each part k comes in b colors, where b is apositive integer. A Stanley–Elder–Fine theorem is as easy to obtain in this contextas the b = 1 case. Let h ( n ) represent the number of b -colored partitions of n . Let k , k , . . . , k b represent the colored parts. Let F k ( n ) be the frequency of k in allthe partitions of n . From (2.1), we have F k ( n ) = bh ( n − k ) + bh ( n − k ) + bh ( n − k ) + · · · . (3.1)Let π be a partition. As before, let g k ( π ) := the number of parts which appear at-least k times in π ,and G k ( n ) = X | π | = n g k ( π ) . Thus G k ( n ) is the number of times a part appears at-least k times in a partition,summed over all the partitions of n . Then we have Theorem 3.1.
Let F k ( n ) and G k ( n ) be as above. Then, for all n = 1 , , . . . , F k ( n ) = G k ( n ) , for k = 1 , , . . . , n .Proof. The proof is virtually the same as the b = 1 case. Note that if we add a r (of any color) to any partition of n − r , we obtain a partition of n which hasat-least one r as a part. We can add any one of the b r ′ s. This can be done foreach r = 1 , , , . . . . Thus G ( n ) = bh ( n −
1) + bh ( n −
2) + · · · + bh (0) , since every part from every partition of n which is repeated at-least once will beaccounted for (uniquely) in this way.In general, we see that if we add c + 1 c + · · · + 1 c ( k -times) to any partition of n − k , we obtain a partition of n where c appears as a part at-least k times; if we TANLEY–ELDER–FINE THEOREMS FOR COLORED PARTITIONS 5 add c + 2 c + · · · + 2 c ( k -times) to any partition of n − k , we obtain a partition of n where c appears as a part at-least k times; and so on. Thus G k ( n ) = bh ( n − k ) + bh ( n − k ) + bh ( n − k ) + · · · for k = 1 , , , . . . . In view of (3.1), G k ( n ) and F k ( n ) are equal. (cid:3) Next, we consider the quantity H k ( n ) , defined as the sum of parts divisibleby k , counted without multiplicity, in all the b -colored partitions of n . As anexample, consider an ordinary partition (that is, b = 1 ) represented by u + 4 u or . This contributes to the sum. On the otherhand, a 2-colored partition + 3 + 6 + 6 + 6 + 6 ∈ u + 4 u contributes to H (30) . Theorem 3.2.
Let F k ( n ) and H k ( n ) be as above. Then for all n , we have kF k ( n ) = H k ( n ) − H k ( n − k ) . Remark.
When the number of colors b = 1 , i.e., in the case of ordinary partitions,this result reduces to a result of Andrews and Merca [2]. Proof.
We first show H k ( n ) = bkh ( n − k ) + 2 bkh ( n − k ) + 3 bkh ( n − k ) + · · · . (3.2)The argument is similar to the one for G k ( n ) . If we add ( rk ) c (the part rk in color c ) to any partition of n − rk , we obtain a partition with ( rk ) c as a part. Thiscontributes rk to H k ( n ) . Conversely, if we delete ( rk ) c in a partition of n where itcomes as a part, we obtain a partition of n − rk . This shows (3.2).Now it is clear that H k ( n ) − H k ( n − k ) equals kF k ( n ) by (3.1). (cid:3) Partitions with odd and distinct parts
We consider b -colored partitions with all parts odd (which all come in b -colors).Let h ( n ) now denote the number of b -colored partitions with only odd parts. Aneasy extension of Euler’s ODD=DISTINCT theorem (see [4, eq. (2.1)]) says that h ( n ) is also the number of b -colored partitions with distinct parts. Here the h ( n ) are generated by ∞ Y k =1 (cid:0) q k (cid:1) b = ∞ Y k =1 (cid:0) − q k − (cid:1) b . Let F ok ( n ) denote the corresponding frequency function. Then we have F ok ( n ) = ( F o ( n − k ) + bh ( n − k ) , if k is odd; , if k is even. (4.1a) = ( h ( n − k ) + h ( n − k ) + h ( n − k ) + · · · , if k is odd; , if k is even. (4.1b)Let G ok ( n ) be the number of times a part appears at-least k times in a partition,summed over all the b -colored partitions of n with odd parts. Theorem 4.1.
Let F ok ( n ) and G ok ( n ) be as above and let k be an odd number.Then, for all n = 1 , , . . . , F ok ( n ) = G ok ( n ) + G ok ( n − k ) , for k = 1 , , , . . . . H. S. BAL AND G. BHATNAGAR
Remark.
In the case of ordinary partitions, where b = 1 , this theorem reduces toan observation of Gilbert [8, Th. 8]. Proof.
If we add c + 1 c + · · · + 1 c ( k -times) to any partition of n − k , we obtain apartition of n where c appears as a part at-least k times; if we add c + 3 c + · · · + 3 c ( k -times) to any partition of n − k , we obtain a partition of n where c appearsas a part at-least k times; and so on. Thus G ok ( n ) = bh ( n − k ) + bh ( n − k ) + bh ( n − k ) + · · · (4.2)for k = 1 , , , . . . . Note that this applies even if k is not an odd number.When k is odd, we see from (4.1b) that G ok + G ok ( n − k ) = F ok ( n ) , This proves the theorem. (cid:3)
Next let F dk ( n ) denote the frequency of k in b -colored partitions with all partsdistinct. The differently colored parts of the same weight are considered distinct.For example, + 3 + 5 is considered a -partition of with distinct parts. Thispartition contributes to F d (11) . It is easy to see that F dk ( n ) = bh ( n − k ) − F dk ( n − k ) (4.3a) = bh ( n − k ) − bh ( n − k ) + bh ( n − k ) − bh ( n − k ) + · · · . (4.3b)To obtain (4.3a), observe that a b -colored partition of n (with distinct parts) whichcontains k c as a part is obtained by adding k c to any partition of n − k which doesnot have k c as a part. So the number of distinct partitions of n containing k c is h ( n − k ) − F k c ( n ) , where F k c ( n ) is number of k c ’s in the distinct partitions of n − k (and also the number of distinct partitions containing k c as a part). Now summingover all colors, we obtain (4.3a).Since the number h ( n ) of b -colored partitions with odd parts and distinct partsare the same, equation (4.3b), along with (4.2) immediately yields the followingtheorem. Theorem 4.2.
Let F dk ( n ) and G ok ( n ) be as defined above. Then F dk ( n ) = G ok ( n ) − G ok ( n − k ) , for k = 1 , , , . . . .Remark. When b = 1 , Theorem 4.2 reduces to Gilbert [8, Th. 9].5. Partitions with k copies of k Next we consider a special case of colored partitions generated by the product ∞ Y k =1 − q k ) k . We reuse the notation h ( n ) to denote the number of such partitions of n . Thenotations for F k ( n ) and G k ( n ) are also reused. Theorem 5.1.
Consider the set of partitions of n where each part k comes in k colors. Let F k ( n ) be the number of k ’s (of any color) appearing in all such partitions TANLEY–ELDER–FINE THEOREMS FOR COLORED PARTITIONS 7 of n . Let G k ( n ) denote the number of parts that appear at-least k times in such apartition, summed over all such partitions. Then, for all n = 1 , , . . . , F k ( n ) = k (cid:0) G k ( n ) − G k ( n − k ) (cid:1) , for k = 1 , , . . . , n .Proof. Observe that G k ( n ) = 1 · h ( n − k ) + 2 · h ( n − k ) + 3 · h ( n − k ) + · · · for k = 1 , , , . . . .Thus we have G k ( n ) − G k ( n − k ) = h ( n − k ) + h ( n − k ) + · · · , and, by Theorem 2.1, F k ( n ) = k (cid:0) h ( n − k ) + h ( n − k ) + h ( n − k ) + · · · (cid:1) = k (cid:0) G k ( n ) − G k ( n − k ) (cid:1) . (cid:3) Overpartitions
An overpartition of n is a non-increasing sequence of natural numbers whosesum is n , where the first occurrence of a number can be overlined. We denote thenumber of overpartitions of n by p ( n ) . These can be represented as partitions intwo symbols u and v . The partitions represented by u are ordinary partitions andthe partitions represented by v are distinct. In a sense (to be explained shortly),overpartitions are convolutions of these two types of partitions. One of our theoremsin this section shows how one can simply put together the respective results for twopartition functions to obtain a new theorem for their convolution. In addition, wegive another extension of Stanley’s theorem, which also follows by manipulatingrecurrence relations. Its proof is more intricate than what we have encountered sofar.The overpartitions of n can be formed by adding partitions of k in u with adistinct partition of n − k in v . For example, here is a way to list the overpartitionsof . First we list ordinary partitions up to and then add them with partitionswith distinct parts written in reverse order. In Table 1 we have listed the partitionsof and partitions with distinct parts in reverse order in v . For example, u + v m Partitions of m Partitions of n − m into distinct parts Partitions of n − m into odd parts - v + v , v w , w + w u v + v , v w , w u , u v w u , u + u , u v w u , u + u , u + u , u , u - - Table 1.
Generating overpartitions and odd-overlined partitions
H. S. BAL AND G. BHATNAGAR represents and u + v represents . Evidently, X m p ( m ) p ( n − m | distinct parts ) = p ( n ) , a convolution of two sums; thus the generating function of overpartitions is theproduct of the respective generating functions: Q ( q ) = X n ≥ p ( n ) q n = ∞ Y k =1 q k − q k . It is in this sense we describe overpartitions as convolutions of ordinary partitionswith partitions of distinct parts.Since the number of partitions into distinct parts equals the number of oddpartitions, it is natural to consider the partitions formed by adding an ordinarypartition of m in u with a partitions of n − m with odd parts in w (see Table 1).These are equinumerous to overpartitions. We call them odd-overlined partitions.Consider colored overlined partitions (of both kinds) generated by the generatingfunctions ∞ X n =0 h ( n ) q n = ∞ Y k =1 (cid:0) q k (cid:1) s (cid:0) − q k (cid:1) r = ∞ Y k =1 (cid:0) − q k (cid:1) r (cid:0) − q k − (cid:1) s . Here the ordinary parts are r -colored and the distinct/odd parts are colored in s colors.We mix and match the notations of §3 and §4. So, for example, h ( n ) will refer tothe number of overpartitions (respectively, odd-overlined partitions), F k ( n ) refersto the frequency of k in appearing in ordinary partition (in r colors) contained inthe overline partition, and F dk ( n ) and F ok ( n ) are the frequencies of k of the overlinedparts which come in s colors. Similarly, let G k ( n ) , G ok ( n ) be defined as earlier. Thenwe have: Theorem 6.1.
Let F k ( n ) , F ok ( n ) , F dk ( n ) , G k ( n ) , and G ok ( n ) be as defined above,in the context of colored overpartitions/odd-overlined partitions. Then, for all n =1 , , . . . , F k ( n ) = G k ( n ) , for k = 1 , , , . . . ; F dk ( n ) = G ok ( n ) − G ok ( n − k ) , for k = 1 , , , . . . ; F ok ( n ) = G ok ( n ) + G ok ( n − k ) , for k = 1 , , , . . . .Remark. One can get analogous theorems for partitions where each part k comesin k + b colors, by combining the considerations of §3 with §5.Before concluding, we give one more theorem concerning overpartitions, whichis of a different nature than those studied above. Consider overpartitions with allparts of a single color generated by Q ( q ) . For this theorem, we prefer the imageryof overpartitions where the part in v is overlined.Let F k ( n ) and F d ( k ) be as above. Let F k ( n ) = F k ( n ) + F dk ( n ) . So F ( n ) is the number of overpartitions of n with or as parts. Let G k ( n ) be thenumber of overpartitions where a part is repeated at-least k times. For example, TANLEY–ELDER–FINE THEOREMS FOR COLORED PARTITIONS 9 the overpartition has the part repeated three times andcontributes to G k (11) for k = 1 , , .Thus G ( n ) is the total number of parts m in overpartitions of n such that m or m appear at-least once in a partition of n , and G ( n ) is the number of partsrepeated thrice or more. An extension of Stanley’s theorem (that it, the b = 1 and k = 1 case of Theorem 3.1) to overpartitions says that the number of overpartitionsof n with or as a part is the difference of these two quantities. Theorem 6.2.
Let F k ( n ) and G k ( n ) be as above. Then, for n = 1 , , . . . , F ( n ) = G ( n ) − G ( n ) . Proof.
It is easy to see (by the arguments used to obtain (2.1) and (4.3b)) that F ( n ) = p ( n −
1) + p ( n −
2) + p ( n −
2) + p ( n −
3) + · · · , and F d ( n ) = p ( n − − p ( n −
2) + p ( n − − p ( n −
3) + · · · , so F ( n ) = 2 (cid:0) p ( n −
1) + p ( n −
3) + p ( n −
5) + · · · (cid:1) . (6.1)To obtain an expression for the right hand side, we need the following ancillarycounting functions. Let p m ( n ) be the number of overlined partitions which have m as a part. Note that p m ( n ) = p ( n − m ) . (6.2)We also need the following functions: O m ( n ) := the number of overpartitions where the part m or m appears at-least once ; O m ( n ) := the number of overpartitions where the part m appearsat-least once; O m ( n ) := the number of overpartitions where the part m doesnot appear and m appears; T m ( n ) := the number of overpartitions where the part m or m appears at-least thrice .Evidently O m ( n ) = O m ( n ) + O m ( n ); n X m =1 O m ( n ) = G ( n ); and, n X m =1 T m ( n ) = G ( n ) . To prove the theorem, we find expressions for O m ( n ) and T m ( n ) in terms of p ( n ) .Note that O ( n ) = p ( n − O ( n ) = p ( n − − p ( n − − O ( n − p ( n − − p ( n −
2) + 2 p ( n − − p ( n −
4) + · · · . The first of these follows because we obtain an overpartition of n with as a partby adding a to each overpartition of n − . For O ( n ) we note that we can add a to each overpartition of n − , which has neither nor as a part. Finally, thelast line follows from (6.2) and iteration.From the above, we find that O (( n ) = 2 ( p ( n − − p ( n −
2) + p ( n − − p ( n −
4) + · · · ) . Similarly, we have, for m = 1 , , . . . , nO m (( n ) = 2 ( p ( n − m ) − p ( n − m ) + p ( n − m ) − p ( n − m ) + · · · ) . (6.3)Next, we note that T ( n ) = p ( n −
3) + O ( n − p ( n − − p ( n −
4) + p ( n − − p ( n −
7) + · · · ) The first of these is true because any overpartition where comes at-least threetimes is obtained by adding to an overpartition of n − or by adding to an overpartition of n − which does not have a but has an . The secondfollows by using the formula for O ( n ) computed above.Similarly, for m = 1 , , . . . , n , T m ( n ) = 2 ( p ( n − m ) − p ( n − m ) + p ( n − m ) − p ( n − m ) + · · · ) (6.4)Finally, we have G ( n ) − G ( n ) = n X m =1 O m ( n ) − T m ( n )= n X m =1 (cid:0) p ( n − m ) − p ( n − m ) (cid:1) (using (6.3) and (6.4)) = 2 (cid:0) p ( n −
1) + p ( n −
3) + p ( n −
5) + · · · (cid:1) = F ( n ) , using (6.1). This completes the proof. (cid:3) Closing credits
The key idea in our extensions of the Stanley–Elder–Fine theorem is the re-currence (2.2) from which (2.1) follows. From here, it is easy to manipulate theexpression for G k ( n ) corresponding to the choice of b k . Even so, the correspondingtheorems of Andrews and Merca [2] and Gilbert [8] have motivated the form of ourtheorems. In particular, the definition of H k ( n ) for ordinary partitions in [2] wasvery useful.As we saw when considering overpartitions, we can mix and match and findStanley–Elder–Fine theorems for partitions that can be constructed by the convo-lution of two different kinds of partitions. In addition to overpartitions, many suchpartitions have appeared in the literature, and this technique can be used to givesuch theorems of them.We mention some related work. Banerjee and Dastidar [6] have given a proofby combinatorial means too, but it is much more intricate than the one given here.Their colored partitions are different from ours; they are closer in spirit to the workin § 6. Dastikar and Sen Gupta [7] has given an extension of Stanley’s theorem TANLEY–ELDER–FINE THEOREMS FOR COLORED PARTITIONS 11 which comes from summing (1.3) for k = 1 , , . . . , k and noting that the sum equals F ( n ) . Their results can be immediately extended (as in this paper) to the contextof colored partitions. Andrews and Deutsch [3] have a different generalization ofElder’s theorem. Their starting point and key argument is not far from ours, butthey have taken a different path to generalization; see also [1].The relations (2.2) and (2.1) have number-theoretic consequences. These arestudied by the authors in [5]. References [1] A. M. Alanazi and A. O. Munagi. On partition configurations of Andrews-Deutsch.
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