ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS

Transcription

1 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS ALEXANDER BERKOVICH AND ALI KEMAL UNCU arxiv:500730v3 [mathnt] 9 Apr 206 Abstract This article is an extensive study of partitions with fixed number of odd and even-indexed odd parts We use these partitions to generalize recent results of C Savage and A Sills Moreover we derive explicit formulas for generating functions for partitions with bounds on the largest part the number of parts and with a fixed value of BG-ran or with a fixed value of alternating sum of parts We extend the wor of C Boulet and as a result obtain a four-variable generalization of Gaussian binomial coefficients In addition we provide combinatorial interpretation of the Berovich Warnaar identity for Rogers Szegő polynomials Introduction and Notation A partition π is a non-increasing finite seuence π = (λ λ 2 ) of positive integers The elements λ i that appearin theseuenceπ arecalledpartsofπ Forpositiveintegersi wecallλ 2i odd-indexed parts and λ 2i even indexed parts of π We say π is a partition of n if the sum of all parts of π is eual to n Conventionally the empty seuence is considered as the uniue partition of zero We will abide by this convention Partitions can be represented graphically in multiple ways For consistency we are going to focus on representing partitions using Ferrers diagrams The 2-residue Ferrers diagram of partition π is given by taing the ordinary Ferrers diagram drawn with boxes instead of dots and filling these boxes using alternating 0 s and s starting from 0 on odd-indexed parts and on even -indexed parts We can exemplify 2-residue diagrams with π = (20752) in Table Table 2-residue Ferrers diagram of the partition π = (20752) In this paper we consider partitions with fixed number of odd and even-indexed odd parts We start our discussion by considering partitions into distinct parts In this way we are led to Theorem Theorem For non-negative integers i j and n p(ijn) = p (ijn) where p(ijn) is the number of partitions of n into distinct parts with i odd-indexed odd parts and j even-indexed odd parts and p (ijn) is the number of partitions of n into distinct parts with i parts that are congruent to modulo 4 and j parts that are congruent to 3 modulo 4 Date: April Mathematics Subject Classification 05A5 05A7 05A9 B34 B37 B75 P8 P83 33D5 Key words and phrases Partitions with parity restrictions and bounds; partition identities; -series; BG-ran; alternating sum of parts; Rogers Szegő polynomials; (little) Göllnitz identity

2 2 ALEXANDER BERKOVICH AND ALI KEMAL UNCU We can demonstrate Theorem for special choices i = j = and n = 4 in Table 2 Table 2 p(4) and p (4) with respective partitions for Theorem p(4) = 0 : p (4) = 0 : (3) (3) (03) (95) (932) (85) (752) (742) (653) (643) (2) (03) (932) (832) (76) (752) (742) (653) (643) (5432) Let P(ij) be the generating function for p(ijn): P(ij) = n 0p(ijn) n Let L n m be non-negative integers We will use standard notations in [4] and [2] L (a) L := (a;) L := ( a n ) n=0 (a a 2 a ;) L := (a ;) L (a 2 ;) L (a ;) L (a;) := lim L (a;) L We define the Gaussian binomial (-binomial) and -trinomial coefficients respectively as { () := () n() n for n 0 n 0 otherwise n () := = for n+m and nm 0 n m n m () n () m () n m Next we are going to introduce basic -hypergeometric series [2] Let r and s be non-negative integers and a a 2 a r b b 2 b s and z be variables Basic -hypergeometric series is defined as () ( ) a a 2 a r rφ s ;z := b b 2 b s Using techniues in [4] it is clear that n=0 (a a 2 a r ;) [ n ( ) n 2) ] r+s (n z n (b b s ;) n 22 +( ) µ ( 4 ; 4 ) is the generating function for the number of partitions with distinct parts congruent to 2+( ) µ modulo 4 where µ {0} Therefore Theorem amounts to the identity (2) P(ij) = ( 2 ; 2 2i2 i 2j 2 +j ) ( 4 ; 4 ) i ( 4 ; 4 ) j for a variable with < Euation (2) is going to be discussed further in the next section The BG-ran of a partition π denoted BG(π) is defined as BG(π) := i j where i is the number of odd-indexed odd parts and j is the number of even-indexed odd parts Another euivalent representation of BG-ran of a partition π comes from 2-residue diagrams One can show that BG(π) = r 0 r

3 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 3 where r 0 is the number of 0 s in the 2-residue diagram of π and r is the number of s in the 2-residue diagram BG-ran of the partition π = (20752) in the example of Table is eual to 0 In Section 2 we present and prove explicit formulas for the generating functions for the number of partitions with fixed number of odd-indexed and even-indexed odd parts We end Section 2 by discussing a new partition theorem which extends the worof Savageand Sills in [5] In Section 3 we examine various partitions with a bound on the largest part and fixed values of BG-ran Section 4 deals with some -theoretic implications of Rogers Szegő polynomials In Section 5 we recall the wor of Boulet [8] Ishiawa and Zeng [4] and provide combinatorial interpretation for the identity for the Rogers Szegő polynomials in [7] We conclude this section with a new partition theorem which utilizes alternating sum of parts statistics In Section 6 we extend the wor of Boulet There we derive new formulas for generating functions for weighted partitions with bounds on the number of parts and the size of parts We conclude with an outloo 2 Generating functions and Recurrence relations We start by generalizing our earlier definitions Let p N (ijn) be the number of partitions of n counted by p(ijn) with the extra condition that parts of the partitions that are being counted are less than or eual to N Let P N (ij) be the generating function for p N (ijn): P N (ij) := n 0p N (ijn) n The relation between P N (ij) and P(ij) in (2) is P(ij) = lim N P N(ij) The following theorem gives explicit formulas for the generating functions P N (ij) Theorem 2 Let N i and j be non-negative integers and be a variable then P 2N (ij) = 2i2 i+2j 2 N (2) +j ( 2 ; 2 ) N i j i j (22) P 2N+ (ij) = 2i2 i+2j 2 +j ( 2 ; 2 ) N i j+ [ N + i j 4 ]4 2(N+i j+) 4(N+) We prove this assertion using recurrence relations Using the definitions of P N (ij) recurrence relations are easily attained by extracting the largest parts of partitions counted by these generating functions Lemma 22 Let N i and j be non-negative integers ν {0} Then (23) P 2N+ν (ij) = P 2N+ν (ij)+ 2N+ν χ(i ν)p 2N+ν (ji ν) where N ν The initial conditions are where P 0 (ij) = δ i0 δ j0 { if statement is true χ(statement) := 0 otherwise and δ ij := χ(i = j) the Kronecer delta function We remar that similar generating functions and recurrence relation were discussed in [6] Proof Let ν {0} π = (λ λ 2 λ ) be a partition counted by p 2N+ν (ijn) for some If λ < 2N +ν then π is also counted by p 2N+ν (ij) If λ = 2N +ν then by extracting λ we get a new partition π = (λ 2 λ 3 λ ) into distinct parts with largest part 2N +ν If ν = 0 then the number of odd parts stay the same but the indexing of those parts switch If ν = then on top of the change of parities of odd numbers the number of odd parts in π is one less than the number

4 4 ALEXANDER BERKOVICH AND ALI KEMAL UNCU of odd parts in π Therefore π is a partition that is counted by p 2N+ν (ji ν) This proves the lemma We need to show that the right-hand sides of (2) and (22) satisfy the same recurrence relations of Lemma 22 to prove Theorem 2 The right-hand side of (2) can be rewritten for this purpose: 2i2 i+2j 2 N +j ( 2 ; 2 ) N i j i j 4 = 2i2 i+2j 2 N +j ( 2 ; 2 ) N i j i j 4N = 2i2 i+2j 2 +j ( 2 ; 2 ) N i j [ N i j 2(N+i j) +2(N+i j) ( 2(N i+j) ) 4 ]4 2(N+i j) 4N [ + 2i2 +i+2j 2 N ]4 j+2n ( 2 ; 2 2(N i+j) ) N i j i j 4N This proves that the right-hand side (2) satisfies the recurrence relation of Lemma 22 for ν = 0 Recurrence relation of (22) can be shown in the same manner Moreover the initial condition of Lemma 22 is obviously true for the right-hand side of (2) and (22) This finishes the proof of Theorem 2 Three comments will be noted here First P 2N+ (ij) is a polynomial in for all choices of N i and j from is definition It is also easy to see that the right-hand side of the (22) needs to be a polynomial by combining (2) and (23) with ν = Yet in its current form the right-hand side of (22) comes with a non-trivial rational term with no obvious cancellation This is going to be addressed and a new representation will be given in Section 4 Secondly N in either line of Theorem 2 proves the identity (2) Hence Theorem follows Lastly let be a fixed non-negative integer Taing the limit N in (2) and/or (22) setting j = (i = ) summing over i (j) and finally using -binomial theorem [2 (II3)] we get Theorem 23 Theorem 23 Let be a fixed non-negative integer then P(i) = ( 2 ; i2 i ) ( 4 ; 4 ) ( 4 ; 4 = ( 2 ; 2 ) ( ; (24) ) ) i ( 4 ; 4 ) (25) i 0 j 0 i 0 P(j) = ( 2 ; j2 +j ) ( 4 ; 4 ) ( 4 ; 4 = ( 2 ; 2 ) ( 3 ; ) ) j ( 4 ; 4 ) j 0 Next we can compare combinatorial interpretations of extremes of (24) and (25) The sum on the left-hand side of the identity (24) (or (25)) gives us the generating function for number of partitions into distinct parts with even-indexed (or odd-indexed) odd parts On the right-hand side of (24) (or (25)) we have the generating function for the number of partitions into distinct parts with exactly parts congruent to (or 3) modulo 4 We rewrite these interpretations together Theorem 24 For a fixed non-zero integer the number of partitions of n into distinct parts where there are odd-indexed (even-indexed) odd parts is eual to the number of partitions of n into distinct parts where there are exactly parts congruent to (3) modulo 4 Special case = 0 in the Theorem 23 and Theorem 24 is well nown in literature Setting = 0 we can rewrite products in (24) and (25) as (26) and (27) ( 2 ; 2 ) ( ; 4 ) = ( ;4 ) (; 4 ) ( 2 ; 4 ) (; 4 ) = ( 2 ; 2 ) ( 3 ; 4 ) = ( ; 8 ) ( 2 ; 8 ) ( ; 8 ) = ( 5 6 ; 8 )

5 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 5 where we use the Euler Theorem Far right products of (26) and (27) appear in the little Göllnitz identities [3] Little Göllnitz identities have the combinatorial counterparts Theorem 25 (Göllnitz) The number of partitions of n into parts (greater than ) differing by at least 2 and no consecutive odd parts appear in the partitions is eual to the number of partitions of n into parts congruent to 5 or 6 modulo 8 (2 3 or 7 modulo 8) Comparing Theorem 24 with = 0 and (26) (27) we arrive at the new little Göllnitz theorems of Savage and Sills [5] Theorem 26 (Savage Sills) The number of partitions of n into distinct parts where odd-indexed (even-indexed) parts are even is eual to the number of partitions of n into parts congruent to 2 3 or 7 modulo 8 ( 5 or 6 modulo 8) An interested reader is invited to examine [] for another companion to the Göllnitz identities 3 Partitions with fixed value of BG-ran The explicit generating function formulas for P N (ij) given in Theorem 2 opens the door to various applications and interesting interpretations We can start by setting i = j + for any integer P N (j + j) is the generating function for number of partitions into distinct parts N with j even-indexed odd parts and BG-ran eual to By summing these functions over j we lift the restriction on the even-indexed odd parts Let B N () denote the generating function for number of partitions into distinct parts less than or eual to N with BG-ran eual to Then (3) B N () := j 0P N (j +j) We have a new combinatorial result Theorem 3 Let N and j be non-negative integers be any integer Then (32) B 2N+ν () = 22 2N +ν N + where ν {0} Proof This identity is a conseuence of the -Gauss identity [2 (II8)] We can outline the proof as follows Using the definition of P 2N+ν (j + j) (I0) (I25) in [2] as needed we come to the following B 2N+ν () = 22 (4 ; 4 ) N ( ν 2(N++) ) ( 4 ; 4 ) ( 2 ; 2 ) N +ν 2 ( 2(N +ν) 2(N +ν ) 2φ 4(+) ; 4 4(N+ν)+2 Applying the -Gauss identity and rewriting infinite products in a non-trivial fashion using (I5) in [2] repeatedly yields B 2N+ν () = 22 2N +ν N + 2 Theorem 3 can also be used to prove the similar result for partitions (not necessarily in distinct parts) with the same type of bounds on the largest part and fixed BG-ran Let B N () be the generating function for number of partitions into parts less than or eual to N with BG-ran euals This terminology was introduced by Alladi [] It is used to distinguish between (little) Göllnitz and (big) Göllnitz partition theorems )

6 6 ALEXANDER BERKOVICH AND ALI KEMAL UNCU Theorem 32 Let N be a non-negative integer be any integer Then where ν {0} B 2N+ν () = 22 ( 2 ; 2 ) N+ ( 2 ; 2 ) N +ν The proof of Theorem 32 comes from the combinatorial bijection of extracting doubly repeating parts to partitions Let U N be the set of partitions with parts less than or eual to N and BG-ran eual to Let D N be the set of partitions into distinct parts less than or eual to N with BG-ran being eual to Let E N be the set of partitions with parts less than or eual to N whose parts appear an even number of times Define bijection ρ : U N D N E N where ρ N (π) = (π π ) where this bijection is established as the row extraction of two parts of same size at once from a given partition repeatedly until there are no more repeating parts in the outcome partition π The extracted parts are collected in the partition π Note that the number of odd parts in π and π might be different BG(π) = BG(π ) Table 3 is an example of this map with π = ( ) so ρ N (π) = (π π ) = ((752)(5544)) with BG(π) = BG(π ) = 0 for any N 7 and = 0 Table 3 ρ N (( )) = ((752)(5544)) The generating function for number of partitions from the set E 2N+ν where all parts appear an even number of times and are less than or eual to 2N +ν is ( 2 ; 2 ) 2N+ν where ν {0} Therefore eeping the bijection ρ 2N+ν above in mind the generating function for number of partitions with BG-ran eual to and the bound on the largest part being 2N +ν is the product (33) B2N+ν () = B 2N+ν() ( 2 ; 2 ) 2N+ν proving Theorem 32 The combinatorial interpretation coming from (32) was previously unnown Recall that the expression 22 2N +ν N + 2 is the generating function for number of partitions with BG-ran eual to and the largest part 2N +ν where ν {0} This relation gives a combinatorial explanation of a well nown identity: Theorem 33 Let ν {0} and let N +ν be a positive integer Then N+ν (34) 22 2N +ν = ( ;) 2N+ν N + 2 = N

7 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 7 It is clear that summing B N () defined in (3) over all possible BG-rans yield the generating function for the number of partitions into distinct parts N yielding (34) Identity (34) was discussed in [3 42] by Andrews He showed that (34) is euivalent to an identity for Rogers Szegő polynomials Recall that the Rogers Szegő polynomials are defined as N (35) H N (z) := z l l Then we have the identity [3] (36) l=0 H 2N+ν ( 2 ) = ( ;) 2N+ν where ν {0} In oder to show the euivalence of (34) and (36) we use n+m n+m (37) = nm n n where n and m are positive integers In (34) first we let Then we change the -binomial term using (37) on the left-hand side and rewrite the right-hand side as ( /;/) 2N+ν = (N+ν)(2N+) ( ;) 2N+ν Multiplying both sides with (N+ν)(2N+) and changing the summation variable in the euation (34) with N +( ) ν +ν In ν = case in addition we change the order of summation In this way we arrive at (36) Theorem 32 and Theorem 33 is enough to prove the following combinatorially anticipated corollary Corollary 34 Let N be a non-negative integer Then (38) where ν {0} N+ν = N 22 ( 2 ; 2 ) N+ ( 2 ; 2 = ) N +ν (;) 2N+ν 4 Some Implications of the formula for Rogers Szegő polynomials Let t and z be variables It is obvious that the double sum (4) P N (ij)t i z j ij 0 is the generating function for the number of partitions into distinct parts where exponents of t and z eep trac of the odd-indexed and the even-indexed odd parts respectively We would lie to show that the double sum (4) can be written as a single-fold sum Theorem 4 Let N be a non-negative integer Then 2N+ν (ij)t ij 0P i z j = ( t; 4 ) N i+ν ( z; 4 ) i 2i i i 0 4 where ν {0} Proving this theorem relies on the following new identity Theorem 42 For non-negative integers N i j and a variable N [ N l+ν N l N +ν ]2 (42) N l j ν N+i j+ = ( ;) N i j+ν l i j i j ν 2(N+) 2 l=0 where ν {0} 2 2

8 8 ALEXANDER BERKOVICH AND ALI KEMAL UNCU We prove these identities by appeal to Rogers Szegő polynomial summation formula (36) We divide both sides of (42) with +ν i j ν 2(N+) 2 and simplify the factorials Using H N i j+ν ( 2 ) = ( ;) N i j+ν from (36) for ν {0} we get the final result The identity (42) with ν = provides an explicit polynomial formula for (22) Furthermore (42) is enough to demonstrate Theorem 4 Let t i z j denote the coefficient of t i z j term of the series this notation precedes Proof (Theorem 4) Use the -binomial theorem [2 (II3)] to explicitly write the powers of t and z Extracting the coefficient of t i z j we get t i z j l 0 ( t; 4 ) N l+ν ( z; 4 ) l 2l = ω(ij) l 4 N l+ν l i 4 l=0 4 l j 4 2(N l j) where ω(ij) = 2i 2 i+2j 2 +j and ν {0} This gives us the result by comparing (42) with 2 4 and eualities (2) and (22) Theorem 4 yields interesting corollaries Setting (tz) = (0) and (0) gives us the new -series identities of Corollary 43 Corollary 43 Let N and be non-negative integers Then +ν ( 2 ; 2 ) N +ν ν2(n +) = (43) ( ; 4 ) 0 ν 4(N+) ν ( 2 ; 2 ) N +ν 2 2 ν2(n++) = (44) ( ; 4 ) 0 ν 4(N+) N +ν where ν {0} It is amusing to observe that there is no dependence on ν on the right-hand side of (43) With a later theorem in Section 6 we can also prove the following Theorem 44 Let N be a non-negative integer then (45) ( ; 4 ) 2 = ( 3 ; 4 ) N This theorem is easily proven by choosing (abcd) = (t/tz/z) in (64) and setting (tz) = (0) Notethat theright-handsidesof (44)and(45)areverysimilar Infact thechoice(abcd) = (t/tz/z) and then (tz) = (0) in (64) yields the right-hand side of (44) Johann Cigler brought to our attention that (45) is an easy corollary of a more general identity Theorem 45 Let N be a non-negative integer then z 0 (y;) = [ N 0 ] (yz;) N z This theorem can easily be proven by appeal to -binomial theorem [2 II3]

9 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 9 5 Partitions with Boulet-Stanley weights In [6] Stanley made a suggestion for suitable weights to be used on diagrams Boulet in [8] extensively utilized this suggestion on what we will denote a four-variable decoration of Ferrers diagram of a partition We can decorate any Ferrers diagram of a partition with variables a b c and d We fill the boxes on the odd-indexed rows with alternating variables a and b starting from a and boxes on the even-indexed rows filled with alternating variables c and d starting from c One can define a weight on these four-variable decorated diagrams as ω π (abcd) = a #a b #b c #c d #d where #a denotes the number of boxes decorated with variable a in partition π s four-variable decorated diagram etc One example of a four-variable decoration and the weight is given in Table 4 Table 4 Four-variable decorated Ferrers diagram of the partition π = ( ) a b a b a b a b a b a b c d c d c d c d c d a b a b a b a c d c d c a b ω π (abcd) = a b 0 c 8 d 7 The generating function for the weighted count of Ferrers diagrams of partitions from a set S with weight ω π (abcd) is ω π (abcd) π S Let ρ N be the obvious bijective map U N D N E N Let U N be the set of partitions with parts less than or eual to N and D N be the set of partitions into distinct parts with parts less than or eual to N Recall that E N is the set of partitions into parts less than or eual to N where every part repeats an even number of times Let ρ : U D E where ρ U D and E are the same sets as above where we remove the bound on the largest part N In [8] Boulet proved identities for the generating functions for weighted count of partitions with four-decorated Ferrers diagrams from the sets D and U Theorem 5 (Boulet) For variables a b c and d and := abcd we have (5) (52) Ψ(abcd) := π D Φ(abcd) := π U ω π (abcd) = ( a abc;) (ab;) ω π (abcd) = ( a abc;) (abac;) The transition from (52) to (5) can be done with the aid of the bijective map ρ The generating function for the weighted count of four-variable Ferrers diagrams which exclusively have an even number of rows of the same length is (ac;) Hence we get (53) Φ(abcd) = Ψ(abcd) (ac;) which is similar to (33)

10 0 ALEXANDER BERKOVICH AND ALI KEMAL UNCU It should be noted that these generating functions are consistent with the ordinary generating functions for number of partitions when all variables are selected to be For example Define the generating functions (54) (55) Φ() = Ψ N (abcd) := (;) ω π (abcd) π D N Φ N (abcd) := ω π (abcd) π U N which are finite analogues of Boulet s generating functions for the weighted count of four-variable decorated Ferrers diagrams In [4] Ishiawa and Zeng write explicit formulas for (54) and (55) Theorem 52 (Ishiawa Zeng) For a non-zero integer N variables a b c and d we have N (56) Ψ 2N+ν (abcd) = ( a;) N i+ν ( c;) i (ab) i i i=0 N (57) Φ 2N+ν (abcd) = ( a;) N i+ν ( c;) i (ab) i (ac;) N+ν (;) N i where ν {0} and = abcd We would lie to point out the special case of (57) with (abcd) = (zy y/z z/y /zy) was first discovered and proven by Andrews in [2] We note in passing that (56) together with the -binomial theorem [2 II3] yields Theorem 53 For ν {0} where = abcd N 0 i=0 x N (;) N Ψ 2N+ν (abcd) = (+aν) ( axν ;) ( abcx;) (x;) (abx;) Similar to (53) the connection between (56) and (57) can be obtained by means of the bijection ρ N In this way we have (58) Φ 2N+ν (abcd) = Ψ 2N+ν(abcd) (ac;) N+ν (;) N where N is a non-negative integer and ν {0} We can now rewrite Theorem 4 as N (59) Ψ 2N+ν (t/tz/z) = ( t; 4 ) N i+ν ( z; 4 ) i 2i i 4 i=0 This may be viewed as a special case of Theorem 52 The sum similar to that on the right-hand side of (56) was seen in the literature before In fact Berovich and Warnaar [7] utilized that sum in their identity for the Rogers Szegő polynomials Theorem 54 (Berovich Warnaar) Let N be a non-negative integer Then the Rogers Szegő polynomials can be expressed as N (50) H 2N+ν (z 2 ) = ( z; 4 ) N l+ν ( /z; 4 ) l (z) 2l l 4 for ν {0} l=0

11 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS We would lie to mention that recently Cigler provided a new proof of (50) in [] Let π = (λ λ 2 λ ) be a partition We want to define the weights π = λ + λ λ called the norm of the partition π and γ(π) = λ λ 2 +λ 3 λ 4 + +( ) + λ the alternating sum of parts of the partition π We see that the right-hand sides of (56) and (50) coincide with the particular choice (abcd) = (zz/z/z) in (56) (5) H N (z 2 ) = Ψ N (zz/z/z) Obviously Ψ N (zz/z/z) is the generalting function for the number of partitions into distinct parts N where exponent of z is the alternating sum of the parts of partition That is Ψ N (zz/z/z)= π D N π z γ(π) Therefore extraction of the coefficient of z would give us the generating function for the number of partitions with the alternating sums of parts eual to Using definition for H N (z 2 ) given in (35) one can easily extract the coefficient of z for a fixed non-negative integer This way we get ] π (52) [ N This can be interpreted as 2 = π D N γ(π) = Theorem 55 Let N n and be non-negative integers The number of partitions of n into solely odd parts 2N 2 + is eual to the number of partitions of n into distinct parts N with the alternating sum of parts being eual to This result can be explained with the aid of the Sylvester bijection as in Bressoud s boo ([9] p52 ex 225) Thus Theorem 54 may be interpreted as an analytical proof of Theorem 55 Moreover the connection (58) used in (52) yields (53) z Φ N (zz/z/z)= ( 2 ; 2 ) ( 2 ; 2 ) N where z Φ N (zz/z/z) denotes the coefficient of z term in the power series expansion of Φ N (zz/z/z) This leads us to a new combinatorial theorem Theorem 56 Let N n and be non-negative integers Then A N (n) = B N (n) where A N (n) is the number of partitions of n into no more than N parts where exactly parts are odd and B N (n) is the number of partitions of n into parts N where alternating sum of parts is eual to We illustrate Theorem 56 in Table 5 Table 5 A 3 (02) and B 3 (02) with respective partitions for Theorem 56 A 3 (02) = 9 : B 3 (02) = 9 : (9) (8) (73) (72) (63) (55) (54) (532) (433) (333) (332) (3222) (322) (32) (3) (22222) (222) (2)

12 2 ALEXANDER BERKOVICH AND ALI KEMAL UNCU It is clear that the partitions counted by A N (n) and B N (n) are conjugates of each other This follows easily from the observation that the number of odd parts in a partition turns into the alternating sum of parts in the conjugate of this partition 6 Further Observations In this section we will extend Boulet s approach to the weighted partitions with bounds on the number of parts and largest parts Theorem 5 Let U N be defined as in Section 3 as the set of partitions with largest part less than or eual to N Let D N be the set of partitions into parts N where the difference between an odd-indexed part and the following non-zero even-indexed part is Let ẼN be the set of partitions into parts N where the Ferrers diagrams of these partitions exclusively have odd-height columns and every present column size repeats an even number of times Define D and Ẽ similar to the sets D N and Ẽ N where we remove the restriction on the largest part Let ρ : U D Ẽ (ρ N : U N D N ẼN) be the similar map to ρ (ρ N ) Let π be a partition in U N The image ρ N ( π) = ( π π ) is obtained by extracting even number of odd height columns from π s Ferrers diagram repeatedly until there are no more repetitions of odd height columns in the Ferrers diagram of π We put these extracted columns in π and the partition that is left after extraction is π An example of this map is ρ ((074)) = ( π π ) = ((43)(644)) as demonstrated in Table 6 Table 6 ρ ((074)) = ((43)(644)) With the definition of the bijection ρ N we can finitize Boulet s combinatorial approach Let π be a fixed partition with largest part less than or eual to 2N + ν for a non-negative integer N and ν {0} We loo at ρ 2N+ν ( π) = ( π π ) Here π is a partition in D 2(N )+ν with the specified difference conditions In this construction is half the number of parts in π which is a partition in E 2 The generating function for the weighted count of four-variable decorated Ferrers diagrams of such a partition π is (6) ( a;) N +ν ( abc;) N (ac;) N +ν (;) N Similarly the generating function for the weighted count of four-variable decorated Ferrers diagrams of such π is (ab) (62) (;) Hence for ν {0} the generating function Φ 2N+ν (abcd) for the weighted count of partitions with parts less than or eual to 2N +ν is the sum over of the product of two functions in (6) and (62) In this way we arrive at Theorem 6 For a non-zero integer N variables a b c d and = abcd we have N ( a;) i+ν ( abc;) i (63) Φ 2N+ν (abcd) = (ab) N i (;) N i i=0 (ac;) i+ν N (ac;) N+ν (64) Ψ 2N+ν (abcd) = ( a;) i+ν ( abc;) i (ab) N i i (ac;) i+ν i=0

13 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 3 where ν {0} We remar that we use (58) to derive (64) Observe that Theorem 6 is a perfect companion to Theorem 52 However our derivation of Theorem 6 unlie Theorem 52 is completely combinatorial Next we rewrite (56) and (64) using hypergeometric notations as (65) and (66) Ψ 2N+ν (abcd) = (ab) N ( c;) N (+νa) 2 φ ( N a ν N c Ψ 2N+ν (abcd) = ( a 2 b) N Nν (+νa) ( cbd N ν ;) N ( N c ;) N 3φ ( N a ν abc ac ν ) ; d ) ; N ab Comparing (65) and (66) we get ( N a ν ) ( (67) 2 φ ; d = ( a ν ) N(bd N ν ;) N N a ν ) abc 3φ N c ( N c ;) N ac ν ; N ab It is easy to chec that (67) is nothing else but [2 (III8)] with the choice of variables b a ν c N /c and z d for ν {0} We can further generalize (63) of Theorem 6 by putting bounds on the number of parts in the given partitions Let Φ NM (abcd) be the generating function of partitions with weights where every part is less than or eual to N and the number of non-zero parts is less than or eual to M Theorem 62 Let ν {0} and 2N +ν M be positive integers then N l+m l Φ 2N+ν2M (abcd) = (ab) N l (abc) m2 (m 2 2 ) l N l (68) l+ν m =0 l=0 [ a m (m 2 ) l+ν m ] M m m 2 n=0 m 2=0 M +l n m m 2 M n m m 2 m 2 ( l+ν ;) n (;) n (ac) n Proof of Theorem 62 utilizes the same maps ρ N and ρ N as in the proof of Theorem 6 To account for the bounds on the number of parts the previously used generating functions are being replaced with appropriate -binomial coefficients Summing over the possibilities as we did in the proof of Theorem 6 yields Theorem 62 For example we replace product in (62) with M + (ab) This is the generating function for the number of partitions into even parts such that the total number of parts is odd and 2M with the largest part being exactly 2 where we count these partitions with Boulet-Stanley weights Fixing (abcd) = () we get +2M (69) Φ N2M () = 2M So it maes sense to view the sum in (68) as a four-parameter generalization of the -binomial coefficient (69) We can combinatorially see that Φ 2N+ν2M (abcd) satisfies similar recurrence relations to - binomial coefficients Using these recurrences we can write the Φ NM (abcd) for an odd M We have the relations (60) Φ 2N+ν2M+ (abcd) = Φ 2N+ν2(M+)(cdab) Φ 2N +ν2(m+) (cdab) c ν (cd) N

14 4 ALEXANDER BERKOVICH AND ALI KEMAL UNCU where N M are non-negative integers and ν {0} This gives us the full list of possibilities for the bounds of Φ NM (abcd) Yee in [7] wrote a generating function for some weighted count of partitions with bounds on the largest part and the number of parts We can also remove Yee s restrictions on weights In other words Yee s combinatorial study can be generalized to deal with the four-variable decorated Ferrers diagrams Theorem 63 Let 2N +ν and 2M +µ be positive integers Then M + +ν N Φ 2N+ν2M+µ (abcd) = (ac) (ab) N j =0 j=0 ( j (+νµa)a m (m 2 )+νµm M +µ ν (6) m =0 ( N j m 2=0 [ c m2 (m 2 2 ) M m 2 for ν µ {0} where (νµ) (0) and = abcd ] m ( M +µ ;) N j m2 (;) N j m2 ) M +j m j m Theorem 62 and Theorem 63 give different expressions for the same function Also note that the idea behind (60) can be applied to Theorem 63 for the missing combination of bounds This is left as an exercise for the interested reader Another important result comes from nowing the combinatorial generating function interpretations of the function Ψ N (abcd) By looing at different choices of the variables we can find -series identities that were not combinatorially interpreted before One finding of this type is a -hypergeometric identity of Berovich and Warnaar [7 (330)] and it s analogue For ν {0} it is obvious that Ψ 2N+ν (a/aa/a) is the generating function of partitions into distinct parts less than or eual to 2N +ν where the exponent of a counts the number of odd parts in the partitions Clearly (62) Ψ 2N+ν (a/aa/a) = ( a; 2 ) N+ν ( 2 ; 2 ) N Using (56) yields ( (63) Ψ 2N+ν (a/aa/a) = ( a; 4 4N ) a ) N+ν 2 φ (a) 4N ν;4 +4δν0 /a Comparing (62) and (63) gives Theorem 64 For a non-negative integer N and variables a and ( 4N ) a 2φ (a) 4(N δν0);4 +4δν0 /a = ( 2 ; 2 ) N ( a 3 ; 2 ) N +ν ( a 5 ; 4 ) N +ν where ν {0} and δ ij is the Kronecer delta function The case ν = with N n a a/ and 2 in Theorem 64 is [7 (330)] and ν = 0 is the easy analogue we get using (62) 7 Outloo Theorems 3 and 32 call for combinatorial proofs which we will discuss elsewhere We have found the doubly bounded versions of Theorem 3 and Theorem 55 Let N and M be non-negative integers Note that Φ NM (t/t/tt) is the generating function for the number of partitions into parts N and number of parts M where the exponent of t counts the BG-ran of counted partitions Recall that Φ NM (zz/z/z) is the generating function for the number of partitions into parts N and number of parts M where the exponent of z is the alternating sum of parts of counted partitions )

15 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 5 Proposition 7 Let ν µ {0} and 2N +ν 2M +µ be positive integers then N+ν (7) Φ 2N+ν2M+µ (t/t/tt) = t j 2j2 j N +M +µ N +ν +M N +j M +j j= N We remar that the products of -binomial coefficients that are similar to those on the right-hand side of (7) were studied by Burge in [0] Proposition 72 Let µ {0} and N 2M +µ be positive integers then N M +µ+j M +N j Φ N2M+µ (zz/z/z)= z j j j M 2 j=0 Both of these results compared with either (68) or (6) with their appropriate choices of variables give us new -series identities relating single-fold sums and four-fold sums These identities will be discussed elsewhere ItiseasytochecthatM inthepropositions7and72implies(38)and(53) respectively Proposition 7 is symmetric with respect to its bounds on the number of parts and largest parts which is expected because the BG-ran of a partition is invariant under conjugation Hence as N we see that Proposition 7 implies (38) as in the case of M In Proposition 72 by extracting the coefficient of z we get a new partition identity Proposition 73 Let µ {0} N and 2M + µ positive integers and n and be non-negative integers Then A 2M+µN (n) = B N2M+µ (n) where A 2M+µN (n) is the number of partitions into parts 2M + µ with no more than N parts where exactly parts are odd and B N2M+µ (n) is the number of partitions of n into parts N with no more than 2M +µ parts with alternating sum of parts eual to We illustrate this result in Table 7 Table 7 A 53 (02) and B 35 (02) with respective partitions for Proposition 73 A 53 (02) = 4 : (55) (54) (532) (433) B 35 (02) = 4 : (333) (332) (3222) (22222) The set of partitions that appears in Table 7 is a subset of the ones in Table 5 Proposition 73 is a generalized version of Theorem 56 with the extra bound on the number of parts There is an obvious connection between notations of Theorem 56 and Proposition 73 A MN (n) A N (n) (B NM (n) B N (n)) as M Similar to the case of Theorem 56 the partitions counted by A N2M+µ (n) and B 2M+µN (n) are conjugates of each other The approach used in Section 2 can be generalized to apply for more general generating functions Let P N (ijm) be the generating function for the number of partitions into distinct parts N with i odd-indexed j even-indexed odd parts and m even parts Obviously we have P N (ijm) = P N (ij) m 0 where P N (ij) was defined in Section 2 Authors intend to discuss these more general generating functions elsewhere To this end we already proved that

16 6 ALEXANDER BERKOVICH AND ALI KEMAL UNCU Proposition 74 Let i j m and N be non-negative integers Then [ N +j m+j P 2N+ν (0jm) = j(j+)+m(m+) j( )m+j 2 ] j +m j 2 [ 4 N +i m+i P 2N+ (i0m) = i(i+)+m(m+)+i( )m+i 2 ] i+m i 2 [ 4 N +i m+i P 2N (i0m) = i(i+)+m(m+)+i( )m+i 2 ] i+m i [ 2 ] + i(i+)+m(m )+i( )m+i +2N N +i i+m where ν {0} We illustrate Proposition 74 in Table 8 Table 8 Various choices of (Nijm) in Proposition [ m+i ] 2 i P N (ijm) Expression Relevant partitions 4 P 7 (02) = : (432) (632) (652) (654) = P 7 (0) = : (32) (52) (54) (72) (74) (76) = P 6 (20) = = : (32) (52) (54) (543) We comment that Proposition 74 provides significant refinement of the Savage-Sills generating functions in [5] 8 Acnowledgment Authors would lie to than Krishnaswami Alladi George Andrews Johann Cigler Sinai Robins and Li-Chien Shen for their ind interest and helpful comments Special thans to Fran Patane for careful reading of the manuscript and thoughtful suggestions References [] K Alladi On a Partition Theorem of Göllnitz and uartic Transformations J Number Theory 69(998) [2] G E Andrews On a partition function of Richard Stanley Electron J Combin (2004) no-2 Research paper [3] G E Andrews Partitions and the Gaussian sum The mathematical heritage of C F Gauss World Sci Publ River Edge NJ 99 [4] G E Andrews The theory of partitions Cambridge Mathematical Library Cambridge University Press Cambridge 998 Reprint of the 976 original MR (99c:26) [5] A Berovich and F G Garvan On the Andrews-Stanley refinement of Ramanujan s partition congruence modulo 5 and generalizations Trans Amer Math Soc 358 (2006) no [6] A Berovich and A K Uncu A new Companion to Capparelli s Identities Adv in Appl Math 7 (205)

17 ON PARTITIONS WITH FIXED NUMBER OF EVEN-INDEXED AND ODD-INDEXED ODD PARTS 7 [7] A Berovich and S O Warnaar Positivity preserving transformations for -binomial coefficients Trans Amer Math Soc 357 (2005) no [8] C E Boulet A four-parameter partition identity Ramanujan J 2 (2006) no [9] D M Bressoud Proofs and Confirmations st ed Cambridge: Cambridge University Press 999 [0] W H Burge Restricted partition pairs J Combin Theory Ser A 63 (993) [] J Cigler Some elementary results and conjectures about -Newton binomials preprint 205 [2] G Gasper and M Rahman Basic Hypergeometric Series Vol 96 Cambridge university press 2004 [3] H Göllnitz Partitionen mit Differenzenbedingungen J Reine Angew Math 225 (967) [4] M Ishiawa and J Zeng The Andrews-Stanley partition function and Al-Salam-Chihara polynomials Discrete Math 309 (2009) no 5-75 [5] C D Savage and A V Sills On an identity of Gessel and Stanton and the new little Göllnitz identities Adv in Appl Math 46 (20) no [6] R P Stanley Some remars on sign-balanced and maj-balanced posets Adv in Appl Math 34 (2005) [7] A J Yee On partition functions of Andrews and Stanley J Combin Theory Ser A 07 (2004) no Department of Mathematics University of Florida 358 Little Hall Gainesville FL 326 USA address: alexb@ufledu Department of Mathematics University of Florida 358 Little Hall Gainesville FL 326 USA address: auncu@ufledu