aa r X i v : . [ m a t h . C O ] N ov ON CERTAIN WEIGHTED 7-COLORED PARTITIONS
SHANE CHERN AND DAZHAO TANG
Abstract.
Inspired by Andrews’ 2-colored generalized Frobenius partitions, we considercertain weighted 7-colored partition functions and establish some interesting Ramanujan-type identities and congruences. Moreover, we provide combinatorial interpretations ofsome congruences modulo 5 and 7. Finally, we study the properties of weighted 7-coloredpartitions weighted by the parity of certain partition statistics. Introduction
In his 1984 Memoir of the American Mathematical Society, Andrews [2] introduced the generalized Frobenius partition or simply the
F-partition of n , which is a two-rowed arrayof nonnegative integers (cid:18) a a · · · a r b b · · · b r (cid:19) , (1.1)wherein each row, which is of the same length, is arranged in non-increasing order with n = r + P ri =1 a i + P ri =1 b i . Furthermore, Andrews studied many general classes of F-partitions. One of them is F-partitions whose parts are taken from k copies of the non-negative integers, which is called k -colored F-partitions . Let cφ k ( n ) denote the number of k -colored F-partitions of n . Andrews derived the following generating function for cφ ( n ) . Theorem 1.1 (Eq. (5.17), [2]) . We have ∞ X n =0 cφ ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ . (1.2)Here and in the sequel, we adopt the following customary notations on partitions and q -series: ( a ; q ) ∞ := ∞ Y n =0 (1 − aq n ) , ( a , a , · · · , a n ; q ) ∞ := ( a ; q ) ∞ ( a ; q ) ∞ · · · ( a n ; q ) ∞ , | q | < . In addition, Andrews obtained the following congruence modulo 5 for cφ ( n ) . Date : October 14, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Weighted 7-colored partition; Ramanujan-type congruence; Unified multirank;Vector crank.
Theorem 1.2 (Corollary 10.1, [2]) . For all n ≥ , cφ (5 n + 3) ≡ . (1.3)Moreover, Andrews defined the D-rank of an F-partition (1.1) to be a − b (if the partsare colored, the D-rank is the numerical magnitude of a − b ). He then conjectured [2,Conjecture 11.1] that the D-rank might explain the congruence (1.3) combinatorially for2-colored F-partitions. Unfortunately, this was asserted untrue by Lovejoy [17].According to the generating function (1.2) for cφ ( n ) and the second author’s recent workwith Shishuo Fu [10] involving classical theta functions, in this paper, we study the followingweighted 7-colored partitions w t ( n ) arithmetically as well as combinatorially, given by ∞ X n =0 w t ( n ) q n := ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ . (1.4)In some sense, this partition function can be viewed as a generalization of Andrews’ cφ ( n ) ,as one may notice that w ( n ) = cφ ( n ) for all n ≥ .Similar to cφ ( n ) , there are some Ramanujan-type congruences for w t ( n ) . Below areseveral examples that we will prove in the later sections. When t ≡ , we have w t (3 n + 1) ≡ ,w t (3 n + 2) ≡ . Furthermore, we have w t (5 n + 3) ≡ w t (5 n + 4) ≡ , if t ≡ ,w t (5 n + 4) ≡ , if t ≡ ,w t (5 n + 3) ≡ , if t ≡ . In addition, we get w (7 n + 4) ≡ ,w (11 n + 10) ≡ ,w (24 n + 23) ≡ . Of course, there are more congruences beyond this list. However, these congruences canbe proved by the standard q -series techniques and hence we do not require complicatedtools like modular forms.The rest of this paper is organized as follows. In Sect. 2, we establish some Ramanujan-type identities and congruences for w t ( n ) . We will then consider a unified multirank anda vector crank which can give combinatorial interpretations of certain congruences modulo5 and 7 for w t ( n ) in Sect. 3. In Sect. 4, we study the properties of weighted 7-coloredpartitions weighted by the parity of certain partition statistics. Some Ramanujan-typecongruences and an analog of Euler’s recurrence relation are obtained. In the last section,we conclude with some remarks and questions for further study. EIGHTED 7-COLORED PARTITIONS 3 Ramanujan-type identities and congruences
In this section, we shall prove some Ramanujan-type identities and congruences forweighted 7-colored partitions w t ( n ) .For notational convenience, we denote ( q k ; q k ) ∞ by f k for positive integers k when ma-nipulating q -series. Recall that Ramanujan’s classical theta functions ϕ ( q ) and ψ ( q ) aregiven by ϕ ( q ) := ∞ X n = −∞ q n = f f f , (2.1) ψ ( q ) := ∞ X n =0 q n ( n +1) / = f f . (2.2)It is also known that ϕ ( − q ) = f f . (2.3)Before stating our results, we require the following -dissections. f = f f f f − q f f f , (2.4) f = f f f − q f f f , (2.5) f = f f f + 4 q f f f . (2.6)These follow respectively from the -dissections of ϕ ( − q ) , ϕ ( − q ) , and ϕ ( q ) (cf. [6, p. 40,Entry 25]).We also need the following -dissections of ψ ( q ) and /ϕ ( − q ) , ψ ( q ) = ψ ( q ) (cid:18) x ( q ) + q (cid:19) , (2.7) ϕ ( − q ) = ϕ ( − q ) ϕ ( − q ) (cid:0) qx ( q ) + 4 q x ( q ) (cid:1) , (2.8)where x ( q ) = ( q ; q ) ∞ ( q ; q ) ∞ . (2.9)Here (2.7) comes directly from the Jacobi’s triple product identity. For (2.8), see [4]. Theorem 2.1.
For t ≡ , ∞ X n =0 w t (2 n ) q n = f f f f t/ , (2.10) S. CHERN AND D. TANG ∞ X n =0 w t (2 n + 1) q n = 4 f f f f t/ . (2.11) Proof.
From (2.6) and (1.4), we have ∞ X n =0 w t ( n ) q n = f f f t = f f f f t + 4 q f f f f t . (2.12)Extracting terms involving q n and q n +1 in (2.12) and replacing q by q , one easily obtains(2.10)–(2.11). This completes the proof. (cid:3) This immediately yields
Corollary 2.2.
For t ≡ , w t (2 n + 1) ≡ . Remark 2.3.
We notice that the congruence w (2 n + 1) = cφ (2 n + 1) ≡ wasfirst proved by Andrews [2]. Following the same line of proving Theorem 2.1, one may alsoobtain the 2-dissection of the generating function of w ( n ) . Theorem 2.4.
For t ≡ , ∞ X n =0 w t (3 n ) q n = f f f f f t/ (cid:18) f f f f + 10 q f f f f (cid:19) , (2.13) ∞ X n =0 w t (3 n + 1) q n = 4 f f f f f t/ (cid:18) f f f f + q f f f f (cid:19) , (2.14) ∞ X n =0 w t (3 n + 2) q n = 9 f f f f f t/ . (2.15)As a consequence of Theorem 2.4, we have Corollary 2.5.
For t ≡ , w t (3 n + 1) ≡ ,w t (3 n + 2) ≡ . To prove Theorem 2.4, we need to show the following lemma.
Lemma 2.6.
We have f f = f f f f x ( q ) + 4 qx ( q ) + 9 q + 10 q x ( q ) + 4 q x ( q ) ! . (2.16) Proof.
By Eqs. (2.7) and (2.8), we have f f = ψ ( q ) ϕ ( − q ) EIGHTED 7-COLORED PARTITIONS 5 = ψ ( q ) (cid:18) x ( q ) + q (cid:19) ϕ ( − q ) ϕ ( − q ) (cid:0) qx ( q ) + 4 q x ( q ) (cid:1) = f f f f x ( q ) + 4 qx ( q ) + 9 q + 10 q x ( q ) + 4 q x ( q ) ! . This establishes (2.16). (cid:3)
By Lemma 2.6 and Eq. (2.9), we have the following
Corollary 2.7.
Let ∞ X n =0 a ( n ) q n = f f . Then ∞ X n =0 a (3 n ) q n = f f f f (cid:18) f f f f + 10 q f f f f (cid:19) , ∞ X n =0 a (3 n + 1) q n = 4 f f f f (cid:18) f f f f + q f f f f (cid:19) , ∞ X n =0 a (3 n + 2) q n = 9 f f f f , and hence a (3 n + 1) ≡ ,a (3 n + 2) ≡ . Proof of Theorem 2.4.
We notice that ∞ X n =0 w t ( n ) q n = f f f t . Since t ≡ , Theorem 2.4 is a direct consequence of Corollary 2.7. (cid:3) With the help of Eq. (2.15), we also have
Theorem 2.8.
For all n ≥ , w (24 n + 23) ≡ . (2.17) Proof.
Notice that ∞ X n =0 w t (3 n + 2) q n = 9 f f f f f t/ . Hence to prove (2.17), it suffices to show a (8 n + 7) ≡ , S. CHERN AND D. TANG where ∞ X n =0 a ( n ) q n = f f f f . With the help of (2.4), it follows that, modulo , ∞ X n =0 a ( n ) q n = f f f f ≡ f f = (cid:18) f f f f − q f f f (cid:19) f = f f f f − q f f f . We now extract terms involving q n +1 and replace q by q , then ∞ X n =0 a (2 n + 1) q n ≡ − f f f . Through a similar argument, we have ∞ X n =0 a (4 n + 3) q n ≡ f f , which contains no terms of the form q n +1 . Hence a (4(2 n + 1) + 3) = a (8 n + 7) ≡ . (cid:3) Remark 2.9.
As pointed out by the referee, (2.17) still holds modulo = 729 , which canbe proved via modular forms. However, it is unclear whether there is an elementary proof.We also have some congruences modulo . Theorem 2.10.
For all n ≥ , w t (5 n + 3) ≡ w t (5 n + 4) ≡ , if t ≡ , (2.18) w t (5 n + 4) ≡ , if t ≡ , (2.19) w t (5 n + 3) ≡ , if t ≡ . (2.20) Proof. If t ≡ , we have ∞ X n =0 w t ( n ) q n ≡ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ (mod 5) . From Euler’s pentagonal number theorem [1, p. 11, Corollary 1.7], which tells ( q ; q ) ∞ = ∞ X n = −∞ ( − n q n (3 n +1) / , (2.21)we notice that ( q ; q ) ∞ has no terms in which the power of q is or mod . Hence w t (5 n + 3) ≡ , EIGHTED 7-COLORED PARTITIONS 7 and w t (5 n + 4) ≡ . We next notice that ∞ X n =0 w t ( n ) q n ≡ ( q ; q ) ∞ ( q t ; q t ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ (mod 5) . Recall that the Jacobi’s identity [7, p. 14, Theorem 1.3.9] tells ( q ; q ) ∞ = ∞ X n =0 ( − n (2 n + 1) q n ( n +1) / . (2.22)From (2.21) and (2.22), one readily has ( q ; q ) ∞ = E + E + E , where E i consists of those terms in which the power of q is i modulo , and ( q ; q ) ∞ = J + J + J , where J i consists of those terms in which the power of q is i modulo . Furthermore, wenote that J ≡ , so ( q ; q ) ∞ ≡ J + J (mod 5) . If t ≡ , we observe that ( q t ; q t ) ∞ ≡ J ∗ + J ∗ (mod 5) , where J ∗ i consists of terms in which the power of q is i modulo . It follows that, ( q ; q ) ∞ ( q t ; q t ) ∞ ≡ ( E + E + E )( J ∗ + J ∗ ) (mod 5) , which contains no terms of the form q n +4 . Hence, w t (5 n + 4) ≡ .If t ≡ , we have ( q t ; q t ) ∞ ≡ J ∗ + J ∗ (mod 5) . The rest of the proof is similar. (cid:3)
For w ( n ) , we have the following mod congruence. Theorem 2.11.
For all n ≥ , w (7 n + 4) ≡ . (2.23) Proof.
Notice that ∞ X n =0 w ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ = ( q ; q ) ∞ ( q ; q ) ∞ ≡ ( q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ (mod 7) . Similarly, (2.22) tells ( q ; q ) ∞ ≡ J + J + J (mod 7) , and hence ( q ; q ) ∞ ≡ J ∗ + J ∗ + J ∗ (mod 7) . S. CHERN AND D. TANG
This time J i and J ∗ i consist of terms in which the power of q is i modulo .It follows that, modulo , ( q ; q ) ∞ ( q ; q ) ∞ ≡ ( J + J + J )( J ∗ + J ∗ + J ∗ ) , which contains no terms of the form q n +4 . Hence, w (7 n + 4) ≡ . (cid:3) Finally, we have a mod congruence for w ( n ) . Theorem 2.12.
For all n ≥ , w (11 n + 10) ≡ . (2.24) Proof.
It is easy to see that ∞ X n =0 w ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ ≡ q ; q ) ∞ ( q ; q ) ∞ ( q ; q ) ∞ (mod 11) . Hence it suffices to prove that a (11 n + 10) ≡ , where ∞ X n =0 a ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ . According to Theorem 2 in [9], let p = 11 and ( r, s ) = ( − , , we have a (11 n + 120) = 11 a (cid:16) n (cid:17) , where a ( α ) = 0 is α is not a nonnegative integer. This along with the verification of thefirst several n ’s finish our proof. (cid:3) Combinatorial interpretations
A unified multirank.
In this section, we give combinatorial interpretations of The-orems 2.10 and 2.11. In doing so, we introduce a multirank and a vector crank for weighted7-colored partitions. This multirank (resp. vector crank) function enables us to divide itscorresponding partition set into five (resp. seven) equivalence classes.For a given partition λ , we let ℓ ( λ ) denote the number of parts in λ and σ ( λ ) denote thesum of all parts in λ with the convention ℓ ( λ ) = σ ( λ ) = 0 for empty partition λ of 0. Let P denote the set of all ordinary partitions, and O (resp. DE , DO ) denote the set of allpartitions into odd parts (resp. distinct even parts, distinct odd parts).Let V t = { ( λ , λ , λ , λ , λ , tλ , tλ ) | λ ∈ DE , λ , λ , λ , λ ∈ O , λ , λ ∈ P} . For a vector (weighted 7-colored) partition −→ λ ∈ V t , we define the sum of parts function s ,the weight function wt , and the multirank function r by s ( −→ λ ) = σ ( λ ) + σ ( λ ) + σ ( λ ) + σ ( λ ) + σ ( λ ) + tσ ( λ ) + tσ ( λ ) , (3.1) EIGHTED 7-COLORED PARTITIONS 9 wt ( −→ λ ) = ( − ℓ ( λ ) , (3.2) r ( −→ λ ) = ℓ ( λ ) − ℓ ( λ ) + 2( ℓ ( λ ) − ℓ ( λ )) + 2 ( ℓ ( λ ) − ℓ ( λ )) . (3.3) Remark 3.1.
We note that for the purpose of combinatorially obtaining Ramanujan-typecongruences, there are a number of alternatives that will work equally well as multirank r . For example, we can use ℓ ( λ ) − ℓ ( λ ) + 2( ℓ ( λ ) − ℓ ( λ )) + h ( ℓ ( λ ) − ℓ ( λ )) for h to replace the right hand side of r in (3.3).The weighted count of vector partitions of n with multirank equal to m , denoted by N V t ( m, n ) , is given by N V t ( m, n ) = X −→ λ ∈V t,s ( −→ λ )= nr −→ λ )= m wt ( −→ λ ) . We also define the weighted count of vector partitions of n with multirank congruent to k modulo m by N V t ( k, m, n ) = ∞ X j = −∞ N V t ( jm + k, n ) = X −→ λ ∈V t,s ( −→ λ )= nr −→ λ ) ≡ k (mod m ) wt ( −→ λ ) . Thus we have the following generating function for N V t ( m, n ) : ∞ X m = −∞ ∞ X n =0 N V t ( m, n ) z m q n = ( q ; q ) ∞ ( zq, z − q, z q, z − q ; q ) ∞ ( z q t , z − q t ; q t ) ∞ . (3.4)According to the definition of N V t ( m, n ) , it is nontrivial that N V t ( m, n ) is always non-negative. However, the following corollary of the q -binomial theorem proved by Berkovichand Garvan [5] gives us an affirmative answer. Proposition 3.2. If | q | , | z | < , then ( az ; q ) ∞ ( a ; q ) ∞ ( z ; q ) ∞ = 1( a ; q ) ∞ + ∞ X n =1 z n ( aq n ; q ) ∞ ( q ; q ) n . This immediately yields the following corollary.
Corollary 3.3.
The coefficients N V t ( m, n ) defined in (3.4) are always nonnegative. Theorem 3.4.
The following relations hold for all n ≥ , j ≡ , k ≡ and l ≡ . N V j (0 , , n + 4) = N V j (1 , , n + 4) = · · · = N V j (4 , , n + 4) = w j (5 n + 4)5 , (3.5) N V k (0 , , n + 3) = N V k (1 , , n + 3) = · · · = N V k (4 , , n + 3) = w k (5 n + 3)5 , (3.6) N V l (0 , , n + 3) = N V l (1 , , n + 3) = · · · = N V l (4 , , n + 3) = w l (5 n + 3)5 , (3.7) N V l (0 , , n + 4) = N V l (1 , , n + 4) = · · · = N V l (4 , , n + 4) = w l (5 n + 4)5 . (3.8)To prove Theorem 3.4, the main ingredient is the following modified Jacobi’s tripleproduct identity [12]: ∞ Y n =1 (1 − q n )(1 − zq n )(1 − z − q n ) = ∞ X n =0 ( − n q n ( n +1) / z − n (cid:18) − z n +1 − z (cid:19) . (3.9) Proof of Theorem 3.4.
We will present the proof of the case j ≡ to illustrate themain idea. The proofs of the remaining cases are similar.Putting z = ζ = e πi/ and t = j = 5 s + 1 in (3.4), we see that ∞ X m = −∞ ∞ X n =0 N V j ( m, n ) ζ m q n = ∞ X n =0 4 X i =0 N V j ( i, , n ) ζ i q n = ( q ; q ) ∞ ( ζ q, ζ − q, ζ q, ζ − q ; q ) ∞ ( ζ q s +1 , ζ − q s +1 ; q s +1 ) ∞ = ( q ; q ) ∞ ( ζ q s +1 , ζ − q s +1 , q s +1 ; q s +1 ) ∞ ( q ; q ) ∞ ( q s +1) ; q s +1) ) ∞ = P ∞ m = −∞ P ∞ n =0 ( − m + n q m (3 m − / n ( n +1)(5 s +1) / ζ − n (cid:0) − ζ n +15 (cid:1) ( q ; q ) ∞ ( q s +1) ; q s +1) ) ∞ (1 − ζ ) . Here the last equality relies on (2.21) and (3.9). Since m (3 m − / ≡ , , and n ( n + 1)(5 s + 1) / ≡ , , , it follows that m (3 m − / n ( n + 1)(5 s + 1) / iscongruent to 4 modulo 5 exactly when m ≡ and n ≡ . This meansthat the coefficient of q n +4 in ( − m + n q m (3 m − / n ( n +1)(5 s +1) / ζ − n (cid:0) − ζ n +15 (cid:1) is zero since ζ − n (cid:0) − ζ n +15 (cid:1) = 0 when n ≡ . Thus X i =0 N V j ( i, , n + 4) ζ i = 0 . (3.10)We note that the left hand side of (3.10) is a polynomial in ζ over Z . It follows that N V j ( i, , n + 4) has the same value for all ≤ i ≤ , since the minimal polynomial for ζ over Q is ζ + ζ + ζ + ζ . We therefore establish (3.5) for j ≡ . This ends our proof. (cid:3) EIGHTED 7-COLORED PARTITIONS 11
Table 1.
Multiank for 7-colored partitions w (3) −→ π ( wt ( −→ π ) , r ( −→ π )) −→ π ( wt ( −→ π ) , r ( −→ π ))3 (1 ,
1) 1 + 1 + 1 (1 , (1 , −
1) 1 + 1 + 1 (1 , − (1 ,
2) 1 + 1 + 1 (1 , (1 , −
2) 1 + 1 + 1 (1 , − + 1 ( − ,
1) 1 + 1 + 1 (1 , + 1 ( − , −
1) 1 + 1 + 1 (1 , + 1 ( − ,
2) 1 + 1 + 1 (1 , + 1 ( − , −
2) 1 + 1 + 1 (1 , − + 1 + 1 (1 ,
3) 1 + 1 + 1 (1 , − + 1 + 1 (1 , −
3) 1 + 1 + 1 (1 , − + 1 + 1 (1 ,
6) 1 + 1 + 1 (1 , + 1 + 1 (1 , −
6) 1 + 1 + 1 (1 , − + 1 + 1 (1 ,
1) 1 + 1 + 1 (1 , + 1 + 1 (1 ,
4) 1 + 1 + 1 (1 , − Example 3.5.
In Table 1, we list the multirank r for the total 28 weighted 7-coloredpartitions of 3. Note that the weighted count of multirank divides w (3) = 20 into fiveresidue classes, each with equal size 4.3.2. A vector crank.
For a given partition λ , the crank c ( λ ) of λ is given by [3]: c ( λ ) := ( λ , if n ( λ ) = 0; µ ( λ ) − n ( λ ) , if n ( λ ) > , where n ( λ ) is the number of 1’s in λ , λ is the largest part in λ , and µ ( λ ) is the numberof parts larger than n ( λ ) . By extending the partition set P to a new set P ∗ , in which twoadditional copies of the partition 1 (say ∗ and ∗∗ ) are added, B. Kim [14, 15] obtained ( q ; q ) ∞ ( zq, z − q ; q ) ∞ = X λ ∈P wt ∗ ( λ ) z c ∗ ( λ ) q σ ∗ ( λ ) , where wt ∗ ( λ ) , c ∗ ( λ ) and σ ∗ ( λ ) are defined as follows: wt ∗ ( λ ) := ( , if λ ∈ P , λ = 1 ∗ or λ = 1 ∗∗ ; − , if λ = 1 ,c ∗ ( λ ) := c ( λ ) , if λ ∈ P ;0 , if λ = 1;1 , if λ = 1 ∗ ; − , if λ = 1 ∗∗ , σ ∗ ( λ ) := ( σ ( λ ) , if λ ∈ P ;1 , otherwise . Let W = { ( λ , λ , λ , λ , λ , λ , λ ) | λ ∈ DE , λ , λ , λ , λ ∈ DO , λ , λ ∈ P ∗ } . For a vector partition −→ λ ∈ W , we define the sum of parts function e s , the weight function f wt and the vector crank c by e s ( −→ λ ) = σ ( λ ) + σ ( λ ) + σ ( λ ) + σ ( λ ) + σ ( λ ) + 2 σ ∗ ( λ ) + 2 σ ∗ ( λ ) , f wt ( −→ λ ) = ( − ℓ ( λ ) wt ∗ ( λ ) wt ∗ ( λ ) ,c ( −→ λ ) = ℓ ( λ ) − ℓ ( λ ) + 2 ( ℓ ( λ ) − ℓ ( λ )) + c ∗ ( λ ) + 2 c ∗ ( λ ) . Let M ∗ ( m, n ) denote the number of weighted 7-colored partitions in W ( n ) with vectorcrank equal to m , and M ∗ ( k, m, n ) denote the number of partitions in W ( n ) with vectorcrank congruent to k modulo m . We have ∞ X m = −∞ ∞ X n =0 M ∗ ( m, n ) z m q n = ( q ; q ) ∞ ( zq, z − q, z q, z − q ; q ) ∞ . (3.11) Theorem 3.6.
The following relation holds for all n ≥ , M ∗ (0 , , n + 4) = M ∗ (1 , , n + 4) = · · · = M ∗ (6 , , n + 4) = w (7 n + 4)7 . Proof.
Setting z = ζ = e πi/ in (3.11), we have ∞ X m = −∞ ∞ X n =0 M ∗ ( m, n ) ζ m q n = ( q ; q ) ∞ ( ζ q, ζ − q, ζ q, ζ − q ; q ) ∞ = ( q ; q ) ∞ ( ζ q, ζ − q, q ; q )( ζ q, ζ − q, ζ q, ζ − q, ζ q, ζ − q, q ; q ) ∞ = P ∞ m =0 P ∞ n = −∞ ( − m + n (2 m + 1) q m ( m +1)+ n ( n +1) / (1 − ( ζ ) n +1 )( q ; q ) ∞ (1 − ζ ) ζ n . Here the last equality follows from (2.22) and (3.9) again. The rest of proof is similar toour proof of Theorem 3.4. We therefore omit the details here. (cid:3) Partitions weighted by the parity of multirank and vector crank
In [8], Choi, Kang, and Lovejoy studied arithmetic properties of the ordinary partitionfunction weighted by the parity of the crank. Later, Kim [16] also studied arithmeticproperties of cubic partition pairs weighted by the parity of the crank analog. As an analog
EIGHTED 7-COLORED PARTITIONS 13 of their work, we consider the number of weighted 7-colored partitions w t ( n ) weighted bythe parity of the multirank and vector crank, i.e., ∞ X n =0 c t ( n ) q n := ∞ X n =0 ∞ X m = −∞ ( − m N V t ( m, n ) ! q n = ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ , ∞ X n =0 d ( n ) q n := ∞ X n =0 ∞ X m = −∞ ( − m M ∗ ( m, n ) ! q n = ( q ; q ) ∞ , Interestingly, c t ( n ) satisfies the following equalities and congruences. Theorem 4.1.
For all n ≥ , c t (5 n + 3) = c t (5 n + 4) = 0 , if t ≡ , (4.1) c t (5 n + 4) ≡ , if t ≡ , (4.2) c t (5 n + 3) ≡ , if t ≡ . (4.3) Proof.
Notice that, when t ≡ , ∞ X n =0 c t ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ = ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ = 1( q t ; q t ) ∞ ∞ X n =0 q n ( n +1) , where we use (2.2). Since n ( n + 1) is not congruent to and modulo , (4.1) followsimmediately.When t , we have, modulo , ∞ X n =0 c t ( n ) q n = ( q ; q ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ ≡ ( q ; q ) ∞ ( q t ; q t ) ∞ ( q ; q ) ∞ ( q t ; q t ) ∞ . It follows from (2.2) that ( q ; q ) ∞ ( q ; q ) ∞ = P + P + P , where P i consists of those terms in which the power of q is i modulo . Similar to the proofsof (2.19) and (2.20), one can easily obtain (4.2) and (4.3). This finishes the proof. (cid:3) Remark 4.2.
We first note that (4.1)–(4.3) are interesting as it appears to be very rarethat the original partition function w t ( n ) and its weighted versions (which are weighted bythe parity of its partition statistics) satisfy the same Ramanujan-type congruences. (Here(4.1) can be viewed as mod congruences like (2.18).)On the other hand, one readily sees that c ( n ) = p (cid:0) n (cid:1) where p ( n ) is the number ofpartitions of n and p ( x ) = 0 if x is not an integer. From Ramanujan’s congruences for p ( n ) modulo powers of 5, it is easy to get c (5 α n + λ α ) ≡ α ) , where λ α is the leastpositive reciprocal of 12 modulo α . Interestingly, this family of congruences resemblesSellers’ result [19] for cφ ( n ) : Theorem 4.3 (Corollary 2.12, [18]) . For all n ≥ and α ≥ , cφ (5 α n + λ α ) ≡ α ) , where λ α is the least positive reciprocal of 12 modulo α . Relying on the combinatorial version of Euler’s pentagonal number theorem [1, p. 10,Theorem 1.6], we obtain the following interesting corollary.
Corollary 4.4.
Let M ∗ e ( n ) (resp. M ∗ o ( n ) ) denote the number of weighted 7-colored parti-tions counted by w ( n ) with even (resp. odd) vector crank. Then d ( n ) = M ∗ e ( n ) − M ∗ o ( n ) = ( ( − m , if n = m (3 m ± , otherwise . (4.4)5. Final remarks
At last, we collect several questions here to motivate further investigation.1) Yee [20] provided a combinatorial proof of the generating function for k -colored F-partitions, which was independently established by Garvan [13]. To the best of ourknowledge, there is no combinatorial proof for k -colored F-partition congruences up tonow. In this paper, we provide a multirank that can explain the congruence modulo 5combinatorially for w ( n ) . However, the combinatorial correspondence between w ( n ) and cφ ( n ) is unclear. New ideas are required to emerge for solving these questions.2) As shown in Sect. 3, we are able to combinatorially interpret mod and congruences for w t ( n ) by using a unified multirank or a vector crank. However, for mod congruence(2.24), we cannot find such an interpretation along this line and hence we cry out for acombinatorial proof.3) It is well known that Euler’s pentagonal number theorem has a beautiful combinatorialproof, which was found by Franklin [11] in 1882. Hence it is also interesting to find aFranklin-type proof of the difference between M ∗ e ( n ) and M ∗ o ( n ) in (4.4). Acknowledgement
The authors would like to thank George E. Andrews, Shishuo Fu, Michael D. Hirschhornand Ae Ja Yee for their helpful comments and suggestions that have improved this paper toa great extent. The authors also acknowledge the helpful suggestions made by the referee.The second author was supported by the National Natural Science Foundation of China(No. 11501061).
References [1] G. E. Andrews,
The theory of partitions , Encyclopedia of Mathematics and its Applications, Vol. 2.Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. xiv+255 pp. (Reprinted:Cambridge University Press, London and New York, 1984). 6, 14[2] G. E. Andrews, Generalized Frobenius partitions,
Mem. Amer. Math. Soc. (1984), no. 301, iv+44pp. 1, 2, 4 EIGHTED 7-COLORED PARTITIONS 15 [3] G. E. Andrews and F. G. Garvan, Dyson’s crank of a partition,
Bull. Amer. Math. Soc. (N.S.) (1988), no. 2, 167–171. 11[4] N. D. Baruah and K. K. Ojah, Some congruences deducible from Ramanujan’s cubic continued fraction, Int. J. Number Theory (2011), no. 5, 1331–1343. 3[5] A. Berkovich and F. G. Garvan, K. Saito’s conjecture for nonnegative eta products and analogousresults for other infinite products, J. Number Theory (2008), no. 6, 1731–1748. 9[6] B. C. Berndt,
Ramanujan’s notebooks. Part III , Springer-Verlag, New York, 1991. xiv+510 pp. 3[7] B. C. Berndt,
Number theory in the spirit of Ramanujan , Student Mathematical Library, . AmericanMathematical Society, Providence, RI, 2006. xx+187 pp. 7[8] D. Choi, S.-Y. Kang, and J. Lovejoy, Partitions weighted by the parity of the crank, J. Combin. TheorySer. A (2009), no. 5, 1034–1046. 12[9] S. Cooper, M. D. Hirschhorn, and R. Lewis, Powers of Euler’s product and related identities,
Ramanu-jan J. (2000), no. 2, 137–155. 8[10] S. Fu and D. Tang, Multiranks and classical theta functions, Int. J. Number Theory , in press. 2[11] F. Franklin, Sur le développement du produit infini (1 − x )(1 − x )(1 − x )(1 − x ) ... , Comptes rendus (1881), 448–450. 14[12] F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod , and , Trans. Amer. Math. Soc. (1988), no. 1, 47–77. 10[13] F. G. Garvan,
Generalizations of Dyson’s rank , Thesis (Ph.D.), The Pennsylvania State University.1986. 136 pp. 14[14] B. Kim, An analog of crank for a certain kind of partition function arising from the cubic continuedfraction,
Acta Arith. (2011), no. 1, 1–19. 11[15] B. Kim, Partition statistics for cubic partition pairs,
Electron. J. Combin. (2011), no. 1, Paper128, 7 pp. 11[16] B. Kim, Cubic partition pairs weighted by the parity of the crank, Korean J. Math. (2015), no. 4,637–642. 12[17] J. Lovejoy, Ramanujan-type congruences for three colored Frobenius partitions, J. Number Theory (2000), no. 2, 283–290. 2[18] P. Paule and C.-S. Radu, The Andrews–Sellers family of partition congruences, Adv. Math. (2012),no. 3, 819–838. 14[19] J. Sellers, Congruences involving F-partition functions,
Internat. J. Math. Math. Sci. (1994), no.1, 187–188. 13[20] A. J. Yee, Combinatorial proofs of generating function identities for F-partitions, J. Combin. TheorySer. A (2003), no. 1, 217–228. 14(Shane Chern)
Department of Mathematics, The Pennsylvania State University, Univer-sity Park, PA 16802, USA
E-mail address : [email protected]; [email protected] (Dazhao Tang) College of Mathematics and Statistics, Chongqing University, Huxi Cam-pus LD206, Chongqing 401331, P.R. China
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