k-Marked Dyson Symbols and Congruences for Moments of Cranks
aa r X i v : . [ m a t h . C O ] D ec k -Marked Dyson Symbols andCongruences for Moments of Cranks William Y.C. Chen , Kathy Q. Ji and Erin Y.Y. Shen , , Center for Combinatorics, LPMC-TJKLCNankai University, Tianjin 300071, P. R. China Center for Applied MathematicsTianjin University, Tianjin 300072, P. R. China [email protected], [email protected], [email protected] Abstract.
By introducing k -marked Durfee symbols, Andrews found a combinatorialinterpretation of 2 k -th symmetrized moment η k ( n ) of ranks of partitions of n . Recently,Garvan introduced the 2 k -th symmetrized moment µ k ( n ) of cranks of partitions of n inthe study of the higher-order spt-function spt k ( n ). In this paper, we give a combinatorialinterpretation of µ k ( n ). We introduce k -marked Dyson symbols based on a representationof ordinary partitions given by Dyson, and we show that µ k ( n ) equals the number of( k + 1)-marked Dyson symbols of n . We then introduce the full crank of a k -markedDyson symbol and show that there exist an infinite family of congruences for the fullcrank function of k -marked Dyson symbols which implies that for fixed prime p ≥ r and k ≤ ( p − /
2, there exist infinitely many non-nested arithmeticprogressions An + B such that µ k ( An + B ) ≡ p r ) . Dyson’s rank [9] and the Andrews-Garvan-Dyson crank [2] are two fundamental statisticsin the theory of partitions. For a partition λ = ( λ , λ , . . . , λ ℓ ), the rank of λ , denoted r ( λ ), is the largest part of λ minus the number of parts. The crank c ( λ ) is defined by c ( λ ) = ( λ , if n ( λ ) = 0 ,µ ( λ ) − n ( λ ) , if n ( λ ) > , where n ( λ ) is the number of ones in λ and µ ( λ ) is the number of parts larger than n ( λ ).Andrews [3] introduced the symmetrized moments η k ( n ) of ranks of partitions of n given by η k ( n ) = + ∞ X m = −∞ (cid:18) m + ⌊ k − ⌋ k (cid:19) N ( m, n ) , (1.1)1here N ( m, n ) is the number of partitions of n with rank m .In view of the symmetry N ( − m, n ) = N ( m, n ), we have η k +1 ( n ) = 0. As for the evensymmetrized moments η k ( n ), Andrews [3] showed that for fixed k ≥ η k ( n ) is equal tothe number of ( k +1)-marked Durfee symbols of n . Kursungoz [15] and Ji [13] provided thealternative proof of this result respectively. Bringmann, Lovejoy and Osburn [7] definedtwo-parameter generalization of η k ( n ) and k -marked Durfee symbols. In [3], Andrewsalso introduced the full rank of a k -marked Durfee symbol and defined the full rankfunction N F k ( r, t ; n ) to be the number of k -marked Durfee symbols of n with full rankcongruent to r modulo t .The full rank function N F k ( r, t ; n ) have been extensively studied and they possesmany congruence properties, see for example, [5–8,14]. Recently, Bringmann, Garvan andMahlburg [6] used the automorphic properties of the generating functions of N F k ( r, t ; n )to prove the existence of infinitely many congruences for N F k ( r, t ; n ). More precisely, forgiven positive integers j , k ≥
3, odd positive integer t , and prime Q not divisible by 6 t ,there exist infinitely many arithmetic progressions An + B such that for every 0 ≤ r < t ,we have N F k ( r, t ; An + B ) ≡ Q j ) . (1.2)Since η k ( n ) = t − X r =0 N F k +1 ( r, t ; n ) , by (1.2), we see that there exist an infinite family of congruences for η k ( n ), namely,for given positive integers k and j , prime Q >
3, there exist infinitely many non-nestedarithmetic progressions An + B such that η k ( An + B ) ≡ Q j ) . Analogous to the symmetrized moments η k ( n ) of ranks, Garvan [12] introduced the k -th symmetrized moments µ k ( n ) of cranks of partitions of n in the study of the higher-orderspt-function spt k ( n ). To be more specific, µ k ( n ) = + ∞ X m = −∞ (cid:18) m + ⌊ k − ⌋ k (cid:19) M ( m, n ) , (1.3)where M ( m, n ) denotes the number of partitions of n with crank m for n >
1. For n = 1and m = − , ,
1, we set M ( m,
1) = 0; otherwise, we define M ( − ,
1) = 1 , M (0 ,
1) = − , M (1 ,
1) = 1 . It is clear that µ k +1 ( n ) = 0, since M ( m, n ) = M ( − m, n ).In this paper, we give a combinatorial interpretation of µ k ( n ). We first introduce thenotion of k -marked Dyson symbols based on a representation for ordinary partitions given2y Dyson [9]. We show that for fixed k ≥ µ k ( n ) equals the number of ( k + 1)-markedDyson symbols of n . Moreover, we define the full crank of a k -marked Dyson symboland define full crank function N C k ( r, t ; n ) to be the number of k -marked Dyson symbolsof n with full crank congruent to r modulo t . We prove that for fixed prime p ≥ r and k ≤ ( p + 1) /
2, there exists infinitely many non-nested arithmeticprogressions An + B such that for every 0 ≤ i ≤ p r − N C k ( i, p r ; An + B ) ≡ p r ) . (1.4)Note that µ k ( n ) = p r − X i =0 N C k +1 ( i, p r ; n ) , so that from (1.4) we can deduce that there exist an infinite family of congruences for µ k ( n ), that is, for fixed prime p ≥
5, positive integers r and k ≤ ( p − /
2, there existinfinitely many non-nested arithmetic progressions An + B such that µ k ( An + B ) ≡ p r ) . k -marked Dyson symbols In this section, we introduce the notion of k -marked Dyson symbols. A 1-marked Dysonsymbol is called a Dyson symbol, which is a representation of a partition introducedby Dyson [10]. For 1 ≤ i ≤ k , we define the i -th crank of a k -marked Dyson symbol.Moreover, we define the function F k ( m , m , . . . , m k ; n ) to be the number of k -markedDyson symbol of n with the i -th crank equal to m i for 1 ≤ i ≤ k . The following theoremshows that the number of k -marked Dyson symbols of n can be expressed in terms of thenumber of Dyson symbols of n . Theorem 2.1.
For fixed integers m , m , . . . , m k , we have F k ( m , . . . , m k ; n ) = + ∞ X t ,..., t k − =0 F k X i =1 | m i | + 2 k − X i =1 t i + k − n ! . (2.1)For a partition λ = ( λ , . . . , λ ℓ ), let ℓ ( λ ) denote the number of parts of λ and | λ | denote the sum of parts of λ . A Dyson symbol of n is a pair of restricted partitions ( α, β )satisfying the following conditions:(1) If ℓ ( α ) = 0, then β = β ;(2) If ℓ ( α ) = 1, then α = 1;(3) If ℓ ( α ) >
1, then α = α ; 3 N M
ABC
Figure 2.1: The decomposition of λ .(4) n = | α | + | β | + ℓ ( α ) ℓ ( β ).When we display a Dyson symbol, we shall put α on the top of β in the form of a Durfeesymbol [3] or a Frobenius partition [1].For example, there are 5 Dyson symbols of 4: (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) , (cid:18) (cid:19) . Theorem 2.2 (Dyson) . There is a bijection Ω between the set of partitions of n and theset of Dyson symbols of n . For completeness, we give a proof of the above theorem.
Proof of Theorem 2.2 : Let λ = ( λ , . . . , λ ℓ ) be a partition of n . A Dyson symbol ( α, β )of n can be constructed via the following procedure. There are two cases.Case 1: One is not a part of λ . We set α = ∅ and β = λ ′ .Case 2: One is a part of λ . Assume that one occurs M times in λ . We decom-pose the Ferrers diagram of λ into three blocks as illustrated in Figure 2.1, where N is the number of parts of λ that are greater than M . In this case, we see that λ =( λ , . . . , λ N , λ N +1 , . . . , λ s , M ), where λ N > M , λ N +1 ≤ M and 1 M means M occurrencesof 1. Then remove all parts equal to one from λ and insert a new part M , so that we geta partition µ = ( λ , . . . , λ N , M, λ N +1 , . . . , λ s ) as shown in Figure 2.2.4 + 1 M β β β β α α α ABC
Figure 2.2: The Dyson symbol ( α, β ) . Now the partitions α and β can be obtained from µ . First, let β = ( λ − M, λ − M, . . . , λ N − M ), and let ν = ( M, λ N +1 , . . . , λ s ). Then we get α = ( ν ′ , ν ′ , . . . , ν ′ M ), where ν ′ the conjugate of ν , see Figure 2.2.It is easy to verify that ( α, β ) is a Dyson symbol of n and the above procedure isreversible, and hence the proof is complete.For a Dyson symbol ( α, β ), Dyson [10] considered the difference between the numberof parts of α and β , which we call the crank of ( α, β ). Let F ( m ; n ) denote the numberof Dyson symbols of n with crank m . Dyson [10] observed the following relation based onthe construction in Theorem 2.2. Corollary 2.3 (Dyson) . For n ≥ and integer m , M ( − m, n ) = F ( m ; n ) . (2.2)A k -marked Dyson symbol is defined as the following array η = α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p ,β ( k ) , β ( k − , . . . , β (1) , consisting of k pairs of partitions ( α ( i ) , β ( i ) ) and a partition p = ( p k − , p k − , . . . , p ) subjectto the following conditions:(1) The smallest part of p equals 1, that is, p k − ≥ · · · ≥ p ≥ p = 1.52) For 1 ≤ i ≤ k −
1, each part of α ( i ) and β ( i ) is between p i − and p i , namely, p i ≥ α ( i )1 ≥ α ( i )2 ≥ · · · ≥ α ( i ) ℓ ≥ p i − and p i ≥ β ( i )1 ≥ β ( i )2 ≥ · · · ≥ β ( i ) ℓ ≥ p i − . (3) Each part of α ( k ) and β ( k ) is no less than p k − , namely, α ( k )1 ≥ α ( k )2 ≥ · · · ≥ α ( k ) ℓ ≥ p k − and β ( k )1 ≥ β ( k )2 ≥ · · · ≥ β ( k ) ℓ ≥ p k − . (4) If ℓ ( α ( k ) ) = 1 , then α ( k )1 = p k − ;If ℓ ( α ( k ) ) > , then α ( k )1 = α ( k )2 ;If ℓ ( α ( k ) ) = 0 and ℓ ( β ( k ) ) = 1, then β ( k )1 = p k − ;If ℓ ( α ( k ) ) = 0 and ℓ ( β ( k ) ) ≥
2, then β ( k )1 = β ( k )2 ;If ℓ ( α ( k ) ) = 0 and ℓ ( β ( k ) ) = 0, then p k − = max { α ( k − , β ( k − } .For example, the array below η = (5 , ,
4) (3 , ,
2) (1 , , ,
2) (2 , , (2.3)is a 3-marked Dyson symbol.We next define the weight of a k -marked Dyson symbol. Recall that for a pair ofpartitions ( α, β ) with ℓ ( α ) ≥ ℓ ( β ), a balanced part β i of β is defined recursively as follow.If the number of parts greater than β i in α is equal to the number of unbalanced partsbefore β i in β , that is, the number of unbalanced parts β j with 1 ≤ j < i ; otherwise, wecall β i is an unbalanced part, see [13, p.992]. We use b ( α, β ) to denote the number ofbalanced parts of ( α, β ).For example, for the pair of partitions (cid:18) αβ (cid:19) = (cid:18) (cid:19) , the first part 3 of β is balanced, and the second part 2 and the third part 2 are unbalanced.Therefore, b ( α, β ) = 1 . We now define the i -th crank and the i -th balanced number of a k -marked Dsyonsymbol. Let η = α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p β ( k ) , β ( k − , . . . , β (1) be a k -marked Dyson symbol. The pair of partitions ( α ( i ) , β ( i ) ) is called the i -th vectorof η . For 1 ≤ i ≤ k , we define c i ( η ), the i -th crank of η , to be the difference between thenumber of parts of α ( i ) and β ( i ) , that is, c i ( η ) = ℓ ( α ( i ) ) − ℓ ( β ( i ) ).6or 1 ≤ i < k , we define b i ( η ), the i -th balanced number of η by b i ( η ) = ( b ( α ( i ) , β ( i ) ) , if ℓ ( α ( i ) ) ≥ ℓ ( β ( i ) ) ,b ( β ( i ) , α ( i ) ) , if ℓ ( α ( i ) ) < ℓ ( β ( i ) ) . For i = k , we set b k ( η ) = 0.For the 3-marked Dyson symbol η in (2.3), we have c ( η ) = − , c ( η ) = 0 , c ( η ) = 2and b ( η ) = 1 , b ( η ) = 1 , b ( η ) = 0.For 1 ≤ i ≤ k , we define l i ( η ), the i -th large length of η by l i ( η ) = ( ℓ ( α ( i ) ) , if ℓ ( α ( i ) ) ≥ ℓ ( β ( i ) ) ,ℓ ( β ( i ) ) , if ℓ ( α ( i ) ) < ℓ ( β ( i ) ) . Similarly, we define the i -th small length s i ( η ) of η by s i ( η ) = ( ℓ ( β ( i ) ) , if ℓ ( α ( i ) ) ≥ ℓ ( β ( i ) ) ,ℓ ( α ( i ) ) , if ℓ ( α ( i ) ) < ℓ ( β ( i ) ) . The weight of k -marked Dyson symbol is defined by | η | = k X i =1 ( | α ( i ) | + | β ( i ) | ) + k − X i =1 p i + ( l ( η ) + D + k − s ( η ) − D ) , (2.4)where l ( η ) = k X i =1 l i ( η ) , s ( η ) = k X i =1 s i ( η ) , and D = k X i =1 b i ( η ) . (2.5)For example, the weight of the 3-marked Dyson symbol η in (2.3) equals 97.For a k -marked Dyson symbol η , if the weight of η equals n , we call η a k -markedDyson symbol of n . We can now define the function F k ( m , . . . , m k ; n ) as the number of k -marked Dyson symbols of n with the i -th crank equal to m i for 1 ≤ i ≤ k . Note thata 1-marked Dyson symbol is a Dyson symbol and F ( m ; n ) = M ( − m, n ) . The followingtheorem shows the function F k ( m , . . . , m k ; n ) has the mirror symmetry with respect toeach m j . Theorem 2.4.
For n ≥ , k ≥ and ≤ j ≤ k , we have F k ( m , . . . , m j , . . . , m k ; n ) = F k ( m , . . . , − m j , . . . , m k ; n ) . (2.6) Proof.
The above identity is trivial for m j = 0. We now assume that m j >
0. Let H k ( m , . . . , m k ; n ) denote the set of k -marked Dyson symbols of n counted by F k ( m , . . . , k ; n ). We aim to build a bijection Λ between the set H k ( m , . . . , m j , . . . , m k ; n ) and theset H k ( m , . . . , − m j , . . . , m k ; n ).Let η = α ( k ) , α ( k − , . . . , α ( j ) , . . . , α (1) p k − , p k − , · · · p j · · · p β ( k ) , β ( k − , . . . , β ( j ) , . . . , β (1) be a k -marked Dyson symbol in H k ( m , . . . , m j , . . . , m k ; n ). To define the map Λ, we needto construct a new j -th vector ( ¯ α ( j ) , ¯ β ( j ) ) from ( α ( j ) , β ( j ) ). There are four cases.Case 1: 1 ≤ j ≤ k −
1. Set ¯ α ( j ) = β ( j ) and ¯ β ( j ) = α ( j ) .Case 2: j = k and ℓ ( α ( k ) ) = 1. In this case, we have α ( k )1 = p k − and β ( k ) = ∅ . Set¯ α ( k ) = ∅ and ¯ β ( k ) = α ( k ) . Case 3: j = k , ℓ ( α ( k ) ) ≥ ℓ ( β ( k ) ) = 1. Let t = β ( k )1 − β ( k )2 . Set¯ α ( k ) = ( β ( k )1 − t, β ( k )2 , . . . , β ( k ) ℓ ) and ¯ β ( k ) = ( α ( k )1 + t, α ( k )2 , . . . , α ( k ) ℓ ) . Case 4: j = k , ℓ ( α ( k ) ) ≥ ℓ ( β ( k ) ) = 1. Let t = β ( k )1 − p k − . Set¯ α ( k ) = ( β ( k )1 − t ) and ¯ β ( k ) = ( α ( k )1 + t, α ( k )2 , . . . , α ( k ) ℓ ) . From the above construction, it can be checked that ℓ ( ¯ α ( j ) ) − ℓ ( ¯ β ( j ) ) = − ( ℓ ( α ( j ) ) − ℓ ( β ( j ) )) . Then Λ( η ) is defined as α ( k ) , α ( k − , . . . , ¯ α ( j ) , . . . , α (1) p k − , p k − , · · · p j · · · p β ( k ) , β ( k − , . . . , ¯ β ( j ) , . . . , β (1) . Hence Λ( η ) is a k -marked Dyson symbol in H k ( m , . . . , − m j , . . . , m k ; n ). Furthermore, itcan be seen that the above process is reversible. Thus Λ is a bijection.We are now ready to prove Theorem 2.1, which says that the number of k -markedDyson symbols of n can be expressed in terms of the number of Dyson symbols of n . Thistheorem is needed in the combinatorial interpretation of µ k ( n ) given in Theorem 3.1. ByTheorem 2.4, we see that Theorem 2.1 can be deduced from the following formula. Theorem 2.5.
For n ≥ and m , m , . . . , m k ≥ , we have F k ( m , . . . , m k ; n ) = + ∞ X t ,..., t k − =0 F k X i =1 m i + 2 k − X i =1 t i + k − n ! . (2.7)8o prove the above theorem, we introduce the structure of strict k -marked Dysonsymbols. Recall that a strict bipartition of n is a pair of partitions ( α, β ) such that α i > β i for i = 1 , , . . . , ℓ ( β ) and | α | + | β | = n. Note that for a strick bipartition ( α, β )we have ℓ ( α ) ≥ ℓ ( β ). For example, (cid:18) (cid:19) is a strict bipartition.Strict bipartitions are the building blocks of strict k -marked Dyson symbols. For k ≥
2, let η = α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p β ( k ) , β ( k − , . . . , β (1) be a k -marked Dyson symbols of n . If ( α ( i ) , β ( i ) ) is a strict bipartition for any 1 ≤ i < k ,we say that η a strict k -marked Dyson symbol of n .Notice that there is no balanced part in a strict bipartition. Consequently, if η is astrict k -marked Dyson symbol, then the i -th balanced number b i ( η ) of η equals zero for1 ≤ i < k . To prove Theorem 2.5, we define a function F sk ( m , . . . , m k ; n ) as the numberof strict k -marked Dyson symbols of n with the i -th crank equal to m i for 1 ≤ i ≤ k and define a function F k ( m , . . . , m k , t , . . . , t k − ; n ) as the number of k -marked Dysonsymbols of n with the i -th crank equal to m i for 1 ≤ i ≤ k and the i -th balance numberequal to t i for 1 ≤ i ≤ k −
1. The relation stated in Theorem 2.5 can be established viatwo steps as stated in the following two theorems.
Theorem 2.6.
For n ≥ , k ≥ , m , m , . . . , m k ≥ and t , t , . . . , t k − ≥ , we have F k ( m , . . . , m k , t , . . . , t k − ; n ) = F sk ( m + 2 t , . . . , m k − + 2 t k − , m k ; n ) . (2.8) Theorem 2.7.
For n ≥ , k ≥ and m , m , . . . , m k ≥ , we have F sk ( m , . . . , m k ; n ) = F k X i =1 m i + k − n ! . (2.9)To prove Theorem 2.6, we need a bijection in [13, Theorem 2.4]. Let P ( r ; n ) denote theset of pairs of partitions ( α, β ) of n where there are r balanced parts and ℓ ( α ) − ℓ ( β ) ≥ Q ( r ; n ) denote the set of strict bipartitions ( ¯ α, ¯ β ) of n with ℓ ( ¯ α ) − ℓ ( ¯ β ) ≥ r. Given two positive integers n and r , there is a bijection ψ between P ( r ; n ) and Q (2 r ; n ).Furthermore, the bijection ψ possesses the following properties. For ( α, β ) ∈ P ( r ; n ), let( ¯ α, ¯ β ) = ψ ( α, β ). Then we have¯ α = max { α , β } , ¯ α ℓ = α ℓ , and ¯ β ℓ ≥ β ℓ . (2.10) ℓ ( ¯ α ) = ℓ ( α ) + r and ℓ ( ¯ β ) = ℓ ( β ) − r. (2.11)9e next give a proof of Theorem 2.6 by using the bijection ψ . Proof of Theorem 2.6.
Let P k ( m , . . . , m k , t , t , . . . , t k − ; n ) denote the set of k -markedDyson symbols of n with the i -th crank equal to m i and the i -th balanced numberequal to t i , and let Q k ( m , . . . , m k ; n ) denote the set of strict k -marked Dyson sym-bols of n with the i -th crank equal to m i . We proceed to define a bijection Ω between P k ( m , . . . , m k , t , t , . . . , t k − ; n ) and Q k ( m + 2 t , . . . , m k − + 2 t k − , m k ; n ).Let η = α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p β ( k ) , β ( k − , . . . , β (1) be a k -marked Dyson symbol in P k ( m , . . . , m k , t , t , . . . , t k − ; n ). For 1 ≤ i < k , weapply the bijection ψ described above to ( α ( i ) , β ( i ) ) to get a pair of partitions ( ¯ α ( i ) , ¯ β ( i ) ).From the properties of the bijection ψ , we see that( ¯ α ( i ) , ¯ β ( i ) ) is a strict bipartition and¯ α ( i )1 = max { α ( i )1 , β ( i )1 } , ¯ α ( i ) ℓ = α ( i ) ℓ , ¯ β ( i ) ℓ ≥ β ( i ) ℓ (2.12)and ℓ ( ¯ α ( i ) ) = ℓ ( α ( i ) ) + t i , ℓ ( ¯ β ( i ) ) = ℓ ( β ( i ) ) − t i . (2.13)Then Ω( η ) is defined to be α ( k ) , ¯ α ( k − , . . . , ¯ α (1) p k − , p k − , · · · p β ( k ) , ¯ β ( k − , . . . , ¯ β (1) . By (2.12), we see that that for 1 ≤ i < k −
1, each part of ¯ α ( i ) and ¯ β ( i ) is between p i − and p i , namely, p i ≥ ¯ α ( i )1 ≥ ¯ α ( i )2 ≥ · · · ≥ ¯ α ( i ) ℓ ≥ p i − and p i ≥ ¯ β ( i )1 ≥ ¯ β ( i )2 ≥ · · · ≥ ¯ β ( i ) ℓ ≥ p i − . It is also clear from (2.13) that the i -th crank of Ω( η ) is equal to m i + 2 t i for 1 ≤ i < k and the k -th crank of Ω( η ) is equal to m k . Using (2.13) again, we get l (Ω( η )) = k − X i =1 ℓ ( ¯ α ( i ) ) + ℓ ( α k ) = k X i =1 ( ℓ ( α ( i ) ) + t i ) = k X i =1 ℓ ( α ( i ) ) + D = l ( η ) + D and s (Ω( η )) = k − X i =1 ℓ ( ¯ β ( i ) ) + ℓ ( β k ) = k X i =1 ( ℓ ( β ( i ) ) − t i ) = k X i =1 ℓ ( α ( i ) ) − D = s ( η ) − D. Thus the weight of Ω( η ) is equal to k X i =1 ( | ¯ α ( i ) | + | ¯ β ( i ) | ) + k − X i =1 p i + ( l (Ω( η )) + k − · s (Ω( η ))= k X i =1 ( | α ( i ) | + | β ( i ) | ) + k − X i =1 p i + ( l ( η ) + k − D ) · ( s ( η ) − D ) , | η | . So Ω( η ) is in Q k ( m + 2 t , . . . , m k − +2 t k − , m k ; n ). Since ψ is a bijection, it is readily verified that Ω is also a bijection, andhence the proof is complete.We now turn to the proof of Theorem 2.7. Proof of Theorem 2.7.
Recall that Q k ( m , . . . , m k ; n ) denotes the set of strict k -markedDyson symbols of n with the i -th crank equal to m i and H ( m ; n ) denotes the set ofDyson symbols of n with crank m . To establish a bijection Φ between Q k ( m , . . . , m k ; n )and H ( m + · · · + m k + k − n ), let η = α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p β ( k ) , β ( k − , . . . , β (1) be a strict k -marked Dyson symbol in Q k ( m , . . . , m k ; n ). Let α be the partition consistingof all parts of α (1) , α (2) , . . . , α ( k ) together with p , . . . , p k − , and let β be the partitionconsisting of all parts of β (1) , β (2) , . . . , β ( k ) . Then Φ( η ) is defined to be ( α, β ) . From thedefinition of k -marked Dyson symbols, we see that ( α, β ) is a Dyson symbol. It is alsoeasily seen that ℓ ( α ) = l ( η ) + k − , ℓ ( β ) = s ( η ) (2.14)and | α | = k X i =1 | α ( i ) | + k − X i =1 p i , | β | = k X i =1 | β ( i ) | . (2.15)It follows from (2.14) that ℓ ( α ) − ℓ ( β ) = k X i =1 m i + k − . Combining (2.14) and (2.15), we deduce that the weight of ( α, β ) equals | α | + | β | + ℓ ( α ) ℓ ( β ) = k X i =1 | α ( i ) | + k − X i =1 p i + k X i =1 | β ( i ) | + ( l ( η ) + k − s ( η ) = | η | . This proves that ( α, β ) is a Dyson symbol in H ( m + · · · + m k + k − n ).We next describe the reverse map of Φ. Let (cid:18) αβ (cid:19) = (cid:18) α α . . . α ℓ β β . . . β ℓ (cid:19) be a Dyson symbol in H ( m + · · · + m k + k − n ). We proceed to show that a strict k -marked Dyson symbol η can be recovered from the Dyson symbol ( α, β ).11irst, we see that the k -th vector ( α ( k ) , β ( k ) ) of η and p k − can be recovered from( α, β ). Let j k be largest nonnegative integer such that β j k ≥ α m k + j k +1 , that is, for any i ≥ j k + 1, we have β i < α m k + i +1 . Define (cid:18) α ( k ) β ( k ) (cid:19) = (cid:18) α α . . . α m k + j k β β . . . β j k (cid:19) and p k − = α m k + j k +1 . Obviously, ℓ ( α ( k ) ) − ℓ ( β ( k ) ) = m k . To recover ( α ( k − , β ( k − ) and p k − , we let (cid:18) α ′ β ′ (cid:19) = (cid:18) α m k + j k +2 α m k + j k +3 . . . α ℓ β j k +1 β j k +2 . . . β ℓ (cid:19) . By the choice of j k , we find that α m k + j k + i +1 > β j k + i for any i , in other words, α ′ i >β ′ i . Consequently, ( α ′ , β ′ ) is a strict bipartition. Then ( α ( k − , β ( k − ) and p k − can beconstructed from ( α ′ , β ′ ). Let j k − be the largest nonnegative integer such that β ′ j k − ≥ α ′ m k − + j k − +1 . Define (cid:18) α ( k − β ( k − (cid:19) = (cid:18) α ′ α ′ . . . α ′ m k − + j k − β ′ β ′ . . . β ′ j k − (cid:19) and p k − = α ′ m k − + j k − +1 . Now we have ℓ ( α ( k − ) − ℓ ( β ( k − ) = m k − . Since ( α ′ , β ′ ) is a strict bipartition, we deducethat ( α ( k − , β ( k − ) is a strict bipartition.The above procedure can be repeatedly used to determine ( α ( k − , β ( k − ) , p k − , . . . , ( α (2) , β (2) ) , p , ( α (1) , β (1) ). The k -marked Dyson symbol η can be defined as α ( k ) , α ( k − , . . . , α (1) p k − , p k − , · · · p β ( k ) , β ( k − , . . . , β (1) . It can be checked that η is a strict k -marked Dyson symbol in Q k ( m , . . . , m k ; n ). More-over, it can be seen that Φ( η ) = ( α, β ), that is, Φ is indeed a bijection. This completesthe proof.Here is an example to illustrate the reverse map Φ − . Assume that m = 1 , m =1 , m = 0, and (cid:18) αβ (cid:19) = (cid:18) (cid:19) , which a Dyson symbol of 127, that is, ( α, β ) ∈ H (4; 127). From ( α, β ), we get (cid:18) α (3) β (3) (cid:19) = (cid:18) (cid:19) , p = 3 , (cid:18) α ′ β ′ (cid:19) = (cid:18) (cid:19) . α ′ , β ′ ), we get (cid:18) α (2) β (2) (cid:19) = (cid:18) (cid:19) , p = 1 , (cid:18) α (1) β (1) (cid:19) = (cid:18) (cid:19) . Finally, we obtain η = (6 6 3) (3 3 2 2 1) (1)3 1(5 5 4) (2 1 1 1) . It can be checked that η ∈ Q (1 , ,
0; 127). µ k ( n ) In this section, we use Theorem 2.1 to give a combinatorial interpretation of µ k ( n ) interms of k -marked Dyson symbols. Theorem 3.1.
For k ≥ and n ≥ , µ k ( n ) is equal to the number of ( k + 1) -markedDyson symbols of n .Proof. By definition of F k ( m , . . . , m k ; n ), the assertion of the theorem can be stated asfollows ∞ X m ,...,m k +1 = −∞ F k +1 ( m , . . . , m k +1 ; n ) = µ k ( n ) . (3.1)Using Theorem 2.1, we see that the left-hand side of (3.1) equals ∞ X m ,m ,...,m k +1 = −∞ F k +1 ( m , . . . , m k +1 ; n )= ∞ X m ,m ,...,m k +1 = −∞ ∞ X t ,...,t k =0 F k +1 X i =1 | m i | + 2 k X i =1 t i + k ; n ! . (3.2)Given k and n , let c k ( j ) denote the number of integer solutions to the equation | m | + · · · + | m k +1 | + 2 t + · · · + 2 t k = j in m , m , . . . , m k +1 and t , t , . . . , t k subject to the further condition that t , t , . . . , t k arenonnegative. It can be shown that generating function of c k ( j ) is equal to ∞ X j =0 c k ( j ) q j = 1 + q (1 − q ) k +1 ,
13o that c k ( j ) = (cid:18) k + j k (cid:19) + (cid:18) k + j − k (cid:19) . Substituting j by m − k , we get c k ( m − k ) = (cid:18) m + k − k (cid:19) + (cid:18) m + k k (cid:19) . Thus (3.2) simplifies to ∞ X m ,m ,...,m k +1 = −∞ F k +1 ( m , . . . , m k +1 ; n )= ∞ X m =1 (cid:20)(cid:18) m + k − k (cid:19) + (cid:18) m + k k (cid:19)(cid:21) F ( m ; n ) . Using Corollary 2.3 and noting that M ( − m, n ) = M ( m, n ), we conclude that ∞ X m ,m ,...,m k +1 = −∞ F k +1 ( m , . . . , m k +1 ; n )= ∞ X m =1 (cid:20)(cid:18) m + k − k (cid:19) + (cid:18) m + k k (cid:19)(cid:21) M ( m, n ) , which equals µ k ( n ), as claimed.For example, for n = 5 and k = 1, we have µ (5) = 35, and there are 35 2-markedDyson symbols of 5 as listed in the following table. (1) 1(1) (2 2) 1 (1)2 (2) (1)1(1 1) (1) (1 1)1 (1 1) (1 1 1 1) 1 (1 1) (1)1 (1) (1 1 1 1)1 (1 1) (1 1)1 (1)1(1 1 1) (2) (1)2 (1 1 1)1(1) (1 1 1) (1)1 (1) (1 1)1 (1) (1) (1 1 1)1 (1 1)1(1 1) (2 1)2 (1 1 1)1 (1) (1 1)1(1) (1) (1)2(2) (1) (1)1 (1 1) (1)1 (1 1 1) (1) 1 (1 1 1) (1 1) 1 (1 1) (2) 2 (1) (1 1 1) 1 (1) (1)1(1) (1 1) µ k ( n ) In this section, we introduce the full crank of a k -marked Dyson symbol. We show thatthere exist an infinite family of congruences for the full crank function of k -marked Dysonsymbols.To define the full crank of a k -marked Dyson symbol η , denoted F C ( η ), we recall that c k ( η ) denotes the k -th crank of η , l ( η ) denotes the large length of η and s ( η ) denotes the15mall length of η and D denotes the balanced number of η . Then F C ( η ) is given by F C ( η ) = ( l ( η ) − s ( η ) + 2 D + k − , if c k ( η ) > , − ( l ( η ) − s ( η ) + 2 D + k − , if c k ( η ) ≤ . It is clear that for k = 1, the full crank of a 1-marked Dyson symbol reduces to the crankof a Dyson symbol.Analogous to the full rank function for a k -marked Durfee symbol defined by Andrews[3], we define the full crank function N C k ( i, t ; n ) as the number of k -marked Dyson symbolsof n with the full crank congruent to i modulo t . The following theorem gives an infinitefamily of congruences of the full crank function. Theorem 4.1.
For fixed prime p ≥ and positive integers r and k ≤ ( p + 1) / . Thenthere exist infinitely many non-nested arithmetic progressions An + B such that for each ≤ i ≤ p r − , N C k ( i, p r ; An + B ) ≡ p r ) . Since µ k ( n ) = p r − X i =0 N C k +1 ( i, p r ; n ) , Theorem 4.1 implies the following congruences for µ k ( n ). Theorem 4.2.
For fixed prime p ≥ , positive integers r and k ≤ ( p − / . Then thereexists infinitely many non-nested arithmetic progressions An + B such that µ k ( An + B ) ≡ p r ) . To prove Theorem 4.1, let
N C k ( m ; n ) denote the number of k -marked Dyson symbolsof n with the full crank equal to m . In this notation, we have the following relation. Theorem 4.3.
For n ≥ , k ≥ and integer m , N C k ( m ; n ) = (cid:18) m + k − k − (cid:19) M ( m, n ) . (4.1) Proof.
Recall that F k ( m , . . . , m k , t , . . . , t k − ; n ) is the number of k -marked Dyson sym-bols of n such that for 1 ≤ i ≤ k , the i -th crank equal to m i and the i -th balance numberequal to t i . By the definition of N C k ( m, n ), we see that if m ≥
1, then we have
N C k ( m ; n ) = X F k ( m , m , . . . , m k , t , t , . . . , t k − ; n ) , (4.2)where the summation ranges over all integer solutions to the equation | m | + · · · + | m k | + 2 t + · · · + 2 t k − = m − k + 1 (4.3)16n m , m , . . . , m k and t , t , . . . , t k − subject to the further condition that m k is positiveand t , t , . . . , t k − are nonnegative.Combining Theorem 2.6 and Theorem 2.7, we find that F k ( m , m , . . . , m k , t , t , . . . , t k − ; n ) = F k X i =1 | m i | + 2 k − X i =1 t i + k − n ! . (4.4)Substituting (4.4) into (4.2), we get N C k ( m ; n ) = X F k X i =1 | m i | + 2 k − X i =1 t i + k − n ! , (4.5)where the summation ranges over all solutions to the equation (4.3). Let ¯ c k ( m − k + 1)denote the number of integer solutions to the equation (4.3). It is not difficult to verifythat ¯ c k ( m − k + 1) = (cid:18) m + k − k − (cid:19) . Thus, (4.5) simplifies to
N C k ( m ; n ) = (cid:18) m + k − k − (cid:19) F ( m ; n ) . Using Corollary 2.3 and noting that M ( − m, n ) = M ( m, n ), we conclude that N C k ( m ; n ) = (cid:18) m + k − k − (cid:19) M ( m, n ) , as required. Similarly, it can be shown that relation (4.1) also holds for m ≤ M ( i, t ; n ) denote the number of partitions of n with the crank congruent to i modulo t . The following congruences for M ( i, t ; n ) given by Mahlburg [16] will be usedin the proof of Theorem 4.1. Theorem 4.4 (Mahlburg) . For fixed prime p ≥ and positive integers τ and r , there areinfinitely many non-nested arithmetic progressions An + B such that for each ≤ m ≤ p r − , M ( m, p r ; An + B ) ≡ p τ ) . We are now ready to complete the proof of Theorem 4.1 by using Theorems 4.3 and4.4.
Proof of Theorem 4.1.
For 0 ≤ i ≤ p r −
1, by the definition of
N C k ( i, p r ; n ), we have N C k ( i, p r ; n ) = + ∞ X t = −∞ N C k ( p r t + i ; n ) . (4.6)17eplacing m by p r t + i in (4.1), we get N C k ( p r t + i ; n ) = (cid:18) p r t + i + k − k − (cid:19) M ( p r t + i, n ) . (4.7)Substituting (4.7) into (4.6), we find that N C k ( i, p r ; n ) = + ∞ X t = −∞ (cid:18) p r t + i + k − k − (cid:19) M ( p r t + i, n ) . (4.8)Since p is a prime and k ≤ ( p + 1) /
2, we see that (2 k − p . It followsthat (cid:18) p r t + i + k − k − (cid:19) ≡ (cid:18) i + k − k − (cid:19) (mod p r ) . Thus (4.8) implies that
N C k ( i, p r ; n ) ≡ + ∞ X t = −∞ (cid:18) i + k − k − (cid:19) M ( p r t + i, n ) (mod p r )= (cid:18) i + k − k − (cid:19) M ( i, p r ; n ) . Setting τ = r in Theorem 4.4, we deduce that there are infinitely many non-nestedarithmetic progressions An + B such that for every 0 ≤ i ≤ p r − M ( i, p r ; An + B ) ≡ p r ) . Consequently, there are infinitely many non-nested arithmetic progressions An + B suchthat for every 0 ≤ m ≤ p r − N C k ( i, p r ; An + B ) ≡ p r ) , and hence the proof is complete. Acknowledgments.
This work was supported by the 973 Project and the NationalScience Foundation of China.
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