SKEW DOUBLED SHIFTED PLANE PARTITIONS: CALCULUS AND ASYMPTOTICS

. In this paper, we establish a new summation formula for Schur processes, called the complete summation formula . As an application, we obtain the generating function and the asymptotic formula for the number of doubled shifted plane partitions , which can be viewed as plane partitions “shifted at the two sides”. We prove that the order of the asymptotic formula depends only on the diagonal width of the doubled shifted plane partition, not on the proﬁle (the skew zone) itself. By using similar methods, the generating function and the asymptotic formula for the number of symmetric cylindric partitions are also derived.

In the literature there are several approaches to deriving the generating functions for various defective plane partitions, such as: (1) Determinant evaluation and nonintersecting lattice paths [1,10]; (2) Hook lengths and combinatorial proofs [21,23]; (3) Schur functions and Schur processes [3,22]; (4) Lozenge tilings and Kuo condensation [6]. The Schur process was first introduced by Okounkov and Reshetikhin [18,19] in 2001. Later, they used the Schur process to derive the cyclic symmetry of the topological vertex by considering a certain type of plane partitions [20]. This result was further developed by Iqbal et al., for providing a short proof of the Nekrasov-Okounkov formula [12,17]. Borodin [4] used the Schur process to derive the generating function for cylindric partitions, introduced by Gessel and Krattenthaler [10]. The Macdonald process, which is a (q, t)-generalization of the Schur process, was first introduced by Vuletić [27], and further developed by Corteel, Savelief, Vuletić and Langer [7,14,26] in the study of weighted cylindric partitions and plane overpartitions. Finally, a survey of Macdonald processes was published by Borodin and Corwin [5].
The Schur process approach is shown to be a powerful tool in the study of various kinds of defective plane partitions. In fact, the above generating functions for defective plane partitions (A-E) are specializations of two general summation formulas for Schur processes, namely, the open summation formula (8) and the cylindric summation formula (10). Formulas (8) and (10) have been developed by Okounkov, Reshetikhin, Borodin, Corteel, Savelief, Vuletić and Langer [4,7,14,18,19,26,27]. For convenience, they are also reproduced in Theorem 2.1.
In the present paper we establish a new summation formula for Schur processes, called the complete summation formula (9). As an application, we obtain the generating functions for the skew doubled shifted plane partitions and the symmetric cylindric partitions (see Fig. 1 (F)/(G)/(H)).
Let us reproduce some classical formulas in this introduction (see, e.g., [21,24]). The generating functions for ordinary plane partitions (PP), shifted plane partitions (ShiftPP) and symmetric plane partitions (SPP) are the following respectively: As a byproduct of our complete summation formula (9) we can further derive the following generating function for doubled shifted plane partitions of width m (see Fig. 1 (F) and Section 3 for the definition).
Inspired by the works of Dewar, Murty and Kotěšovec [8,13], we establish some useful theorems for asymptotic formulas in [11] (see Theorem 4.2). Furthermore, the following asymptotic formula for the number of doubled shifted plane partitions can also be obtained.
where C(m) is a constant independent of n given by the following expression: For example, the generating function and asymptotic formula for doubled shifted plane partitions of width m = 3 (see Fig. 2 The proofs of Theorems 1.1 and 1.2 will be given in Section 4. In fact, Theorems 1.1 and 1.2 can be extended to skew doubled shifted plane partitions (see Sections 3 and 4). The asymptotic formulas for two other skew doubled shifted plane partitions, together with some ordinary plane partitions (PP), cylindric partitions (CP) and symmetric cylindric partitions (SCP) are also reproduced next. The proofs of those asymptotic formulas can be found in Sections 4 and 5 for DSPP and SCP respectively, and in [11] for PP and CP.  The skew doubled shifted plane partitions have some nice properties. (1) The order of the asymptotic formula depends only on the width of the doubled shifted plane partition, not on the profile (the skew zone) itself. The similar property holds for ordinary plane partitions. We may think that this is natural by intuition. However, the cylindric partitions (CP and SCP) show that this is not always the case. (2) We empirically observe that, the asymptotic formula for doubled shifted plane partitions gives already good approximative values for the numbers of DSPP, even for small integer n. While the asymptotic formula for PP needs a large integer n to produce an acceptable value, as shown in the following It is amazing how the orders of the asymptotic formulas for CP and SCP differ. For the CP, the exponents of n in the denominator are always 1, but the exponents of e differ. While for the SCP, the exponents of n differ, but the exponents of n are constant. Let us summarize these observations in the following The rest of the paper is arranged in the following way. In Section 2 we establish the complete summation formula for Schur processes. The basic notation and the trace generating functions for skew doubled shifted plane partitions can be found in Section 3. We derive the generating functions and the asymptotic formulas for the numbers of skew doubled shifted plane partitions and symmetric cylindric partitions, in Sections 4 and 5, respectively.
First we use (15) to prove some lemmas.
Proof. Let F (X) be the left-hand side of (16). Then, by (15) we have Hence, we obtain Since We have that (16) is proved.
Proof. By (15) and Lemma 2.3 we have (17) is similar to the following formula stated in Macdonald's book [15] (see p. 94, ex. 28(a)), which has two free partitions λ and γ: Now we are ready to give the proof of Theorem 2.2.
The set of all DSPP with profile δ is denoted by DSPP δ . Recall that the Schur process for plane partitions was first introduced by Okounkov and Reshetikhin [19] (see also [14]); the main idea was to read the plane partitions along the diagonals. When reading the DSPP ω with profile δ along the diagonals from left to right, we obtain a sequence of integer partitions (λ 0 , λ 1 , . . . , λ h ) such that λ i−1 ≺ λ i (resp. λ i−1 λ i ) if δ i = 1 (resp. δ i = −1), and |ω| = h i=0 |λ i |. For simplicity, we identify the skew doubled shifted plane partition ω and the sequence of integer partitions by writing ω = (λ 0 , λ 1 , . . . , λ h ).
The (diagonal) width of the skew doubled shifted plane partition ω is defined to be h + 1.
For example, with the DSPP ω given in Fig. 3, we obtain a sequence of partitions: For a sequence of parameters u i (i ≥ 0), write U j = u 0 u 1 · · · u j−1 (j ≥ 0). Let DSPP δ (λ 0 , λ h ) denote the set of the skew doubled shifted plane partitions ω = (λ 0 , λ 1 , . . . , λ h ) starting from λ 0 and ending at λ h with profile δ. Letting Z i = {U −δi i } in Theorem 2.1, we obtain the following trace generating functions for skew doubled shifted plane partitions.
Theorem 3.2. Let δ = (δ i ) 1≤i≤h be a profile. We have The above theorem implies many classical results on various defective plane partitions, including the trace generating function of ordinary plane partitions (PP) (Stanley [23]), and the generating functions of symmetric plane partitions (SPP) (Andrews [2], Macdonald [15]), skew plane partitions (Sagan [21]) and skew shifted plane partitions (Sagan [21]). 4. Formulas for skew doubled shifted plane partitions. Let u i = z for i ≥ 0 in (20). We then derive the generating function for DSPP δ : The right-hand side of the above identity can be further simplified. For each profile δ = (δ i ) 1≤i≤m−1 , we define the following multisets as Therefore, Eq. (22) implies the following theorem.
By applying the above two results, our next theorem gives asymptotic formulas for the number of skew doubled shifted plane partitions. For simplicity, write Then we obtain the following result.
where C 1 (δ) and C 2 (m) are two constants with respect to n: Proof. By the definitions of W 1 (δ) and W 2 (δ) we have Hence, Furthermore, If we exchange any two adjacent letters in δ, the sum of the last two terms doesn't change, therefore it is equal to 1 Then we obtain By Theorems 4.2 and 4.1 the number of DSPP with profile δ and size n is asymptotic to This achieves the proof.
Remark 2. In Theorem 1.2, we derive an asymptotic formula for bounded m and large n. We would like to mention that, a recent paper of Fang, Hwang and Kang [9] generalized our Theorem 1.2 and obtained stronger asymptotic formulas for all possible values of m, n with m ≤ n.

5.
Formulas for symmetric cylindric partitions. Cylindric partitions were introduced by Gessel and Krattenthaler [10]. They obtained the generating function for cylindric partitions of some given shape that satisfy certain row bounds as some summation of determinants related to q-binomial coefficients. Later, Borodin gave an equivalent definition [4] and obtained the generating function for cylindric partitions. A cylindric partition (CP) with profile δ is a DSPP ω = (λ 0 , λ 1 , . . . , λ h−1 , λ h ) with profile δ such that λ 0 = λ h . The size of such partition is defined by |ω| = h−1 i=0 |λ i | (notice that λ h is not counted here). The following generating function for cylindric partitions, first proved by Borodin [4], later by Tingley [25] and Langer [14], can be obtained by letting u i = z (0 ≤ i ≤ h − 1) and u h = 1 in (21).
Theorem 5.1 (Borodin [4]). Let δ = (δ i ) 1≤i≤h be a profile. Then the generating function for the cylindric partitions with profile δ is where W 3 (δ) is the following multiset A symmetric cylindric partition (SCP) with profile δ = (δ 1 , δ 2 , . . . , δ h ) is a DSPP Notice that a symmetric cylindric partition is always a cylindric partition, and when λ h = ∅, the SCP ω becomes an SyPP. The size of the symmetric cylindric partition ω is defined by Theorem 5.2. The generating function for symmetric cylindric partitions with profile δ is Proof. By (19), we have By Lemma 2.4, this is equal to the right hand side of (27).
Remark 3. The term r depends only on the width of the symmetric cylindric partitions, not on the profile itself, while the term b depends on the profile. It is interesting to compare these phenomena with the asymptotic formula for cylindric partitions [11]. There, b depends only on the width, and r depends on the profile.
For example, consider the three symmetric cylindric partitions (SCPa)-(SCPc) given in Fig. 2. Their profiles, generating functions and asymptotic formulas are respectively:  .