An infinite family of internal congruences modulo powers of 2 for partitions into odd parts with designated summands

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part. In that same work, Andrews, Lewis, and Lovejoy also studied such partitions wherein all parts must be odd, and they denoted the number of such partitions of size $n$ by the function $PDO(n)$. Since then, numerous authors have proven a variety of divisibility properties satisfied by $PDO(n)$. Recently, the second author proved the following internal congruences satisfied by $PDO(n)$: For all $n\geq 0$, \begin{align*} PDO(4n)&\equiv PDO(n) \pmod{4},\\ PDO(16n)&\equiv PDO(4n) \pmod{8}. \end{align*} In this work, we significantly extend these internal congruence results by proving the following new infinite family of congruences: For all $k\geq 0$ and all $n\geq 0$, $$PDO(2^{2k+3}n) \equiv PDO(2^{2k+1}n) \pmod{2^{2k+3}}.$$ We utilize several classical tools to prove this family, including generating function dissections via the unitizing operator of degree two, various modular relations and recurrences involving a Hauptmodul on the classical modular curve $X_0(6)$, and an induction argument which provides the final step in proving the necessary divisibilities. It is notable that the construction of each $2$-dissection slice of our generating function bears an entirely different nature to those studied in the past literature.


Introduction
In 2002, Andrews, Lewis, and Lovejoy [2] introduced the combinatorial objects which they called partitions with designated summands.These are built by taking unrestricted integer partitions and designating exactly one of each occurrence of a part.For example, there are ten partitions with designated summands of size 4: Andrews, Lewis, and Lovejoy denoted the number of partitions with designated summands of size n by the function P D(n).Hence, using this notation and the example above, we know P D(4) = 10.
In the same paper, Andrews, Lewis, and Lovejoy [2] also considered the restricted partitions with designated summands wherein all parts must be odd, and they denoted the corresponding enumeration function by P DO(n).Thus, from the example above, we see that P DO(4) = 5, where we have counted the following five objects: Andrews, Lewis, and Lovejoy noted in [2, eq. (1.6)] that the generating function for P DO(n) is given by ∞ n=0 P DO(n)q n = E(q 4 )E(q 6 ) 2 E(q)E(q 3 )E(q 12 ) , where E(q) = (q; q) ∞ := (1 − q)(1 − q 2 )(1 − q 3 )(1 − q 4 ) • • • is the usual q-Pochhammer symbol.
Beginning with [2], a wide variety of Ramanujan-like congruences have been proven for P D(n) and P DO(n) under different moduli.See [1, 3-7, 10, 12, 18] for such work.Recently, Sellers [15] proved a number of arithmetic properties satisfied by P DO(n) modulo powers of 2 in response to a conjecture of Herden et.al. [11].As part of that work, Sellers proved the following internal congruences on the way to proving infinite families of divisibility properties satisfied by P DO(n): For all n ≥ 0, P DO(4n) ≡ P DO(n) (mod 4), ( P DO(16n) ≡ P DO(4n) (mod 8). (1.3) While searching for such internal congruences computationally, it became clear that the above internal congruences are part of a much larger family.The ultimate goal of this paper is to prove the existence of the following infinite family: Theorem 1.For all k ≥ 0 and all n ≥ 0, P DO(2 2k+3 n) ≡ P DO(2 2k+1 n) (mod 2 2k+3 ). (1.4) It should be noted that replacing n by 2n in the above immediately yields the following corollary: Corollary 2. For all k ≥ 0 and all n ≥ 0, P DO(2 2k+4 n) ≡ P DO(2 2k+2 n) (mod 2 2k+3 ). (1.5) Notice that the k = 0 case of this corollary states that, for all n ≥ 0, P DO(16n) ≡ P DO(4n) (mod 8) which is (1.3) mentioned above.It is also necessary to point out that the above families of congruences are not optimal in several isolated cases, especially when k is small.For instance, we will show in (6.1) that P DO(32n) ≡ P DO(8n) (mod 64), which also yields P DO(64n) ≡ P DO(16n) (mod 64).
In order to prove Theorem 1, we introduce the following auxiliary functions: ) 2 E(q)E(q 3 )E(q 12 ) , γ = γ(q) := E(q) 5 E(q 2 ) 5 E(q 6 ) 5 E(q 3 ) 15 , ξ = ξ(q) := E(q 2 ) 5 E(q 6 ) E(q)E(q 3 ) 5 , and We further define for k ≥ 2, Finally, let U be the unitizing operator of degree two, given by These will allow us to represent each 2-dissection slice of the generating function of P DO(n), accompanied by a certain multiplier: as a polynomial in the Hauptmodul ξ on the classical modular curve X 0 (6) of genus 0. We then complete our analysis on these functions in order to prove Theorem 1.This paper is organized as follows: Sect. 2 is devoted to proving several required modular equations.In particular, one important component of this section, as discussed in Sect.2.2, concerns the representation of the degree two unitizations ζ i,j = U κ i ξ j in Z[ξ] for arbitrary exponents i and j, where κ is one of the auxiliary functions we introduced and ξ is the aforementioned Hauptmodul.The next two sections will then be devoted to the 2-adic behavior for these ζ i,j series.Recall that in Theorem 1, we are indeed considering internal congruences for P DO(n).Hence, we introduce another family of auxiliary functions in Sect.5: so as to capture this internal nature.Here the new multipliers λ ′ k are closely related to the original λ k .We will then move on to the divisibility properties for the above new family of series, and in particular, we offer our proof of Theorem 1 in Sect.6.It is notable that the construction of each 2-dissection slice of our generating function bears an entirely different nature to those studied in the past literature, in the sense that the multipliers λ k are pairwise distinct.This fact makes our 2-adic analysis far more complicated but in the meantime very unique.We present a discussion of the difference between our machinery and past work in Sect.7.

Modular equations
In this section, we collect a number of necessary modular relations that will be utilized in the sequel.
2.1.Modular relations for γ and ξ.Let us begin with the functions γ and ξ.It is clear that both γ 6 and ξ are modular functions on the classical modular curve X 0 (6).
2.2.Modular relations for κ and ξ.Our objective in this subsection is as follows: Theorem 4. For any i, j ≥ 0, (2.2) We briefly postpone the proof of Theorem 4 in order to complete some necessary analysis.
Now to show the above expression equals 5ξ − 4ξ 2 as claimed in (2.4), we only need to examine that the alleged linear combination of eta-products is identical to 0, i.e., 5ξ − 4ξ 2 − 15t − t 2 + t 3 + t 4 • q 4 E(q) 4 E(q 12 ) 12 E(q 3 ) 12 E(q 4 ) 4 = 0.This task can be performed readily by Garvan's Maple package ETA [8], as pointed out in the proof of Theorem 3. The remaining identities can be proven in the same way.
Theorem 7. We have (2.12) Proof.This relation can be shown in a way similar to that for Theorem 5.
Now we move to relations for Λ k and ξ for each k ≥ 2.
More precisely, if we write then and for k ≥ 3, we recursively have where ζ 2 k−3 ,ℓ is given by (2.8).
Proof.We begin with the proof of (2.15).It was already shown in [2, Theorem 21] that Thus, Invoking (2.12) gives the claimed expression for Λ 2 .
For (2.16), we make use of the fact that, for k ≥ 3, Noting that κ = γ(q 2 ) 2 /γ(q) and writing Λ k−1 in the above as a polynomial in ξ by virtue of (2.14), we finally obtain that Recalling (2.8), the required result follows.

Minimal ξ-power in ζ
For each i, j ≥ 0, let the coefficients Z i,j (m) with m ≥ 0 be such that It is notable that Z i,j (m) eventually vanishes as ζ i,j ∈ Z[ξ], so the above summation is indeed finite.From the evaluations in Sect.2.2, we have also seen that for each ζ i,j as a polynomial in ξ, the terms ξ m with a lower degree usually vanish.In the next theorem, we characterize the minimal ξ-power in the polynomial expression of ζ i,j .Theorem 9.For any i, j ≥ 0, define Then for any m with 0 ≤ m < d i,j , we have Furthermore, the coefficient Z i,j (d i,j ) is an odd integer.

2-Adic analysis for ζ
With all of the above preparations in hand, we are now in a position to begin thinking about the divisibility of various objects by powers of 2. Throughout the remainder of this work, we denote by ν(n) the 2-adic evaluation of n, that is, ν(n) is the largest nonnegative integer α such that 2 α | n.We also adopt the convention that ν(0) = ∞.
By Theorem 9, we have d 4I+2,0 = d 4I+0,0 + 5, so as to give us Likewise, the 2-adic evaluation of the coefficient of the This process can be continued to all remaining coefficients Z 4I+2,0 .For i = 4I + 3, we shall make use of the following table and argue in the same vein: It is notable that for the 2-adic evaluation of we shall use the second case of (4.1) so as to get as given in the table.
Finally, for i = 4I + 5, we require the table: and then perform a similar analysis to that for the above i = 4I + 2 case.
In the same fashion, we have parallel results for Z i,1 (m).
Theorem 12.For any i ≥ 0, we have Furthermore, for M ≥ 2, Proof.By (2.4) and (2.6), the theorem holds true for i = 0 and 1.We may then apply a similar inductive argument to that for Theorem 11.
Now we are ready to perform the 2-adic evaluations for the coefficients Z 2 k ,j (m) for each k ≥ 0 and j ≥ 0 whenever m ≥ d 2 k ,j .Theorem 13.For any k ≥ 2 and j ≥ 0, we have Furthermore, for M ≥ 2, Proof.In view of Theorems 11 and 12, it is known that the results are true for Z 2 k ,0 (m) and Z 2 k ,1 (m) with any k ≥ 2. Now we apply induction on j and prove for j = 4J + 2, 4J + 3, 4J + 4 and 4J + 5 under the assumption of validity for j = 4J + 0 and 4J + 1.Here a similar strategy to that for Theorem 11 will be used, with (2.10) being invoked: For j = 4J + 2, we require this table: For j = 4J + 3, we require this table: For j = 4J + 4, we require this table: For j = 4J + 5, we require this table: Concrete analyses can be mimicked by consulting the i = 4I + 2 case in the proof of Theorem 11, and we will omit the details.

New auxiliary functions and the associated minimal ξ-powers
All of the work above has revolved around the generating function for the function P DO(n).However, we keep in mind that Theorem 1 is really focused on the internal congruences for the P DO function.To capture this nature, let us introduce a new family of auxiliary functions for k ≥ 3, In light of (1.6), we have More precisely, if we write then and for k ≥ 4, we recursively have where Finally, we recall (5.2) and invoke (2.8) to obtain the desired recurrence.
For k = 2K, we know from (5.4) that By virtue of Theorem 9, ζ 2 2K−1 ,τ 2K−1 starts with the power ξ d where Here we make use of the fact that both 2 2K−1 and τ 2K−1 are even.In the meantime, each of the ζ ∈ Z[ξ] component in the remaining summands in (5.7) starts with a power higher than ξ d .Hence, Φ 2K starts with at least ξ τ 2K .
For k = 2K + 1, we also deduce from (5.4) that Invoking Theorem 9 and noting that τ 2K is odd, it follows that ζ 2 2K ,τ 2K starts with the power ξ d where For the second summand in (5.8), we find that ζ 2 2K ,τ 2K +1 starts with the power ξ d ′ where according to a similar computation.Furthermore, each of the ζ ∈ Z[ξ] components in the remaining summands in (5.8) starts with a power higher than ξ d = ξ d ′ .Hence, we can claim that Φ 2K+1 starts with at least ξ τ 2K+1 .

2-Adic analysis for Φ
We are now in a position to prove our main result, Theorem 1.To do so, we only need to confirm that for each K ≥ 1, In what follows, we first manually analyze the initial cases where K ∈ {1, 2} and then move on to general K by induction.
6.1.Initial cases.Recall that Φ 3 was already formulated in (5.3).Meanwhile, we may obtain an explicit expression in Z[ξ] (containing 43 terms!) for Φ 5 by applying the recurrence (5.4) twice.Consequently, we get the following 2-adic evaluations, where τ stands for τ 3 = 14 or τ 5 = 54 accordingly: In the above table, the last column is true for all M ≥ 3.So we indeed have the following two congruences, with the former being even stronger than the general scenario in Theorem 1: P DO(2 5 n) ≡ P DO(2 7 n) (mod 2 7 ).(6.2) 6.2.Induction.Now we perform induction on K ≥ 2 and establish the following lower bounds for the 2-adic evaluations.
Theorem 16.For any K ≥ 2, it is true that and that for M ≥ 1, Note that the K = 2 case was already covered in Sect.6.1.So in the sequel, we first prove the inequality (6.4) for K ≥ 3 under the inductive assumption that it is true for K − 1.For the sake of notational convenience, let us denote Also, we write Proof of (6.4).To produce Φ 2K+1 from Φ 2K−1 , we need to employ the recurrence (5.4) twice.Writing we deduce from (5.4) that Recall that τ ′′ is even according to (5.5).It then follows from Theorems 9 and 15 that the minimal degree of the ξ-powers in Applying the recurrence (5.4) once again gives us that Noting that τ ′ is odd according to (5.6), Theorems 9 and 15 then imply that the minimal degree of the ξ-powers in Hence, where C(M, M ′′ , M ′′ ) := F 2K−1 (τ ′′ + M ′′ )Z ′′ (τ ′′ + M ′′ , τ ′ + M ′ )Z ′ (τ ′ + M ′ , τ + M).
Meanwhile, the constant term on the right-hand side of (6.8) is F 2K+1 (τ 2K+1 ) as the constant term of ξ(q) is 1.Since the congruence (6.8) is valid, we must have that their constant terms are also congruent under the same modulus, so as to give us which is equivalent to (6.3).

Conclusion
We close with two sets of comments.First, we remind the reader of one of the original goals for proving such internal congruences.As was mentioned above, the second author [15] used two corollaries of Theorem 1 to prove the following Ramanujan-like congruences via induction on α: Theorem 17.For all α ≥ 0 and all n ≥ 0, P DO 2 α (4n + 3) ≡ 0 (mod 4), P DO 2 α (8n + 7) ≡ 0 (mod 8).
It would be gratifying to see other cases of Theorem 1 used to assist in proving divisibility properties satisfied by P DO(n) for higher powers of 2.
Secondly, in the course of proving congruences modulo arbitrary powers for the coefficients of an eta-product H(q) = ∞ n=0 h(n)q n , the usual strategy is to find a suitable basis {ξ 1 , ξ 2 , . . ., ξ L } of the corresponding modular space such that each dissection slice, accompanied by a certain multiplier (usually an eta-product), h(p m n + t m )q n , can be represented as a polynomial in Z[ξ 1 , ξ 2 , . . ., ξ L ].For example, when proving the congruences modulo powers of 5 for the partition function [17] or the congruences modulo powers of 7 for the distinct partition function [9], two specific multipliers typically take turns showing up, i.e., λ 2M −1 = λ and λ 2M = λ ′ for two certain series λ and λ ′ .However, in our study here, the multipliers γ, γ 2 , γ 4 , γ 8 , . . .never overlap.Meanwhile, an important outcome of cycling the multipliers in the previous studies is that it is typically sufficient to represent each degree p unitization U p κ i ξ j as a polynomial in ξ for a certain series κ, with the exponent i restricted to {0, 1}; here we use the case where the basis of the modular space is given by {ξ} as an illustration.However, as shown in Sect.2.2, when there are endless possibilities for the multipliers, we have to extend the consideration of i to infinity, thereby substantially increasing the amount of required p-adic analysis.These striking facts distinguish our work from the past literature.