On the maximal part in unrefinable partitions of triangular numbers

A partition into distinct parts is refinable if one of its parts a can be replaced by two different integers which do not belong to the partition and whose sum is a, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a O(n1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n^{1/2})$$\end{document}-upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if n is even there exists only one maximal unrefinable partition of n(n+1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n(n+1)/2$$\end{document}, and that if n is odd the number of such partitions equals the number of partitions of ⌈n/2⌉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lceil n/2\rceil $$\end{document} into distinct parts. In the second case, an explicit bijection is provided.


Introduction
Integer partitions into distinct parts may appear in several areas of mathematics, sometimes unexpectedly. For example, they have been recently shown to be linked to the set of generators of groups in a group-theoretical problem related to cryptography [ACGS19,ACGS21a]. In particular, Aragona et al. showed that the generators of a given group are linked to partitions into distinct parts which satisfy a condition of non-refinability [ACGS21b] together with a condition on the minimal excludant. This motivates us to investigate some combinatorial aspects of unrefinable partitions, i.e. those in which no part can be written as the sum of two different integers which do not belong to the partition, which to our knowledge have not been investigated so far (cf. the On-Line Encyclopedia of Integer Sequences for the first values [OEI, https://oeis.org/A179009]).
Computational results suggest that the maximal part in an unrefinable partition of n is approximatively √ n. In this paper, we first prove a matching upper bound for the maximal part and then we define maximal unrefinable partitions as those which reach the bound. As a main contribution, we provide a complete classification of maximal unrefinable partitions for triangular numbers. We constructively prove, denoting by T n the n-th triangular number, that for even n there exists exactly one maximal unrefinable partition of T n . For odd n, we obtain a lower bound for the minimal excludant for the maximal unrefinable partitions of T n , defined to be the least integers which is not a part [FP15] and which has been investigated also recently by other authors [AN19,HSS22]. The knowledge of a bound on the minimal excludant, among other considerations, allows us to show an explicit bijection between the set of the maximal unrefinable partitions of T n and the set of partitions of ⌈n/2⌉ into distinct parts in the classical sense [And76].
The remainder of the paper is organized as follows: in Sec. 2 we introduce the notation and define unrefinable partitions. In Sec. 3 we prove two upper bounds for the maximal part in an unrefinable partition of n, distinguishing the case when n is a triangular number and when it is not. The classification theorem, i.e. Theorem 4.1, is proved in Sec. 4, which also contains the result on triangular numbers of an even number. The odd case is developed in Sec 5, which concludes the paper. In particular, we show in Theorem 5.11 a bijective proof that the number of maximal unrefinable partitions of T n equals the number of partitions of ⌈n/2⌉ into distinct parts.
Definition 2.1. Let N ∈ N. Let λ = (λ 1 , λ 2 , . . . , λ t ) be a partition of N into distinct parts and let µ 1 < µ 2 < · · · < µ m be its missing parts. The partition λ is refinable if there exist 1 ≤ ℓ ≤ t and 1 ≤ i < j ≤ m such that µ i + µ j = λ ℓ , and unrefinable otherwise. The set of unrefinable partitions is denoted by U, and by U N we denote those whose sum of the parts is N .
Definition 2.2. Let n ∈ N. We denote by T n the n-th triangular number, i.e.
The complete partition π n def = (1, 2, . . . , n) is the partition of T n with no missing parts.
Notice that every complete partition is unrefinable. The same holds, by definition, for partitions with a single missing part. In particular, if N = T n for some n, then π n is an unrefinable partition of N . Otherwise, if n is the least integer such that N < T (n), then is an unrefinable partition of N , where d = T n − N . In general, the admissible number of missing parts in an unrefinable partition is bounded as in the following result.
Lemma 2.3. Let λ = (λ 1 , λ 2 , . . . , λ t ) be unrefinable and let µ 1 < µ 2 < · · · < µ m be the missing parts. Then the number of missing parts m is bounded by (2) Proof. Let us start by observing that and thus λ is refinable. We prove the claim considering the complete partition π λt and removing from this the maximum number of parts different from λ t . For the previous observation, each candidate part µ i to be removed has a counterpart λ t − µ i in the partition. The bound of Eq.
(2) depends on the fact that this process can be repeated no more than ⌊λ t /2⌋ times.
As already anticipated, our focus is on the maximal part in a partition as in Definition 2.1. In the next section, using Lemma 2.3, we show that the maximal part in an unrefinable partition of n is O(n 1/2 ).
It is clear that the property of being unrefinable imposes on the one hand an upper limitation on the size of the largest part which is admissible in the partition, and on the other a lower limitation on the minimal excludant. We address in this section the natural question of determining what is the maximal part in an unrefinable partition of N . In the case where N is a triangular number the following result provides an answer. The notation introduced in the proof will be used throughout the paper. Equivalently, Proof. Let us start by considering the complete partition π n ⊢ N . Other unrefinable partitions of N are obtained from π n by removing some parts smaller than or equal to n and replacing them with parts larger than n. Hence, the lower bound for the maximal part in any partition of N is n, obtained when no part is removed. Since N = n(n + 1)/2, n is the positive solution of n 2 + n − 2N = 0 and so we have Let h, j ∈ N and let us denote by 1 ≤ a 1 < a 2 < · · · < a h ≤ n the candidate parts to be removed from π n to obtain a new unrefinable partition of N , and by n + 1 ≤ α 1 < α 2 < · · · < α j the corresponding replacements. Since a i = α i we have h > j. Moreover j > 1, otherwise from a i = α 1 the obtained partition is refinable. Hence we obtain h ≥ 3, j ≥ 2, and h > j .
Notice that the upper bound of Eq. 3 is tight. Indeed, let us define the following partition: It is easy to notice that π n ⊢ N and that π n is unrefinable, since its least missing parts are n− 2 and n− 1, and 2n− 4 < (n− 2)+ (n− 1). In the notation of the proof of Proposition 3.1, π n is obtained in the case h = 3 and j = h − 1 = 2. The counterpart of Proposition 3.1 in the case of non-triangular numbers is obtained in a similar way.
Proposition 3.2. Let N ∈ N be such that T n−1 < N < T n for some n ∈ N. For every unrefinable partition λ = (λ 1 , λ 2 , . . . , λ t ) of N we have n ≤ λ t ≤ 2n − 2. (5) Proof. Let us start by considering d and the partition π n,d ⊢ N as in Eq. (1). Other partitions of N are obtained from π n,d by removing some parts smaller than or equal to n which are replaced by d and other parts larger than n or only by other parts larger than n. Proceeding as in the proof of Proposition 3.1, let h, j ∈ N and let us denote by 1 ≤ a 1 < a 2 < · · · < a h ≤ n the candidate parts to be removed from π n,d to obtain a new partition of N , and by α 1 < α 2 < · · · < α j the corresponding replacements. Since a i = α i we have h ≥ j > 1, and we may obtain h = j only if α 1 = d. For this reason, we need to consider the two cases separately.
Let us assume α i > n, for every 1 ≤ i ≤ j. Reasoning as in the proof of Proposition 3.1 we can count m = (h + 1) + α j − n − j. On the other hand, if α 1 = d and α i > n for every 2 ≤ i ≤ j, then we obtain just h missing parts in the interval {1, 2, . . . , n} and exactly j − 1 parts appear in the interval {n + 1, n + 2, . . . , α j }, therefore we obtain the same formula for the number of missing parts m = h + α j − n − (j − 1). By Lemma 2.3 we obtain If α j > 2n − 2, then ⌈α j /2⌉ ≥ n and from Eq. (6) we obtain h ≤ j − 1, a contradiction.
We denote by U N the set of the maximal unrefinable partitions of N .
In the case of triangular numbers N = T n for some n ≥ 6, by virtue of Proposition 3.1, we have that λ = (λ 1 , λ 2 , . . . , λ t ) ∈ U N is maximal if and only if λ t = 2n − 4.
Remark 2. As already observed in the proof of Proposition 3.1, for each λ ∈ U there exist 1 < j < h, 1 ≤ a 1 < a 2 < · · · < a h ≤ n and α 1 , α 2 , . . . , α j ≥ n + 1 such that λ is obtained from π n by removing the parts a i s which are replaced by the parts α i s. Consequently, # U N coincides with the number of choices which lead to partitions meeting the mentioned conditions and, in addition, the condition λ t = 2n − 4. In the remainder of the work, when λ ∈ U we will refer to a i s, α i s, j and h as intended here.

Classification of maximal unrefinable partitions of triangular numbers
We are now ready to prove our first main contribution. Using arguments similar to those of the proofs in the previous section, we classify maximal unrefinable partitions for triangular numbers.
Theorem 4.1. Let n ∈ N, n ≥ 6, and N = T n . Then (2) if n is odd, then π n ∈ U N and the other partitions λ ∈ U N , λ = π n , are such that j = h − 2 and the following conditions hold: Proof. Let λ ∈ U N and let a 1 , a 2 , . . . , a h and α 1 , α 2 , . . . , α j = 2n − 4 as in Remark 2. We already know that h ≥ 3. From the hyphotheses on λ we have that By Lemma 2.3 we have h − j ≤ 2, and, since h > j, we obtain j ∈ {h − 1, h − 2}. Notice that if a ∈ {a 1 , . . . , a h } is such that a < n − 4, then α def = 2n − 4 − a must belong to {α 1 , . . . , α j−1 }, otherwise α + a = 2n − 4 = α j ∈ λ, and so λ is refinable. Then each removed part a i such that a i < n − 4 is in one-to-one correspondence with its replacement which, for the sake of simplicity, we will denote from now on by α i . On the other side, for the same symmetry argument, no part in the interval {n − 4, . . . , n} has a replacement. In such an interval we may choose at most 5 parts. However, we are not allowed to remove, at the same time, parts from one of the pairs (n − 4, n) and (n − 3, n − 1) without contradicting the unrefinability of λ. Analogously, we are not allowed to remove more than three parts. Moreover, we cannot choose to pick only one part to be removed in that interval, otherwise we would obtain h − 1 replacements but at most j − 1 are allowed, and h > j.
We are left to consider the cases of two or three parts to be removed in the interval {n − 4, . . . , n}, both with the assumptions j = h − 1 or j = h − 2. In particular, we will show that in both settings of j, there exists no maximal partition with two removed parts in the selected interval. Moreover, in the case j = h − 1 and three removed parts, we show that the only admissible partition is π n . Finally, partitions in the case j = h − 2 and three removed parts are only possible for odd n as claimed in (2). Let us address each of the four cases separately.
Let us suppose j = h − 1 and 1 ≤ a 1 < a 2 < · · · < a h−2 ≤ n − 5 and n − 4 ≤ a h−1 < a h ≤ n. For each 1 ≤ i ≤ j − 1 = h − 2 we have α i = α j − a i . We will now show that this configuration leads to a contradiction. To do this, we estimate a i and α i from above and from below, respectively. This is clearly accomplished by noticing that For a i = α i we obtain an inequality which is satisfied for h < 3 , which is a contradiction.
The third case j = h − 2 with two removed parts is immediately contradictory, since the parts a 1 , a 2 , . . . , a h−2 determine h − 2 = j replacements but at most j − 1 are possible.
The last case to be considered is the one where j = h − 2 and the three largest parts a i s are chosen in the interval {n − 4, . . . , n}. As already observed, since λ is unrefinable, the only possible choices are Since the right side of Eq. (7) is even and a h−2 + a h−1 + a h is even only if n is odd, Eq. (7) can be satisfied only in the case when n is odd. This proves (2) when n is odd and that the partition π n of Eq. (4) is the only maximal unrefinable partition of T n when n is even, i.e. (1).
From Theorem 4.1 we obtain that the description of maximal unrefinable partitions for the triangular number of an even integer is completed. The odd case is addressed in the following section.

Triangular numbers of odd integers
Throughout this last section, N will denote the triangular number of an odd integer. More precisely, let n = 2k − 1 ∈ N be such that N = T n .
From Theorem 4.1 we have that the set of maximal unrefinable partitions of triangular numbers of odd integers can be partitioned in the following way The following consideration is a trivial but important consequence of Theorem 4.1.
In the case of B h we have From a 1 < a 2 we have n ≥ h 2 − h − 3 and from α 1 < a 1 + a 2 = 3n + h 2 − 5h − 5 2 we obtain λ ∈ B h , i.e. the claim (3) is proved.
Finally, assuming the conditions of A h we have From a 1 < a 2 we have n ≥ h 2 − h − 1 and from α 1 < a 1 + a 2 = 3n + h 2 − 5h − 3 2 we obtain λ ∈ A h , from which the desired result (4) follows.
By interchanging the role of n and h in the statements of Proposition 5.6, we obtain the following description of the set of maximal unrefinable partitions of triangular numbers of an odd number, where we can read the upper bound for h in each different class.
Corollary 5.7. Let n ≥ 7 be odd. Then Remark 4. In the proof of Proposition 5.6 we have exhibited an example of unrefinable partition for each class, constructed by maximizing a 2 + a 3 + . . . a h−3 and consequently by determining a 1 . The unrefinability of the obtained partition is then granted from the fact that a 1 + a 2 > α 1 . Notice that, each other partition λ ′ of the same class is determined by the removed parts a ′ 1 , a ′ 2 , . . . , a ′ h−3 such that a ′ 1 = a 1 + i and a ′ s = a s − i s−1 for s > 1 and i s ≥ 0, where i = h−4 s=1 i s , provided that a ′ i < a ′ s for i < s. The unrefinability of λ ′ is then easily proved, since a ′ 1 + a ′ 2 = a 1 + i + a 2 − i i ≥ a 1 + a 2 ≥ α 1 ≥ α ′ 1 . Example 5.8. Let n = 49. For the bound in the previous corollary, when considering partitions of class D we have 5 ≤ h ≤ (1 + √ 29 + 4n)/2 = 8. Let us fix h = 7 and construct all the partitions in U T49 ∩ D 7 . We recall that, for Theorem 4.1, a partition of class D 7 is given when a 1 , a 2 . . . , a h−3 = a 4 are specified. Therefore, for the sake of simplicity, we denote the partitions just by listing the removed parts (a 1 , a 2 , a 3 , a 4 ). Let us start, as in Proposition 5.6, from the partition n + h 2 − 3h − 9 2 , n − 7, n − 6, n − 5 = (34, 42, 43, 44) .
All the remaining partitions in D 7 , obtained as in Remark 4, are: The partitions in other classes are obtained analogously.
We have already highlighted in Example 5.2 and in Example 5.3 what min λ∈ UT n mex(λ) looks like. The intuition can now be easily proved as a consequence of the previous propositions.
class Table 3. Values of a 1 in the construction of Propositions 5.4, 5.5 and 5.6, for h = 4, h = 5 and h ≥ 6.
Corollary 5.9. Let n ≥ 7 be odd. For each λ ∈ U Tn we have Proof. Notice that µ 1 = a 1 . The claim is trivial if λ = π n . Otherwise it follows from Propositions 5.4, 5.5 and 5.6, recalling that a 1 was calculated in order to be minimal, since a 2 + a 3 + · · · + a h−3 was maximized. The results are summarized in Table 3, where it is not hard to check that (n − 3)/2 is the smaller value that a 1 can assume.

The bijection.
In this conclusive section we prove the main contribution of this work, i.e. we show that, when n is odd, the number of maximal unrefinable partitions of T n equals the number of partitions of ⌈n/2⌉ into distinct parts by means of a bijective proof. Notice that, by the anti-symmetric property (Remark 3) and by the bound on the minimal excludant (Corollary 5.9), a partition in U Tn is determined by at most parts. The following theorem is used to establish a bijection between U T 2k−1 and D k .
Theorem 5.10. Let a 1 , a 2 , . . . , a u be the missing parts smaller than or equal to n − 3 of an unrefinable partition λ = π n of T n , for some odd integer n ≥ 7. Then n, λ and its class are uniquely determined.
Proof. Let us start by proving that n can be obtained from knowing a 1 , a 2 , . . . , a u . In particular, let us prove that by distinguishing the four possible classes. Let us first assume λ ∈ D h for some h ≥ 5.
Recalling that a u ≤ n − 3 and, by the definition of D h and by Remark 3, since a h−1 = n − 1 and a h = n, we have a u / ∈ {n − 3, n − 4}. Therefore a u ≤ n − 5, and so u = h − 3. Recalling that the following conditions hold (i) 1 ≤ a 1 < a 2 < · · · < a h−3 ≤ n − 5, from which we determine n as claimed. Let us consider the class C h . In this case, reasoning as above, we have a u = n − 3 and a u−1 ≤ n − 5, which means u − 1 = h − 3. Therefore from which we obtain again Eq. (10). When λ ∈ B h , we have a u = n − 4, which means h = u + 2, so from which the same n is determined.
We are now ready to prove our last result. Denoting by D the set of all the partitions into distinct parts, let us define the following subsets of D: It is not hard to notice that where Let us conclude the paper by proving a bijection between U T 2k−1 and D k .
where i = h−4 s=1 i s , and so since the sum of the first h − 3 terms is (h 2 − 3h)/2 + h−4 s=1 i s . Finally, in the case when we obtain noticing that the sum of the first h − 2 terms is (h 2 − 3h)/2 + 1 + h−4 s=1 i s . We proved that σ is well defined. Notice also that σ is trivially injective. Therefore it remains to prove that σ is surjective. In particular, it suffices to check that for each partition λ * ∈ (A * ∪ B * ∪ C * ∪ D * ) ∩ D k , λ * = (3, k − 3), there exists λ ∈ (A∪ B∪ C∪ D) ∩ U T 2k−1 such that σ(λ) = λ * , since σ( π n ) = (3, k − 3) by definition. Given λ * = (λ * 1 , λ * 2 , . . . , λ * t ) ∈ D k , by the definition of σ we have that the partition λ denoted by its missing parts (n − 2 − λ * t , . . . , n − 2 − λ * 2 , n − 2 − λ * 1 ) is such that σ(λ) = λ * . It remains to prove that such λ is a maximal unrefinable partition of n. The full details of the proof are here omitted since they can be obtained by arguments very similar to those used for proving that σ is well defined. As an example, let us consider the case when λ * ∈ A * t ∩ D k , for t ≥ 5, and let us prove that λ * is the image of an unrefinable partition λ of class A. Since λ * is a partition of k into t distinct parts and contains 1 and 2 by definition we can write λ * = 1, 2, 3 + i 1 , . .
Remark 5. The bijection σ is not well defined when k < 7. However, it can be easily shown that the result of Theorem 5.11 is still valid when k = 4 and k = 5, where we have # U T7 = #D 4 = 1 and # U T9 = #D 5 = 2, respectively. The claim is false instead in the case k = 6, where we have # U T11 = 4 and #D 6 = 3.
In the proof of Theorem 5.11 we showed that σ is a bijection from U T 2k−1 to D k . Moreover, we also proved that σ is bijective when it is restricted to each class.
Corollary 5.12. The function σ of Theorem 5.11 maps in a bijective way Example 5.13. Coming back to the case of Example 5.2, we represent in Tab. 4 the bijection σ between maximal unrefinable partitions of 13 obtained in the case h = j − 2 (hence those different from π 13 ), represented by black dots, and the partitions of 7 into distinct parts, represented by blue dots. Notice that the partition (3, 4) is not displayed since it corresponds to π 13 . Here x corresponds to x def = min λ∈ UT 13 mex(λ).
Equivalently, by the anti-symmetric property, partitions of 7 into distinct parts can be read looking at the black dots on the right side of the table.
Remark 6. Another combinatorial equality can be derived from the provided construction for U N . Indeed, assuming n = 2k − 1 for k ≥ 7, h ≥ 6, and reasoning as in Example 5.8, it can be easily shown that # U Tn ∩ D h equals the number of partitions in h − 3 parts of f (n, h) in which each part is smaller than or equal to g(n, h), where f (n, h) = −h 3 + 6h 2 + (n − 2)h − 4n − 22 2 and g(n, h) = n − h 2 + 3h + 1 2 .
The proof is obtained from Proposition 5.6, considering the bijection a i ↔ a i − a 1 + 1.
In Tab. 6 the result is summarized for each class. Notice that, using the bijection of Eq. (14) on the partitions shown in Example 5.8, one can recover the eleven partitions of 31 in 4 parts, where each part is not larger than 11.