Unrefinable partitions into distinct parts in a normalizer chain

In a recent paper on a study of the Sylow 2-subgroups of the symmetric group with 2^n elements it has been show that the growth of the first (n-2) consecutive indices of a certain normalizer chain is linked to the sequence of partitions of integers into distinct parts. Unrefinable partitions into distinct parts are those in which no part x can be replaced with integers whose sum is x obtaining a new partition into distinct parts. We prove here that the (n-1)-th index of the previously mentioned chain is related to the number of unrefinable partitions into distinct parts satisfying a condition on the minimal excludant.


Introduction
The sequence (b j ) of the number of partitions of integers into distinct parts has been extensively studied in past and recent years and is a well-understood integer sequence [Eul48,And94] appearing in different areas of mathematics. Recently, triggered by a problem in algebraic cryptography related to translation subgroups in the symmetric group with 2 n elements [CDVS06, ACGS19, CBS19, CCS21], it has been shown that such a sequence is related to the growth of the indices of consecutive terms in a chain of normalizers [ACGS21]. More precisely, let Σ n be a Sylow 2-subgroup of the symmetric group Sym(2 n ) and T be an elementary abelian regular subgroup of Σ n . Defining N 0 n = N Σn (T ) and recursively N i n = N Σn (N i−1 n ), the authors proved that the number log 2 N i n : N i−1 n is independent of n for 1 ≤ i ≤ n − 2, and is equal to the (i + 2)-th term of the sequence (a j ) of the partial sums of (b j ) (cfr. Table 1). The result is obtained by proving that the terms in the chain are saturated subgroups, i.e. generated by rigid commutators, a family of left-normed commutators involving a special set of generators of Σ n . We invite the interested reader to refer to Aragona et al. [ACGS21], where rigid commutators and saturated groups are introduced and described in detail. When i > n−2, the behavior of the chain does not seem to show any recognizable pattern and the study of its combinatorial nature is still open. Nonetheless, further j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 b j 0 0 0 1 1 2 3 4 5 7 9 11 14 17 21 a j 0 0 0 1 2 4 7 11 16 23 32 43 57 74 95 Table 1. First values of the sequences (b j ) and (a j ) investigations on experimental evidences led us to notice that the first exception to the rule, i.e. log 2 N n−1 n : N n−2 n , is linked to the number of partitions of n into distinct parts which do not admit a refinement, i.e. those partitions n = p 1 + p 2 + . . . + p t where no integer p i can be replaced by l ≥ 2 distinct integers q 1 , q 2 , . . . , q l , whose sum is a i and such that the resulting partition is still a partition into distinct parts. In our knowledge, unrefinable partitions have not been investigated so far.
We prove here (cfr. Theorem 9) that a transversal of N n−2 n in N n−1 n is made of rigid commutators which are in one-to-one correspondence with unrefinable partitions into distinct parts with a condition on their minimal excludants, where the minimal excludant of a partition is the least positive integer which does not appear in the partition.
A brief summary on rigid commutators and saturated subgroups is presented in Sec. 2, which also contains a representation of the Sylow 2-subgroup of the symmetric group and to the precise definition of the normalizer chain under investigation. The proof of Theorem 9 and considerations on unrefinable partitions and rigid commutators can be found in Sec. 3.

Preliminaries
Let n be a non-negative integer. Let us define a set of permutations {s i | 1 ≤ i ≤ n} which generates a Sylow 2-subgroup of the symmetric group on 2 n letters.
The Sylow 2-subgroup. Let us consider the set of binary words of length n, where T 0 contains only the empty word. The infinite rooted binary tree T is defined as the graph whose vertices are j≥0 T j and where two vertices, say w 1 . . . w n and v 1 . . . v m , are connected by an edge if |m − n| = 1 and w i = v i for 1 ≤ i ≤ min(m, n). The empty word is the root of the tree and it is connected with both the two words of length 1. We can define a sequence {s i } i≥1 of automorphisms of this tree. Each s i necessarily fixes the root, which is the only vertex of degree 2. The automorphism s 1 changes the value w 1 of the first letter of every non-empty word into w 1 def = (w 1 + 1) mod 2 and leaves the other letters unchanged. If i ≥ 2, we define In general, s i leaves a word unchanged unless the word has length at least i and the letters preceding the i-th one are all zero, in which case the i-th letter is increased by 1 modulo 2. If i ≤ n and the word w 1 . . . w n ∈ T n is identified with the integer 1 + n i=1 2 n−i w i ∈ {1, . . . , 2 n }, then s i acts on T n as the the permutation whose cyclic decomposition is which has order 2. In particular, the group s 1 , . . . , s n acts faithfully on the set T n , whose cardinality is 2 n , as a Sylow 2-subgroup Σ n of the symmetric group Sym(2 n ).

Rigid commutators. The commutator of two elements h and k in a group
In this paper we will only focus on left-normed commutators in s 1 , . . . , s n , therefore, for the sake of simplicity, we write [i 1 , . . . , i k ] to denote the left-normed commutator [s i1 , . . . , s i k ], when k ≥ 2. We also write [i] to denote the element s i and we set [ ] to be the identity permutation.
The set of all the rigid commutators of Σ n is denoted by R and we let Definition 2. A subgroup of Σ n is said to be saturated if it is generated by rigid commutators.
Let us define a special set {t 1 , t 2 , . . . , t n } of rigid commutators where Remark 1. The saturated subgroup T def = t 1 , t 2 , . . . , t n is an elementary abelian regular subgroup of Σ n .
Theorem 3 ([ACGS21].). The normalizer N in Σ n of a saturated subgroup of Σ n is also saturated, provided that N contains T .
For the purposes of this work it is more convenient to use the notation to denote the rigid commutator [i 1 , . . . , i k ]. The commutator of two rigid commutators is again a rigid commutator which, in the above notation, can be computed in the following way: The normalizer chain. The normalizer chain starting at T is defined as In order to show where partitions of integers come into play, let us briefly describe the generators of the first n − 2 normalizers of the chain. First, let us determine the permutations in Σ n normalizing T : for 1 ≤ j < i ≤ n let us define and let us set Next, let us define for each 1 ≤ i ≤ n and j, and Note that, if j ≤ i − 2, then |W i,j | = b j , i.e. it corresponds to the number of partitions of j into at least two distinct parts. The previous elements generate the subgroups in the normalizer chain: Theorem 5 ([ACGS21]). For i ≤ n − 2, the group N i n is the i-th term N i n of the normalizer chain. In particular, the subgroup N i n of Σ n is generated by U n and the rigid commutators ∨[a; X] such that • |X| ≥ 2, • x∈X x ≤ i + 2 − (n − a). The following straightforward consequence is derived: n is independent of n. It equals the (i + 2)-th term of the sequence (a j ) of the partial sums of the sequence (b j ) counting the number of partitions of j into at least two distinct parts.

Unrefinable partitions
Every non-empty finite subset X ⊆ N \ {0} represents a partition of the integer X def = x∈X x into distinct parts. Some partitions, e.g. 7 = 1 + 2 + 4, are not refinable, some others are. For example, in 10 = 1 + 4 + 5, the part 5 can be replaced by 2 + 3 obtaining a partition of 10 = 1 + 2 + 3 + 4 into distinct parts. More precisely: Definition 7. A finite non-empty subset X ⊆ N \ {0} is refinable if there exists x ∈ X and a subset Y ⊆ N \(X ∪ {0}) such that x = Y . If so, X induces a refinable partition into distinct parts of X. Equivalently, (X \ {x})∪ Y is again a partition into distinct parts of X, called a refinement of X. We say that X is an unrefinable partition of X into distinct parts if X is not refinable.
Remark 2. Notice that if X is a refinable partition, then there exists Y as in Definition 7, |Y | = 2, such that x = Y .
Let us now recall the concept of minimal excludant, better known from its use in combinatorial game theory [Spr35,Gru39,FP15] and recently considered in the theory of partitions of integers [AN19, AN20, BM20].
Definition 8. The minimal excludant mex(X) of a set X of positive integers is the least positive integer that is not an element of X.
Rigid commutators and unrefinable partitions are linked by the following consideration depending on Proposition 4. Suppose that a > 1 and that X ⊆ {1, . . . , a − 1} is refinable. If x ∈ X and Y are as in Definition 7, then for some Y ′ and if Z = X, then x ∈ X and Z = ( (1) X = a + 1, (2) X is an unrefinable partition, (3) a ≤ n < a + k.  , and in particular Y = b − 1 ≤ a. We already know that a ≤ b − 1, and so we have the equality X = b = a + 1, as claimed in (1). Suppose now that X is refinable. We have already observed that there exist Moreover, c / ∈ N n−2 n since X = a + 1 > a.
Using the GAP implementation of the algorithmic version of Theorem 5, available at GITHUB (https://github.com/ngunivaq/normalizer-chain), it is easy to compute the first normalizers in the chain of Eq.(2.2). The first values of log 2 |N i n : N i−1 n | are shown in Table 2. Notice that for i ≤ n − 2 the blue numbers correspond to those of the sequence (a j ), whereas black bold numbers represent log 2 |N n−1 n : N n−2 n | and correspond to the number of unrefinable partitions as in Theorem 9. Moreover, notice that the diagonals in the blue area of Table 2 contain the same sequence of numbers. This is not the case when looking at the bold diagonal related to the (n−1)-th normalizer. The reason for this is explained below.  only if m is in the interval a ≤ m < a + mex(X). It is natural to ask when k = mex(X) is the largest possible with respect to a. Since X = a + 1, it is clear that this happens exactly when X is a triangular partition, i.e. X = {1, . . . , k − 1}. We then have that a = X − 1 = k(k − 1)/2 − 1 = (k + 1)(k − 2)/2 is the integer preceding the (k − 1)-th triangular number k(k − 1)/2. Since a = (k + 1)(k − 2)/2 > (k − 2) 2 /2 we have that √ 2a + 2 is an upper bound for the largest possible value of k. In this case, if a ≥ 3, then c ∈ N n−1 n \ N n−2 n implies n < a + k ≤ a + √ 2a + 2 ≤ ( √ a + 1) 2 , i.e. a > n − 1 − 2 √ n.
Example 10. When n = 8 we have computed that log 2 |N 7 8 : N 6 n | = 7. Indeed, applying Theorem 9 it can be shown that the normalizer N 7 8 can be generated by the generators of N 6 8 and by the following 7 rigid commutators of the type ∨ It is natural to wonder whether a general closed formula for log 2 |N n−1 n : N n−2 n | may be found, even though at the time of writing this does not seem an easy task. To our knowledge, the problem of determining a closed formula or a generating function for the sequence c j of the number of unrefinable partitions into distinct parts is open. The first values of c j are shown here in Table 3, while a list of the first j 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 c j 1 1 1 1 1 2 1 1 2 3 1 2 2 3 5 Table 3. First values of the sequences (c j ) 1000 integers can be found in the On-Line Encyclopedia of Integer Sequences [OEI, https://oeis.org/A179009]. The related problem of determining a closed formula for the number of unrefinable partitions with a given minimal excludant, related to the number of partitions of Theorem 9, seems at the moment out of reach.