Group Actions on Partitions

We introduce group actions on the integer partitions and their variances. Using generating functions and Burnside’s lemma, we study arithmetic properties of the counting functions arising from group actions. In particular, we find a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.


Introduction
In the theory of integer partitions, the conjugation is one of the most important operations on partitions.The Durfee square of size d × d in the partition λ is defined by the largest integer d such that the partition λ has at least d parts d.Using the Durfee square, we can represent the partition λ by (d, π 1 , π 2 ), where d is the size of the Durfee square and π 1 and π 2 are partitions into parts d.Let D be the set of partitions represented by the Durfee square decomposition D := {(d, π 1 , π 2 ) : d 0, π 1 and π 2 are partitions into parts d}.
For the symmetric group G = S 2 , we define a group action on D by σ(d, π 1 , π 2 ) = (d, π σ(1) , π σ(2) ), for σ ∈ G.When σ = (12), this group action corresponds to the conjugation of the partition.In the light of that we can understand the conjugation of partitions as a group action, it is natural to ask whether there are interesting consequences on partitions in the view of group actions.Moreover, once we define a group action on the set X, then it is also natural to examine how many orbits there are.For an element x in an orbit, we define the weight of x as an element of X and we define |X/G|(n) is the number of orbits of weight n.For example, |D/S 2 |(n) is the number of partitions of n, where if two partitions are conjugate to each other, then they are considered to be the same partition.
The goal of this paper is investigating properties of counting functions arising from group actions on various partition sets X and groups G.To this end, we will use Burnside's lemma frequently, which says that the number of orbits |X/G| is given by where X g is the invariant subset of X under the action of g.One of the main results is a modulo 4 congruence involving the number of ordinary partitions and the number of partitions into distinct parts.Before stating the result, we introduce some notation.Let p(n) be the number of ordinary partitions of n and let d(n) be the number of partitions of n into distinct parts.For a given partition λ, (λ) is defined by the largest part of λ.If λ is an empty partition, then we define (λ) = 0.
In the next section we prove Theorems 1, and we consider multi-partitions and unimodal sequences under group actions in the remainder of the paper.

Proof of Theorem 1
Let C 2 be the set defined by Note that we allow ν i 's can have zeros as parts and recall that two sequences that differ in the order of their terms define different compositions, while they are considered to be the same partition.The weight of an element (d, ν 1 , ν 2 ) ∈ C 2 is defined by d 2 plus the sum of parts in ν 1 and the first part of ν 2 .For the convenience, the empty composition is the electronic journal of combinatorics 24(3) (2017), #P3.58 considered to be {0} so that we can define the first part of an empty composition is zero.For the symmetric group G = S 2 , we define a group action on C 2 by exchanging the first part of ν i and ν σ(i) for i = 1 or 2. By employing Burnside's lemma, we find that Here and in the sequel, for a given set X, |X|(n) denotes the number of elements in the set X with the weight n.Now we will count how many partitions there are among the elements in the invariant subsets.To this end, we define where D is the set of partitions represented by Durfee square decompositions.By employing general combinatorial arguments [1], we obtain that Here we have used the usual q-series notation, for n ∈ N 0 ∪ {∞}.By using the inclusion-exclusion principle, we derive that Therefore, we find that It turns out that we can reformulate f (q) by using a false theta function.
Proof of Theorem 1.We first note that (1) implies that From Euler's pentagonal number theorem [1, Cor 1.7], we find that Therefore, by Lemma 3, we obtain that From ( 3) and the fact that we conclude Theorem 1.
Now we turn to prove Lemma 3.
Proof of Lemma 3. First we recall that a Bailey pair relative to a is a pair of sequences (α n , β n ) n 0 satisfying If (α n , β n ) is a Bailey pair relative to 1, then Bailey Lemma (see [12] for example) implies that From [11], we find that ) for n > 0, the electronic journal of combinatorics 24(3) (2017), #P3.58 consists of a Bailey pair relative to 1. Recall (see [4], for example) that if (α n , β n ) is a Bailey pair relative to a, then is also a Bailey pair relative to a. Using this with (6), we obtain a new Bailey pair relative to 1, α 0 = 1, with α, β → 0 and γ = τ = q, we derive that From ( 7), (8), and (5), we obtain the desirable identity.
Note that where we use the q-binomial coefficient defined by Recall that the q-binomial coefficient n k generates the partitions which fit inside a (n − k) × k rectangle.From (9), we can see that |C 2 /S 2 | D (n) is the number of partitions of n with certain Durfee dissections (See [2] for more details on Durfee dissections.).Then, for all non-negative integers n, the electronic journal of combinatorics 24(3) (2017), #P3.58

Other Results
In this section, we study the number of orbits in the sets of multi-partitions and in the sets of unimodal sequences under group actions.

Multi-partitions
Let P 2 be the set of bi-partitions : λ 1 and λ 2 are ordinary partitions}.
For the group G = S 2 , we define a group action on P 2 by Let |P 2 /S 2 |(n) be the number of orbits with weight n.Then, |P 2 /S 2 |(n) is the number of bi-partitions of n, where bi-partitions in the same orbit define the same bi-partition.By simple combinatorial arguments and Burnside's formula, we find that Let p −2 (n) be the number of bi-partitions of n.Then, we have From the famous Ramanujan congruence p(5n + 4) ≡ 0 (mod 5) and a bi-partition congruence [6] p −2 (5n + 3) ≡ 0 (mod 5), we can easily see that |P 2 /S 2 |(5n + 3) ≡ 0 (mod 5).
We define the set of tri-partitions and λ 3 are ordinary partitions}.
For G = S 3 , we define a group action on P 3 by We may think each partition λ i is on a vertex of a regular triangle.Then, we can understand |P 3 /S 3 |(n) is the number of tri-partitions of n, where the partitions are considered to be the same if they are invariant under a reflection or a rotation fixing the triangle.From Burnside's lemma, we observe that the electronic journal of combinatorics 24(3) (2017), #P3.58 Proposition 5.For all non-negative integers n, Moreover, Proof.H.-C. Chan [5] defined the cubic partition function c(n) by and proved that c(3n + 2) ≡ 0 (mod 3).Therefore, it suffices to show that where p −3 (n) is defined by We define the Dedekind eta function η(z) by η(z) = q 1/24 (q) ∞ with q = exp(2πiz).
Since η 9 (3z) η 3 (z) is a modular form of weight 3 and level 3 with character −3 • and the dimension of such modular forms space is 2, we find that by checking the first two coefficients.Here, U 3 is an operator defined by ( a(n)q n ) |U 3 := a(3n)q n .(For more details on modular forms, see [10] for example.)From ( 12), we find that η 9 (3z) which proves p −3 (3n + 2) ≡ 0 (mod 9) as desired.We also note that η 3 (3z)η 3 (6z) is a modular form of weight 2 and level 6.Proceeding as before (for this time, we need to check the first three coefficients), we deduce that the electronic journal of combinatorics 24(3) (2017), #P3.58 In terms of cubic partitions, the above modular identity is equivalent to From ( 13) and ( 14), we can obtain the desirable generating function for |P 3 /S 3 |(3n+2).

Unimodal sequences
Recall that a unimodal sequence is a sequence which is weakly increasing up to a point (called the peak), and then weakly decreasing thereafter.The weight of such a sequence is the sum of all of its terms.We define the set of unimodal sequeces U := {(p, π 1 , π 2 ) : p 0, π 1 and π 2 are partitions into parts p}.
For an element (p, π 1 , π 2 ), its weight is defined by p plus the sum of parts in π 1 and π 2 .
Recall that C 2 is the set defined by and ν 2 are compositions into non-negative integers d}.
To match the weights, in this subsection, the weight of an element (d, ν 1 , ν 2 ) ∈ C 2 is redefined by d plus the sum of parts in ν 1 and ν 2 .For the symmetric group G = S 2 , we define a group action on C 2 by exchanging the first part of ν 1 and the first part of ν 2 .By employing Burnside's lemma, we find that |(n) .
Here we will count how many unimodal sequences there are among the elements in the invariant subsets.To this end, we define Then, we can see that where we have used (2) for the last equality.From this, we find that |C 2 /S 2 | U (n) is the number of unimodal sequences of weight n such that the largest part in the partition before the peak is larger than or equal to the largest part in the partition after the peak.From [9, Theorem 1.1], we find that the electronic journal of combinatorics 24(3) (2017), #P3.58 where (α n , β n ) is a Bailey pair relative to a. Therefore, by plugging the Bailey pairs (7) and ( 8) into (15), we derive that (−1) n q n(n+1)/2+(2n+1)r+r(2r+1) , where we replace n and r by −n − 1 and −r in the second summation for the last equality.
In summary, we have obtained a Hecke-type double sum expression for the |C 2 /S 2 | U (n) generating function.
Theorem 6.We have We may think the double sum in the right hand side of (16) as a false theta series in the sense of that it has a wrong sign compare to the theta series f a,b,c (x, y, q) = r,s 0 − r,s<0 (−1) r+s x r y s q a( r 2 )+brs+c( s 2 ) , which is closely related to mock theta functions [7].In a recent paper [8], the author and J. Lovejoy introduced a unimodal T-sequence.In a unimodal T-sequence, we have a peak and there are partitions into parts the peak in the right, in the left, and below of the peak.We define the set of unimodal T-sequences T := {(p, π 1 , π 2 , π 3 ) : p 0, π 1 , π 2 , and π 3 are partitions into parts p}.
For an element in T (resp.C 3 ), its weight is defined by the size of the peak (i.e.p) plus the sum of parts in the partitions (resp.compositions).
the electronic journal of combinatorics 24(3) (2017), #P3.58 For G = S 3 , we define a group action on C 3 by switching the first part of ν i and the first part of ν σ(i) .As before, we investigate Then, we find that q n (q) 3 n + 3q n (q) n n k=0 q 2k (q) 2 k + 2q n n k=0 q 3k (q; q) 3 k .
By employing the inclusion-exclusion principle on the largest parts, we find that From ( 2), ( 8), and the above, we derive that In particular, T 1 (n) > T 2 (n) for all positive integers n.

Corollary 4 .
Let D 2 (n) be the number of partitions of n with 2 successive Durfee squares of sizes d 1 × d 1 and d 2 × d 2 such that there is a part of size d 2 below the second Durfee square.

k.Proposition 7 .
Let T 1 (n) be the number of unimodal T-sequences of weight n such that the partition below the peak has the Durfee rectangle of size (k + 1) × k.Let T 2 (n) be the number of unimodal T -sequences of weight n such that if the largest part of the partition in the right of the peak is k, then the largest part of the partition in the left (resp.below) of the peak is less than or equal to (resp.less than) k.Then, for all non-negative integers n,|C 3 /S 3 | T (n) = T 1 (n) − T 2 (n).