Interpreting the truncated pentagonal number theorem using partition pairs

In 2012 Andrews and Merca gave a new expansion for partial sums of Euler’s pentagonal number series and expressed k−1 ∑ j=0 (−1)(p(n− j(3j + 1)/2)− p(n− j(3j + 5)/2− 1)) = (−1)Mk(n) where Mk(n) is the number of partitions of n where k is the least integer that does not occur as a part and there are more parts greater than k than there are less than k. We will show that Mk(n) = Ck(n) where Ck(n) is the number of partition pairs (S,U) where S is a partition with parts greater than k, U is a partition with k − 1 distinct parts all of which are greater than the smallest part in S, and the sum of the parts in S ∪U is n. We use partition pairs to determine what is counted by three similar expressions involving linear combinations of pentagonal numbers. Most of the results will be presented analytically and combinatorially.


Introduction
Euler's pentagonal number theorem gives an easy recurrence for the number of partitions of n, denoted by p(n).Namely, (−1) j+1 (p(n − j(3j − 1)/2) + p(n − j(3j + 1)/2)) where p(k) = 0 if k < 0. An interesting question is to determine how far off from p(n) we are if we truncate this recurrence sum before we reach n − j(3j − 1)/2 < 0 or n − j(3j + 1)/2 < 0. In [1] Andrews and Merca answered this question when we stop the recurrence sum after an odd number of terms.In [3] Kolitsch gave an answer when we stop the recurrence sum with p(n − 1) + p(n − 2).In Section 2 we will use generating functions to prove the general results.In section 3 we will interpret the results combinatorially.

A Generating Function Proof
If we define B k (n) for k 0 to be the number of partition pairs (S, T ) where S is a partition with parts greater than k, T is a partition with k distinct parts all of which are greater than the smallest part in S, and the sum of the parts in S ∪ T is n, then the generating function for B k (n) is given by Theorem 1.
As an immediate consequence of Theorem 1 and Euler's pentagonal number theorem we get Corollary 2.
Comparing coefficients of q n , we get the desired corollary.
To prove Theorem 1 we note that for j > k the generating function for partitions that fulfill the criterion to be a partition S as described above with smallest part j is q j (q j ;q) ∞ and the corresponding generating function for partitions that fulfill the criterion to be a partition T as described above is the electronic journal of combinatorics 22(2) (2015), #P2.55 (−1) j+k+1 (q j(3j−1)/2 + q j(3j+1)/2 ).
To rewrite we are using identity 10 on page 29 in [2] with x = q k .If we define C k (n) for k > 0 to be the number of partition pairs (S, U ) where S is a partition with parts greater than k, U is a partition with k − 1 distinct parts all of which are greater than the smallest part in S, and the sum of the parts in S ∪ U is n then we have the following theorem.
From the proof of Theorem 1 we have (q; q) ∞ which gives the desired result.As an immediate consequence of Theorem 3 we get the electronic journal of combinatorics 22(2) (2015), #P2.55 Corollary 4.
This corollary follows immediately from Corollary 2 by observing that Theorem 3 gives From Theorem 1 in [1] we get where M k (n) is the number of partitions of n where k is the least integer that does not occur as a part and there are more parts greater than k than there are less than k.
where D k (n) is the number of partition pairs (S, T ) where S is a partition with parts greater than k containing at least one part equal to k + 1, T is a partition with k distinct parts all of which are greater than k + 1, and the sum of the parts in S ∪ T is n.
This corollary follows immediately by observing that where E k (n) is the number of partition pairs (S, U ) where S is a partition with one part equal to k and all other parts greater than k, U is a partition with k − 1 distinct parts all of which are greater than k, and the sum of the parts in S ∪ U is n.

A Combinatorial Look at Our Results
In this section we will combinatorially verify the result observed from Theorem 3 that was used to prove Corollary 4 and the companion result that relates C k (n) and B k (n).These two relationships are stated in the next theorem.
Theorem 8.For k > 0, To prove part (i) of Theorem 8 we need to show how the partitions of n−k(3k+5)/2−1 bijectively correspond to the partition pairs (S, T ) for n where S is a partition with all parts greater than k and k + 1 is included as a part and T is a partition into k distinct parts greater than k.Given a partition P = {a 1 , a 2 , . . ., a r } with a 1 a 2 • • • a r and r i=1 a i = n − k(3k + 5)/2 − 1, we will construct a partition pair ({k + 1} ∪ {a i : where α(m) is the number of parts equal to m in P .This gives a partition pair of the desired type since counts the number of partition pairs for n of the type counted by C k+1 (n).
To prove part (ii) of Theorem 8 we need to show that the partitions of n − k(3k + 1)/2 bijectively correspond to the partition pairs counted by B k (n) + C k (n).Given a partition P = {a 1 , a 2 , . . ., a r } with a 1 a 2 • • • a r and r i=1 a i = n − k(3k + 1)/2, we will define t i = (k + i) + i j=1 α(k + 1 − j) for i = 1, 2, . . ., k.If t 1 is less than the smallest part in P that is larger than k, our partition pair will be given by ({t 1 }∪{a i : a i > k}, {t 2 , t 3 , . . ., t k }) (note if k = 1 then T = { }).These partition pairs are counted by C k (n).If t 1 is greater than or equal to the smallest part in P that is larger than k, our partition pair will be given by ({a i : a i > k}, {t 1 , t 2 , t 3 , . . ., t k }).These partition pairs are counted by B k (n).
We now present a combinatorial proof of Corollary 5 by showing how the partitions of n counted by M k (n) bijectively correspond to the partition pairs counted by C k (n).Given a partition . ., u.The corresponding partition pair will be (S, T ) where the k − 1 elements of T are defined by t j = smallest value among the x i 's where a i = j for j = 1, 2, . . ., k − 1 and S = {b i : i v − u} ∪ ({x i : i = 1, 2, . . ., u} − T ).
We now define N k (n) for k > 0 to be the number of of partitions of n where 1, 2, . . ., k all occur as a part and there are more parts greater than k than there are less than or equal to k.The following theorem holds.
We can use a correspondence similar to the one used to prove Corollary 5 to prove Theorem 9. Given a partition u < v and u i=1 a i + v i=1 b i = n, we will define x i = b v−u+i + a i for i = 1, 2, . . ., u.The corresponding partition pair will be (S, T ) where the k elements of T are defined by t j = smallest value among the x i 's where a i = j for j = 1, 2, . . ., k and S = {b i : i v − u} ∪ ({x i : i = 1, 2, . . ., u} − T ).