On Powers of 2 Dividing the Values of Certain Plane Partition Functions

Darrin D. Frey and James A. Sellers^{1}

Department of Science and Mathematics Cedarville University Cedarville, OH 45314

Email addresses: freyd@cedarville.edu and sellersj@cedarville.edu

Abstract

We consider two families of plane partitions: totally symmetric self-complementary plane partitions (TSS-CPPs) and cyclically symmetric transpose complement plane partitions (CSTCPPs). If T (n) and C(n) are the numbers of such plane partitions in a 2n × 2n × 2n box, then

ord_{2}(T (n)) = ord_{2}(C(n))

for all n ≥ 1. We also discuss various consequences, along with other results on ord_{2}(T (n)).

2000 Mathematics Subject Classification: 05A10, 05A17, 11P83

Keywords: alternating sign matrices, totally symmetric self-complementary plane partitions, TSSCPP, cyclically symmetric transpose complement plane partitions, CSTCPP, Jacobsthal numbers

1 Introduction

In his book “Proofs and Confirmations,” David Bressoud [2] discusses the rich history of the Alternating Sign Matrix conjecture and its proof. One of the themes of the book is the connection between alternating sign matrices and various families of plane partitions. (Reference [3] gives a synopsis of this work.)

Pages 197–199 of [2] list ten families of plane partitions which have been extensively studied. The last family in this list is the set of totally symmetric self–complementary plane partitions (TSSCPPs) which fit in a 2n × 2n × 2n box. In 1994, Andrews [1] proved that the number of such partitions is given by

T (n) = ∏_{j=0}^{n−1} \frac{(3j + 1)!}{(n + j)!}.  (1)

1Current address: Department of Mathematics, Eberly College of Science, The Pennsylvania State University, University Park, PA 16802

(This formula also gives the number of n × n alternating sign matrices. See [9]. The values T (n) can be found as sequence \underline{A005130} in [8].)

Another family mentioned by Bressoud is the set of cyclically symmetric transpose complement plane partitions (CSTCPPs). We will let C(n) denote the number of such partitions that fit in a 2n × 2n × 2n box. (The values C(n) make up sequence \underline{A051255} in [8].) In 1983, Mills, Robbins, and Rumsey [7] proved that

C(n) = ∏_{j=0}^{n−1} \frac{(3j + 1)(6i)!(2i)!}{(4i + 1)!(4i)!}  (2)
= ∏_{j=0}^{n−1} \frac{(3j + 1)!(6i)!(2i)!}{(3j)!(4i + 1)!(4i)!}.

The goal of this note is to consider arithmetic properties of, and relationships between, the two functions T (n) and C(n). In particular, we will prove that, for all n ≥ 1,

ord_{2}(T (n)) = ord_{2}(C(n))

where ord_{2}(m) is the highest power of 2 dividing m. This is the gist of Section 2 below. Using this fact and additional tools developed in Section 3, we will prove the following congruences:

1. For all n ≥ 0,

T (n) ≡ C(n) (mod 4).

2. For all n ≥ 0, n not a Jacobsthal number,

T (n) ≡ C(n) (mod 16).

(The Jacobsthal numbers {J_{n}}_{n=0}^{∞} are the numbers satisfying J_{0} = J_{1} = 1 and J_{n+2} = J_{n+1} +2J_{n} for n ≥ 0. The values J_{n} make up sequence \underline{A001045} in [8].)

Indeed, if we ignore those values of n which are Jacobsthal numbers, we will prove that, for fixed k ≥ 1,

T (n) ≡ C(n) (mod 2^{k})

for all but a finite set of values of n. Moreover, the values of n for which this congruence does not hold must satisfy n ≤ J_{2k+1}. This extends earlier work of the authors [4].

The above results imply some interesting arithmetic properties of C′(n), the number of CSTCPPs in a 2n × 2n × 2n box which are not TSSCPPs. In particular, we have

C′(n) ≡ 0 (mod 4)

for all n ≥ 0. Moreover, we have, for fixed k ≥ 1,

C′(n) ≡ 0 (mod 2^{k})

for all but finitely many non-Jacobsthal numbers. There does not appear to be any obvious reason why this property should hold, nor why C′(n) behaves differently when n is a Jacobsthal number.

2 The 2-adic Relationship Between C(n) and T (n) Throughout this note, we make use of the following lemma: Lemma 2.1. For any prime p and positive integer N,

ord_{p}(N!) = ∑_{k≥1} ⌊\frac{N}{p^{k}} ⌋ .

Proof. For a proof, see [6, Theorem 2.29].

The goal of this section is to prove the following theorem: Theorem 2.2. For all n ≥ 1,

ord_{2}(C(n)) = ord_{2}(T (n)).

Proof. The proof simply involves a manipulation of various sums using Lemma 2.1. Given (2), we have

ord_{2}(C(n)) = ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{3j + 1}{2^{k}}⌋ + ⌊\frac{6j}{2^{k}} ⌋ − ⌊\frac{3j}{2^{k}} ⌋ + ⌊\frac{2j}{2^{k}} ⌋ − ⌊\frac{4j}{2^{k}} ⌋ − ⌊\frac{4j + 1}{2^{k}}⌋
= ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{3j + 1}{2^{k}}⌋ + (⌊\frac{3j}{2^{k−1}} ⌋ − ⌊\frac{3j}{2^{k}} ⌋) + ⌊\frac{2j}{2^{k}} ⌋ − 2 ⌊\frac{4j}{2^{k}} ⌋
= ∑_{j=0}^{n−1} ∑_{k≥1} \{⌊\frac{3j + 1}{2^{k}}⌋ − (2 ⌊\frac{2j}{2^{k−1}} ⌋ − ⌊\frac{j}{2^{k−1}} ⌋)\} + ∑_{j=0}^{n−1} 3j
= ∑_{j=0}^{n−1} ∑_{k≥1} \{⌊\frac{3j + 1}{2^{k}}⌋ − (2 ⌊\frac{2j}{2^{k}} ⌋ − ⌊\frac{j}{2^{k}} ⌋ + 2(2j) − j)\} + ∑_{j=0}^{n−1} 3j
= ∑_{j=0}^{n−1} ∑_{k≥1} \{⌊\frac{3j + 1}{2^{k}}⌋ − (2 ⌊\frac{2j}{2^{k}} ⌋ − ⌊\frac{j}{2^{k}} ⌋)\} + ∑_{j=0}^{n−1}(3j − 4j + j)
= ∑_{j=0}^{n−1} ∑_{k≥1} \{⌊\frac{3j + 1}{2^{k}}⌋ − (⌊\frac{2j}{2^{k}} ⌋ + ⌊\frac{2j + 1}{2^{k}}⌋ − ⌊\frac{j}{2^{k}} ⌋)\}
= ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{3j + 1}{2^{k}}⌋ − ∑_{j=0}^{2n−1} ∑_{k≥1} ⌊\frac{j}{2^{k}} ⌋ + ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{j}{2^{k}} ⌋
= ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{3j + 1}{2^{k}}⌋ − ∑_{j=n}^{2n−1} ∑_{k≥1} ⌊\frac{j}{2^{k}} ⌋
= ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{3j + 1}{2^{k}}⌋ − ∑_{j=0}^{n−1} ∑_{k≥1} ⌊\frac{n + j}{2^{k}} ⌋
= ord_{2}(T (n))

again using Lemma 2.1 and (1).

3 A Finiteness Result In this section, we show that there is an upper bound on the values of n for which ord_{2}(T (n)) = k for any positive integer k. To do this, we use insight obtained from our work in [4]. In that paper, we studied the functions

N_{k}^{#}(n) = ∑_{j=0}^{n−1} ⌊\frac{3j + 1}{2^{k}}⌋ and D_{k}^{#}(n) = ∑_{j=0}^{n−1} ⌊\frac{n + j}{2^{k}} ⌋

which implicitly appear in the second-to-last line of the string of equalities in the proof of Theorem 2.2.

Definition 3.1. We define the function c_{k}(n) := N_{k}^{#}(n) − D_{k}^{#}(n) for any positive integer n, so that

ord_{2}(T (n)) = ∑_{k≥1} c_{k}(n).

Theorem 3.2. Suppose 0 ≤ ρ_{k} < 2^{k}. Then

c_{k}(ρ_{k}) = \{ \begin{matrix}0  if & 0 ≤ ρ_{k} ≤ J_{k−1} \\ ρ_{k} − J_{k−1} if & J_{k−1} < ρ_{k} ≤ 2^{k−1} \\ J_{k} − ρ_{k} if & 2^{k−1} ≤ ρ_{k} < J_{k} \\ 0  if & J_{k} ≤ ρ_{k} < 2^{k}. \end{matrix}

Moreover,

c_{k}(n + 2^{k}) = c_{k}(n) for all n, k in ℕ.

Furthermore, if n = 2^{k}q_{k} + ρ_{k}, then

c_{k}(n) = c_{k}(2^{k}(q_{k} + 1) − ρ_{k}).

Proof. This follows from Lemmas 5.1 through 5.4 of [4] for n which are not Jacobsthal numbers (though in the case of Lemmas 5.2 and 5.3 of [4] one has to look inside the proof to get this stronger result), and Theorem 4.1 of [4] for n which are Jacobsthal numbers.

Since the submission of this article, the authors have found a simpler proof of Theorem 3.2, which will appear in [5]. It is clear from Theorem 3.2 that the values of the function c_{k} increase in increments of 1 beginning at J_{k−1}, reach a peak at 2^{k−1}, and then decrease in increments of 1 between 2^{k−1} and J_{k}. Propositions 3.3 and 3.4 show us that when the parities of i and j are the same, the ascents for c_{i} and c_{j} “line up” in such a way that if say j > i, c_{i} is beginning one of its ascents at the same point that c_{j} is beginning an ascent, so that there is an interval where the two agree. Of course c_{i} will reach its peak first, so beyond that point, the two will fail to agree for some time. However, given the periodic nature of these functions, they will realign at some later point. See the table in the Appendix for a demonstration of this phenomenon.

We use this insight to achieve a lower bound for ord_{2}(T (n)) when n is between two Jacobsthal numbers.

Proposition 3.3. For 0 ≤ i ≤ ⌈\frac{k}{2} − 1⌉ ,

J_{k} = J_{k−2i} + J_{2i−1} · 2^{k−2i+1}.

Proof. Recall from [4] that, for all m ≥ 0, J_{m} = \frac{2^{m+1} + (−1)^{m}}{3} . Then

J_{k−2i} + J_{2i−1} · 2^{k−2i+1} = \frac{2^{k−2i+1} + (−1)^{k−2i} + 2^{2i} · 2^{k−2i+1} + (−1)^{2i−1}2^{k−2i+1}}{3}
= \frac{2^{k+1} + (−1)^{k}}{3} upon simplification
= J_{k}.

Proposition 3.3 allows us to show that, for a given n, the functions c_{k} are equal to each other for several values of k.

Proposition 3.4. Suppose J_{k} ≤ n ≤ 2^{k}, say n = J_{k} + r where r ≥ 0. If 0 ≤ i ≤ ⌈\frac{k}{2} − 1⌉ and 0 ≤ r ≤ J_{k−1−2i},

then

c_{k+1−2i}(n) = r.

Proof. This follows from Theorem 3.2 and Proposition 3.3.

A symmetric result is true when 2^{k} ≤ n ≤ J_{k+1}.

Corollary 3.5. Suppose 2^{k} ≤ n ≤ J_{k+1}, say n = J_{k+1} − r where r ≥ 0.

If 0 ≤ i ≤ ⌈\frac{k}{2} − 1⌉ and 0 ≤ r ≤ J_{k−1−2i}, then

c_{k+1−2i}(n) = r.

Proof. This result follows directly from the fact stated in Theorem 3.2 which says that, if n = 2^{k}q_{k} + ρ_{k}, then

c_{k}(n) = c_{k}(2^{k}(q_{k} + 1) − ρ_{k}).

In our case, we replace k by k + 1 and note that q_{k+1} = 0, so we have

c_{k+1}(n) = c_{k+1}(J_{k+1} − r)
= c_{k+1}(2^{k+1} − (J_{k+1} − r))
= c_{k+1}(J_{k} + r) using [4, Lemma 3.2]
= r by Proposition 3.4.

Theorem 3.6. Let i, k ∈ ℕ such that 0 ≤ i ≤ ⌈\frac{k}{2} − 1⌉ and k − i odd. If (J_{i} + 1) ≤ r ≤ 2^{k} − (J_{i} + 1), then

ord_{2}(T (J_{k} + r)) ≥ (J_{i} + 1) (\frac{k − i − 1}{2}) .

Proof. If r ≤ J_{k−2i−1}, where 0 ≤ i ≤ ⌈\frac{k}{2} − 1⌉ , then by Proposition 3.4 or Corollary 3.5,

c_{k+1−2i}(J_{k} + r) = r.

Now, suppose i is such that k −i is odd and let 2j +1 = k −i. If J_{i} +1 ≤ r ≤ J_{k−1}, then r ≰ J_{i} = J_{k−(k−i)} = J_{k−1−2j}, but r ≤ J_{i+2} = J_{k−(k−i−2)} = J_{k−1−2(j−1)}. Hence

c_{k+1}(J_{k} + r) = c_{k−1}(J_{k} + r) = ⋯ = c_{k+1−2(j−1)}(J_{k} + r) = r.

Thus, ord_{2}(T (J_{k} + r)) has j = \frac{k−i−1}{2} summands of value r, so that

ord_{2}(T (J_{k} + r)) ≥ r (\frac{k − i − 1}{2}) ≥ (J_{i} + 1) (\frac{k − i − 1}{2}) .

Corollary 3.7. If J_{m−1} < n < J_{m}, then

ord_{2}(T (n)) ≥ ⌊\frac{m}{2}⌋ .

Proof. We break the proof into two cases. First, assume that m = 2k, so n = J_{2k−1}+r where 0 < r < 2J_{2k−2}. Using Theorem 3.6 with i = 0 yields

ord_{2}(T (n)) ≥ (J_{0} + 1)(k − 1) if 2 ≤ r ≤ 2J_{2k−2} − 2
> k.

If r = 1 or r = 2J_{2k−2} − 1 then

c_{2k}(n) = c_{2k−2}(n) = ⋯ = c_{2}(n) = 1

noting that 2 = 2k − 2(k − 1), so

ord_{2}(T (n)) ≥ k.

Since k = ⌊\frac{m}{2}⌋, we have our result.

Next, assume that m = 2k + 1, so n = J_{2k} + r where 0 < r < 2J_{2k−1}. Using Theorem 3.6 with i = 1 yields

ord_{2}(T (n)) ≥ (J_{1} + 1)(k − 1) if 2 ≤ r ≤ 2J_{2k−1} − 2
> k.

If r = 1 or r = 2J_{2k−1} − 1 then

c_{2k+1}(n) = c_{2k−1}(n) = ⋯ = c_{3}(n) = 1

noting that 3 = 2k + 1 − 2(k − 1), so

ord_{2}(T (n)) ≥ k.

Since k = ⌊\frac{m}{2}⌋, we have our result.

Corollary 3.8. If ord_{2}(T (n)) = k with k ≥ 1, then n < J_{2k+1}.

Proof. Suppose ord_{2}(T (n)) = k. From [4, Theorem 4.1], n is not a Jacobsthal number since ord_{2}(T (J_{i})) = 0 for all i. Moreover, by Corollary 3.7, if J_{2j−1} < n < J_{2j+1}, then ord_{2}(T (n)) ≥ j. So if j > k, n is not between J_{2j−1} and J_{2j+1}. The largest number remaining is J_{2k+1} − 1 and, in fact, ord_{2}(T (J_{2k+1} − 1)) = k.

4 Implications

It is clear from Theorem 2.2 that

T (n) ≡ C(n) (mod 2).

However, much more can be said.

Theorem 4.1. For all n ≥ 1,

T (n) ≡ C(n) (mod 4).

Proof. Given Theorem 2.2, it is clear that Theorem 4.1 is automatically true for those values of n for which ord_{2}(T (n)) ≥ 1. Hence, we only need to focus on those n for which ord_{2}(T (n)) = 0.

As noted in [4], ord_{2}(T (n)) = 0 if and only if n is a Jacobsthal number. Via straightforward calculations based on (1) and (2), it can be proved that, for all m ≥ 1,

T (J_{m}) ≡ C(J_{m}) ≡ (−1)^{m−1} (mod 4).

Theorem 4.2. For all n ≥ 1, n not a Jacobsthal number,

T (n) ≡ C(n) (mod 16).

Proof. We need only check those values of n for which 1 ≤ ord_{2}(T (n)) ≤ 2, so, from Corollary 3.8, only 1 ≤ n ≤ J_{5} − 1 or 1 ≤ n ≤ 20. This is straightforward using Maple and (1) and (2).

One last congruential implication is noted here.

Theorem 4.3. For all positive integers k and all but finitely many n ≥ 1, n not a Jacobsthal number,

T (n) ≡ C(n) (mod 2^{k}).

Proof. This is quickly proved since all that must be checked are those values of n for which 1 ≤ ord_{2}(T (n)) ≤ k − 2. Corollary 3.8 implies that the only nonJacobsthal positive integers n for which ord_{2}(T (n)) ≤ k − 2 satisfy 1 ≤ n ≤ J_{2k+1} − 1.

Finally, we note that results analogous to Corollary 3.8 and Theorem 4.3 do not appear to hold for primes p > 2. We have confirmed this computationally in regards to Theorem 4.3, and have proved that the finiteness result in Corollary 3.8 can only hold for p = 2. In fact, we [5] have proved the following:

Theorem 4.4. If p is a prime greater than 3, then for each nonnegative integer k there exist infinitely many positive integers n for which ord_{p}(A(n)) = k.

A result similar to Theorem 4.4 can be proved for p = 3, although it is a bit weaker. See [5].

References

[1] G. E. Andrews, Plane partitions V. The TSSCPP conjecture, Journal of Combinatorial Theory, Series A, 66 (1994), 28-39.

[2] D. M. Bressoud, “Proofs and Confirmations: The story of the alternating sign matrix conjecture”, Cambridge University Press, 1999.

[3] D. M. Bressoud and J. Propp, How the Alternating Sign Matrix Conjecture Was Solved, Notices of the American Mathematical Society, 46 (6) (1999), 637-646.

[4] D. D. Frey and J. A. Sellers, Jacobsthal Numbers and Alternating Sign Matrices, Journal of Integer Sequences, 3 (2000), Article 00.2.3.

[5] D. D. Frey and J. A. Sellers, On Efficient Calculation of the Number of Alternating Sign Matrices, preprint, 2001.

[6] C. T. Long, “Elementary Introduction to Number Theory”, 3rd edition, Waveland Press, Inc., Prospect Heights, IL, 1995.

[7] W. H. Mills, D. P. Robbins, and H. Rumsey, Alternating sign matrices and descending plane partitions, Journal of Combinatorial Theory, Series A, 34 (1983), 340-359.

[8] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. Published electronically at http://www.research.att.com/∼njas/sequences/.

[9] D. Zeilberger, Proof of the alternating sign matrix conjecture, Electronic Journal of Combinatorics 3 (1996), R13. http://www.research.att.com/∼njas/sequences/

Appendix

The table below includes the values of the functions c_{2}(n), c_{4}(n), c_{6}(n) and c_{8}(n) for n between 85 and 171, which are the Jacobsthal numbers J_{8} and J_{9}. We provide this table to show the periodic nature of the functions c_{k}(n), and to motivate Proposition 3.3. It should be noted that, if we were to build a similar table for values of n between J_{2m−1} and J_{2m}, then we would focus attention on functions c_{k}(n) where k is odd rather than even.

\begin{matrix}n & c_{2}(n) & c_{4}(n) & c_{6}(n) & c_{8}(n) \\ 85 & 0 & 0 & 0 & 0 \\ 86 & 1 & 1 & 1 & 1 \\ 87 & 0 & 2 & 2 & 2 \\ 88 & 0 & 3 & 3 & 3 \\ 89 & 0 & 2 & 4 & 4 \\ 90 & 1 & 1 & 5 & 5 \\ 91 & 0 & 0 & 6 & 6 \\ 92 & 0 & 0 & 7 & 7 \\ 93 & 0 & 0 & 8 & 8 \\ 94 & 1 & 0 & 9 & 9 \\ 95 & 0 & 0 & 10 & 10 \\ 96 & 0 & 0 & 11 & 11 \\ 97 & 0 & 0 & 10 & 12 \\ 98 & 1 & 0 & 9 & 13 \\ 99 & 0 & 0 & 8 & 14 \\ 100 & 0 & 0 & 7 & 15 \\ 101 & 0 & 0 & 6 & 16 \\ 102 & 1 & 1 & 5 & 17 \\ 103 & 0 & 2 & 4 & 18 \\ 104 & 0 & 3 & 3 & 19 \\ 105 & 0 & 2 & 2 & 20 \\ 106 & 1 & 1 & 1 & 21 \\ 107 & 0 & 0 & 0 & 22 \\ 108 & 0 & 0 & 0 & 23 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 128 & 0 & 0 & 0 & 43 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 148 & 0 & 0 & 0 & 23 \\ 149 & 0 & 0 & 0 & 22 \\ 150 & 1 & 1 & 1 & 21 \\ 151 & 0 & 2 & 2 & 20 \\ 152 & 0 & 3 & 3 & 19 \\ 153 & 0 & 2 & 4 & 18 \\ 154 & 1 & 1 & 5 & 17 \\ 155 & 0 & 0 & 6 & 16 \\ 156 & 0 & 0 & 7 & 15 \\ 157 & 0 & 0 & 8 & 14 \\ 158 & 1 & 0 & 9 & 13 \\ 159 & 0 & 0 & 10 & 12 \\ 160 & 0 & 0 & 11 & 11 \\ 161 & 0 & 0 & 10 & 10 \\ 162 & 1 & 0 & 9 & 9 \\ 163 & 0 & 0 & 8 & 8 \\ 164 & 0 & 0 & 7 & 7 \\ 165 & 0 & 0 & 6 & 6 \\ 166 & 1 & 1 & 5 & 5 \\ 167 & 0 & 2 & 4 & 4 \\ 168 & 0 & 3 & 3 & 3 \\ 169 & 0 & 2 & 2 & 2 \\ 170 & 1 & 1 & 1 & 1 \\ 171 & 0 & 0 & 0 & 0 \end{matrix}

9

The following figure gives plots of the functions c_{2}, c_{4}, c_{6}, c_{8} on the same set of axes.

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ............ ..................  (128, 43) /  (85, 0) /  (171, 0) /  ......................................................................................................................................................................................................................................................................................................................................................................................................... ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ....................................................  .................................................... .................................................... ............................................................................................................................................................................................  .........................................................................................................................................................................................................................................................................................................................................................................................................  .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................  ............................................................................................................................................................................................ ....................................................  .................................................... .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. .................. ....................................................  .................................................... .................................................... ............................................................................................................................................................................................
Values of c_{2}, c_{4}, c_{6}, c_{8}
for n = 85 to n = 171

(Concerned with sequences A001045, A005130 and A051255.)

Received May 18, 2001. Published in Journal of Integer Sequences, July 19, 2001. Return to Journal of Integer Sequences home page.