Download EXERCISE SET 1: MAGIC SQUARES The objective of these

EXERCISE SET 1: MAGIC SQUARES
The objective of these exercises is to outline a proof, due to J. Spencer, of R.
Stanley’s theorem concerning the number a(n) = am (n) of m × m magic squares
of weight n:
Theorem 1. For each integer m ≥ 2, the function am (n) is a polynomial function of
n of degree ≤ (m − 1)2 .
For another proof, see Stanley’s book Enumerative Combinatorics I.
P
n
1. Let G(z) = ∞
n=0 b(n)z be the generating function of a sequence b(n) of complex
numbers. Show that if
R(z)
for all |z| < 1
(1)
G(z) =
(1 − z)m
where R(z) is a polynomial in z of degree less than m satisfying R(1) 6= 0, then there
is a polynomial B(z) of degree no greater than m − 1 such that b(n) = B(n) for all
integers n ≥ 0. Hint: First consider the case where R(z) ≡ 1.
2. Let (X , ≤) be a finite poset (partially ordered set), and
to each
P suppose that
n
element x ∈ X is attached a generating function Hx (z) = ∞
h
(n)z
.
Suppose
n=0 x
further that these generating functions are related as follows: For every x ∈ X , there
are constants ax , bx,y such that
X
(2)
Hx (z) = ax + zHx (z) + z
bx,y Hy (z).
y<x
(The notation y < x means that y ≤ x and y 6= x). Show that each of the generating
functions Hx (z) is of the form (1), and therefore that hx (n) is a polynomial function
of n. Hint: Every finite poset must have minimal elements (that is, elements x such
that there is no y 6= x such that y ≤ x). For minimal elements x, the condition (1) is
easy to check.
For any m × m magic square T , define the support set B(T ) of T to be the set of
ordered pairs (i, j) ∈ [m] × [m] such that T (i, j) ≥ 1. Let X be the set of all support
sets of m × m magic squares, partially ordered by inclusion. For each support set B,
define hB (n) to be
number of magic squares T of weight n such that B(T ) = B,
Pthe
∞
and let HB (z) = n=0 hB (n)z n .
3. Show that the generating functions HB (z) satisfy a relation of the form (2), and
conclude that each of the coefficient sequences hB (n) is a polynomial function of
1
2
EXERCISE SET 1: MAGIC SQUARES
n. Hint: By the Birkhoff-von Neumann theorem, each support set B contains the
support set of a permutation matrix. Consequently, it is possible to find a map
B 7→ φ(B) that assigns to each support set B the support set φ(B) of a permutation
matrix such that φ(B) ⊆ B. Now show that (2) holds with constants
(
1 if B = φ(B)
aB =
0 otherwise
and
bB,C
(
1 if B − φ(B) ⊂ C $ B
=
0 otherwise.
4. Show that the degree of the polynomial function hB (n) is no greater than (m − 1)2 .
Hint: Don’t work too hard – the explanation is actually quite simple.
Project. The approach outlined above can be followed somewhat farther. First, it
should be clear that the system (2) can, in principle, be solved exactly. In practice,
however, the computations might be cumbersome except when the poset (X , ≤) is
small or has a simple structure. Nevertheless, it might be possible to write a Mathematica or Maple program that would take as input the constants ax and bx,y and
a suitable description of the poset and output the generating functions Hx (z). It
should then
Pbe possiblento use such a program to determine the generating functions
Am (z) := n≥0 am (n)z for n = 4, 5, and perhaps 6, 7. I believe that these generating
functions are known only to m = 6. A successful symbolic procedure that would do
the case m = 7 might be worth a short publication somewhere.