Ballot numbers, alternating products, and the Erdös-Heilbronn conjecture

20 Pages·1994·0.198 MB·English

by  Melvyn B. Nathanson

DIMACS Technical Report 94-2
January 1994
Ballot numbers, alternating products, and the
Erdo}s-Heilbronn conjecture
by
1
Melvyn B. Nathanson
Department of Mathematics
Lehman College (CUNY)
Bronx, New York 10468
e-mail: [email protected]
1
ThisresearchwassupportedinpartbygrantsfromthePSC-CUNYResearchAwardProgram. The
paper was written while I was a visitor at the Center for Discrete Mathematics and Theoretical
Computer Science (DIMACS) at Rutgers University, and I thank DIMACS for its hospitality.
DIMACS is a cooperative project of Rutgers University, Princeton University, AT&T Bell
Laboratories and Bellcore.
DIMACS is an NSF Science and Technology Center, funded under contract STC{91{19999;
and also receives support from the New Jersey Commission on Science and Technology.
{ 1 {
1 Introduction
Let A be a subset of an abelian group. Let hA denote the set of all sums of h elements of A
^
with repetitions allowed, and let h A denote the set of all sums of h distinct elements of A,
that is, all sums of the form a1 +(cid:1)(cid:1)(cid:1)+ah, where a1;:::;ah 2 A and ai 6= aj for i 6= j.
Let Abea setof k congruenceclassesmodulo a primep. TheCauchy-Davenport theorem
states that
j2Aj (cid:21) min(p;2k (cid:0)1);
and, by induction,
jhAj (cid:21) min(p;hk (cid:0)h+1)
for every h (cid:21) 2. Erdo}s and Heilbronn conjectured 30 years ago that
^
j2 Aj (cid:21) min(p;2k (cid:0)3):
They did not include this conjecture in their paper on addition of residue classes [9], but
Erd}os has frequently mentioned this problem in lectures and papers (for example, Erd}os-
Graham [8, p. 95]). Dias da Silva and Hamidoune recently prove this conjecture. They used
linear algebra and the representation theory of the symmetric group to show that
^ 2
jh Aj (cid:21) min(p;hk (cid:0)h +1)
for every h (cid:21) 2.
The purpose of this paper is to give a complete and elementary exposition of this proof.
Instead of representation theory, we will use the combinatorics of the h-dimensional ballot
numbers.
2 Multi-dimensional ballot numbers
h
The standard basis for R is the set of vectors fe1;:::;ehg, where
e1 = (1;0;0;0;:::;0)
e2 = (0;1;0;0;:::;0)
.
.
.
eh = (0;0;0;:::;0;1):
h h h
The lattice Z is the subgroup of R generated by the set fe1;:::;ehg, so Z is the set of
h
vectors in R with integral coordinates. Let
h
a = (a0;a1;:::;ah(cid:0)1) 2 Z
and
h
b = (b0;b1;:::;bh(cid:0)1) 2 Z :
{ 2 {
h
A path in Z is a (cid:12)nite sequence of lattice points
a = v0;v1;:::;vm = b
such that
vj (cid:0)vj(cid:0)1 2 fe1;:::;ehg
for j = 1;:::;m. Let vj(cid:0)1;vj be successive points on a path. We call this a step in the
direction ei if
vj = vj(cid:0)1 +ei:
The vector a is called nonnegative if ai (cid:21) 0 for i = 0;1;:::;h(cid:0)1. We write
a (cid:20) b
if b(cid:0)a is a nonnegative vector.
Let P(a;b) denote the number of paths from a to b. The path function P(a;b) is
translation invariant in the sense that
P(a+c;b+c) = P(a;b)
h
for all a;b;c 2 Z . In particular,
P(a;b) = P(0;b(cid:0)a):
The path function satis(cid:12)es the boundary conditions
P(a;a) = 1;
and
P(a;b) > 0 if and only if a (cid:20) b;
If a = v0;v1;:::;vm = b is a path, then
vm(cid:0)1 = b(cid:0)ei
for some i = 1;:::;h, and there is a unique path from b(cid:0)ei to b. It follows that the path
counting function P(a;b) also satis(cid:12)es the di(cid:11)erence equation
hX(cid:0)1
P(a;b) = P(a;b(cid:0)ei):
i=0
Let a (cid:20) b. For i = 0;1;:::;k (cid:0)1, every path from a to b contains exactly bi (cid:0)ai steps
in the direction ei+1. Let
hX(cid:0)1
m = (bi (cid:0)ai):
i=0
{ 3 {
Every path from a to b has exactly m steps, and the number of di(cid:11)erent paths is the
multinomial coe(cid:14)cient
(cid:16) (cid:17)
P
h(cid:0)1
i=0(bi (cid:0)ai) ! m!
P(a;b) = Q = Q : (1)
h(cid:0)1 h(cid:0)1
i=0(bi(cid:0)ai)! i=0(bi (cid:0)ai)!
Let h (cid:21) 2. There are h candidates in an election. The candidates will be labelled by the
integers 0;1;:::;h(cid:0)1. Suppose that m0 votes have already been cast, and that candidate i
has received ai votes. Then
m0 = a0 +a1+(cid:1)(cid:1)(cid:1)+ah(cid:0)1:
We shall call
v0 = a = (a0;a1;:::;ah(cid:0)1)
the initial ballot vector. There are m remaining voters, each of whom has one vote, and
these votes will be cast sequentially. Let vi;k denote the number of votes that candidate i
has received after k additional votes have been cast. We represent the distribution of votes
at step k by the ballot vector
vk = (v0;k;v1;k;:::;vh(cid:0)1;k):
Then
v0;k +v1;k +(cid:1)(cid:1)(cid:1)+vh(cid:0)1;k = k +m0
for k = 0;1;:::;m. Let
vm = b = (b0;b1;:::;bh(cid:0)1)
be the (cid:12)nal ballot vector. It follows immediately from the de(cid:12)nition of the ballot vectors
that
vk (cid:0)vk(cid:0)1 2 fe1;:::;ehg
for k = 1;:::;m, and so
a = v0;v1;:::;vm = b
h
is a path in Z from a to b. Therefore, the number of distinct sequences of m votes that can
lead from the initial ballot vector a to the (cid:12)nal ballot vector b is the multinomial coe(cid:14)cient
(cid:16) (cid:17)
P
h(cid:0)1
i=0(bi (cid:0)ai) ! m!
Q = Q :
h(cid:0)1 h(cid:0)1
i=0 (bi(cid:0)ai)! i=0 (bi (cid:0)ai)!
h
Let v = (v1;:::;vh) and w = (w1;:::;wh) be vectors in R . The vector v will be called
increasing if
v1 (cid:20) v2 (cid:20) (cid:1)(cid:1)(cid:1) (cid:20) vh;
and strictly increasing if
v1 < v2 < (cid:1)(cid:1)(cid:1) < vh:
{ 4 {
Now suppose that the initial ballot vector is
a = (0;0;0;:::;0)
and that the (cid:12)nal ballot vector b = (b0;b1;:::;bh(cid:0)1) is nonnegative and increasing. Let
m = b0 +b1+:::+bh(cid:0)1:
Let B(b0;b1;:::;bh(cid:0)1) denote the number of ways that m votes can be cast so that all of the
k-th ballot vectors are nonnegative and increasing. This is the classical h-dimensional ballot
number. Observe that
B(0;0;:::;0) = 1
and
B(b0;b1;:::;bh(cid:0)1) > 0
ifand only if(b0;b1;:::;bh(cid:0)1)is a nonnegative,increasing vector. Theseboundary conditions
and the di(cid:11)erence equation
hX(cid:0)1
B(b0;b1;:::;bh(cid:0)1) = B(b0;:::;bi(cid:0)1;bi(cid:0)1;bi+1;:::;bh(cid:0)1)
i=0
completely determine the function B(b0;b1;:::;bh(cid:0)1).
There is an equivalent combinatorial problem. Suppose that the initial ballot vector is
(cid:3)
a = (0;1;2;:::;h(cid:0)1);
and that the (cid:12)nal ballot vector
b = (b0;b1;:::;bh(cid:0)1)
is nonnegative and strictly increasing. Let
!
hX(cid:0)1 hX(cid:0)1 h
m = (bi (cid:0)i) = bi (cid:0) :
2
i=0 i=0
^
Let B(b0;b1;:::;bh(cid:0)1) denote the number of ways that m votes can be cast so that all of
the ballot vectors vk are nonnegative and strictly increasing. We shall call this the strict
h-dimensional ballot number..
h
A path v0;v1;:::;vm in Z will be called strictly increasing if every lattice point vk on
^
the path is strictly increasing. Then B(b0;b1;:::;bh(cid:0)1) is the number of strictly increasing
(cid:3)
paths from a to b = (b0;:::;bh(cid:0)1):
The strict h-dimensional ballot numbers satisfy the boundary conditions
^
B(0;1;:::;h(cid:0)1) = 1
{ 5 {
and
^
B(b0;b1;:::;bh(cid:0)1) > 0
if and only if (b0;b1;:::;bh(cid:0)1) is a nonnegative, strictly increasing vector. These boundary
conditions and the di(cid:11)erence equation
hX(cid:0)1
^ ^
B(b0;b1;:::;bh(cid:0)1) = B(b0;:::;bi(cid:0)1;bi(cid:0)1;bi+1;:::;bh(cid:0)1)
i=0
^
completely determine B(b0;b1;:::;bh(cid:0)1).
Thereisasimplerelationshipbetweentheh-dimensionalballotnumbersB(b0;b1;:::;bh(cid:0)1)
^
and B(b0;b1;:::;bh(cid:0)1). The lattice point
v = (v0;v1;:::;vh(cid:0)1)
is nonnegative and strictly increasing if and only if the lattice point
0 (cid:3)
v = v(cid:0)(0;1;2;:::;h(cid:0)1) = v(cid:0)a
is nonnegative and increasing. It follows that
(cid:3)
a = v0;v1;v2;:::;vm = b
(cid:3)
is a path of strictly increasing vectors from a to b if and only if
(cid:3) (cid:3) (cid:3)
0;v1 (cid:0)a ;v2 (cid:0)a ;:::;b(cid:0)a
(cid:3)
is a path of increasing vectors from 0 to b(cid:0)a . Thus,
^
B(b0;b1;:::;bh(cid:0)1) = B(b0;b1 (cid:0)1;b2(cid:0)2;:::;bh(cid:0)1 (cid:0)(h(cid:0)1)):
h
For 1 (cid:20) i < j (cid:20) h, let Hi;j be the hyperplane in R consisting of all vectors (x1;:::;xn)
(cid:16) (cid:17)
h
such that xi = xj. There are 2 such hyperplanes. A path
a = v0;v1;v2;:::;vm = b
willbe calledintersecting ifthereexistsat least onevectorvk on thepath such that vk 2 Hi;j
for some hyperplane Hi;j.
h
The symmetric group Sh acts on R as follows: For (cid:27) 2 Sh and v = (v0;v1;:::;vh(cid:0)1) 2
h
R , let
(cid:27)v = (v(cid:27)(0);v(cid:27)(1);:::;v(cid:27)(h(cid:0)1)):
Apath isintersectingifandonly ifthereisa transposition (cid:28) = (i;j) 2 Sh suchthat (cid:28)vk = vk
for some lattice point vk on the path.
Let I(a;b) denote the number of intersecting paths from a to b. Let J(a;b) denote the
number of paths from a to b that do not intersect any of the hyperplanes Hi;j. Then
P(a;b) = I(a;b)+J(a;b):
{ 6 {
h
Lemma 1 Let a be a lattice point in Z , and let b = (b0;:::;bh(cid:0)1) be a strictly increasing
h
lattice point in Z . A path from a to b is strictly increasing if and only if it intersects none
of the hyperplanes Hi;j, and so
^ (cid:3)
B(b0;:::;bh(cid:0)1) = J(a ;b):
Proof. Let a = v0;v1;:::;vm = b be a path, and let
vk = (v0;k;v1;k;:::;vh(cid:0)1;k)
for k = 0;1;:::;m. If the path is strictly increasing, then everyvector on the path is strictly
increasing, and so the path does not intersect any of the hyperplanes Hi;j. Conversely, if
the path is not strictly increasing, then there exists a greatest integer k such that the lattice
point vk(cid:0)1 is not strictly increasing. Then 1 (cid:20) k (cid:20) m, and
vj;k(cid:0)1 (cid:20) vj(cid:0)1;k(cid:0)1
for some j = 2;:::;n. Since the vector vk is strictly increasing, we have
vj(cid:0)1;k (cid:20) vj;k (cid:0)1:
Since vk(cid:0)1 and vk are successive vectors in a path, we have
vj(cid:0)1;k(cid:0)1 (cid:20) vj(cid:0)1;k
and
vj;k (cid:0)1 (cid:20) vj;k(cid:0)1:
Combining these inequalities, we obtain
vj;k(cid:0)1 (cid:20) vj(cid:0)1;k(cid:0)1 (cid:20) vj(cid:0)1;k (cid:20) vj;k (cid:0)1 (cid:20) vj;k(cid:0)1:
This implies that
vj;k(cid:0)1 = vj(cid:0)1;k(cid:0)1
and so the vector vk(cid:0)1 lies on the hyperplane Hj(cid:0)1;j. Therefore, if b is a strictly increasing
vector, then a path from a to b is strictly increasing if and only if it is non-intersecting, and
so J(a;b) is equal to the number of strictly increasing paths from a to b. It follows that
(cid:3) ^
J(a ;b) is equal to the strict ballot number B(b0;:::;bh(cid:0)1). This completes the proof.
Lemma 2 Let a and b be strictly increasing vectors. Then
P((cid:27)a;b) = I((cid:27)a;b)
for every (cid:27) 2 Sh;(cid:27) 6= id.
{ 7 {
Proof. If a is strictly increasing and if (cid:27) 2 Sh;(cid:27) 6= id, then (cid:27)a is not strictly increasing,
and so every path from (cid:27)a to b must intersect at least one of the hyperplanes Hi;j. This
completes the proof.
Lemma 3 Let a and b be strictly increasing lattice points. Then
X
"((cid:27))I((cid:27)a;b) = 0:
(cid:27)2Sh
Proof. Since a is strictly increasing, it follows that there are h! distinct lattice points of
the form (cid:27)a, where (cid:27) 2 Sh, and none of these lattice points lies on a hyperplane Hi;j. Let (cid:10)
be the set of all intersecting paths that start at any one of the n! lattice points (cid:27)a and end
at b. We shall construct an involution from the set (cid:10) to itself.
Let (cid:27) 2 Sh, and let
(cid:27)a = v0;v1;:::;vm = b
be a path that intersects at least one of the hyperplanes. Let k be the least integer such that
vk 2 Hi;j for some i < j. Then k (cid:21) 1 since a is strictly increasing, and the hyperplane Hi;j
is uniquely determined since vk lies on a path. Consider the transposition (cid:28) = (i;j) 2 Sh.
Then
(cid:28)vk = vk 2 Hi;j;
(cid:28)(cid:27)a 6= (cid:27)a;
and
(cid:28)(cid:27)a = (cid:28)v0;(cid:28)v1;:::;(cid:28)vk = vk;vk+1;:::;vm = b
is an intersecting path in (cid:10) from (cid:28)(cid:27)a to b. Moreover, k is the least integer such that a
lattice point on this new path lies in one of the hyperplanes, and Hi;j is still the unique
2
hyperplane containing vk. Since (cid:28) is the identity permutation for every transposition (cid:28),
it follows that if we apply the same mapping to this path from (cid:28)(cid:27)a to b, we recover the
original path from (cid:27)a to b. Thus, this mapping is an involution on the set (cid:10) of intersecting
paths from the h! lattice points (cid:27)a to b. Moreover, if (cid:27) is an even (resp. odd) permutation,
then an intersecting path from (cid:27)a is sent to an intersecting path from (cid:28)(cid:27)a, where (cid:28) is a
transposition and so (cid:28)(cid:27) is an odd (resp. even) permutation. It follows that the number of
intersectingpaths that start at even permutations of a is equal to the number of intersecting
paths that start at odd permutations of a. This means that
X X
I((cid:27)a;b) = I((cid:27)a;b):
(cid:27)2Sh (cid:27)2Sh
"((cid:27))=1 "((cid:27))=(cid:0)1
This statement is equivalent to the Lemma.
Let [x]k denote the polynomial x(x(cid:0)1)(cid:1)(cid:1)(cid:1)(x(cid:0)k+1). If bi;(cid:27)(i)are nonnegative integers,
then
[bi](cid:27)(i) = bi(bi (cid:0)1)(bi (cid:0)2):::(bi (cid:0)(cid:27)(i)+1)
(
bi!=(bi (cid:0)(cid:27)(i))! if (cid:27)(i) (cid:20) bi
=
0 if (cid:27)(i) > bi:
{ 8 {
Theorem 1 Let h (cid:21) 2, and let b0;b1;:::;bh(cid:0)1 be integers such that
0 (cid:20) b0 < b1 < (cid:1)(cid:1)(cid:1) < bh(cid:0)1:
Then (cid:16) (cid:17)
h
(b0+b1 +(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )! Y
^
B(b0;b1;:::;bh(cid:0)1) = (bj (cid:0)bi):
b0!b1!(cid:1)(cid:1)(cid:1)bh(cid:0)1! 0(cid:20)i<j(cid:20)h(cid:0)1
(cid:3) h
Proof. Let a = (0;1;2;:::;h (cid:0) 1), and let b = (b0;b1;:::;bh(cid:0)1) 2 Z . Applying the
previous lemmas, we obtain
^
B(b0;b1;:::;bh(cid:0)1) =
(cid:3)
= J(a ;b)
(cid:3) (cid:3)
= P(a ;b)(cid:0)I(a ;b)
X
(cid:3) (cid:3)
= P(a ;b)+ "((cid:27))I((cid:27)a ;b)
(cid:27)2Sh
(cid:27)6=id
X
(cid:3) (cid:3)
= P(a ;b)+ "((cid:27))P((cid:27)a ;b)
(cid:27)2Sh
(cid:27)6=id
X
(cid:3)
= "((cid:27))P((cid:27)a ;b)
(cid:27)2Sh
(cid:16) (cid:17)
h
X (b0+(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )!
= "((cid:27)) Q
h(cid:0)1
(cid:27)2Sh i=0(bi (cid:0)(cid:27)(i))!
(cid:27)a(cid:3)(cid:20)b
(cid:16) (cid:17)
h
(b0 +(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )! X
= "((cid:27))[b0](cid:27)(0)[b1](cid:27)(1)(cid:1)(cid:1)(cid:1)[bh(cid:0)1](cid:27)(h(cid:0)1)
b0!b1!(cid:1)(cid:1)(cid:1)bh(cid:0)1! (cid:27)2Sh
(cid:27)a(cid:3)(cid:20)b
(cid:16) (cid:17)
h
(b0 +(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )! X
= "((cid:27))[b0](cid:27)(0)[b1](cid:27)(1)(cid:1)(cid:1)(cid:1)[bh(cid:0)1](cid:27)(h(cid:0)1)
b0!b1!(cid:1)(cid:1)(cid:1)bh(cid:0)1! (cid:27)2Sh
(cid:12) (cid:12)
(cid:16) (cid:17) (cid:12)(cid:12) 1 [b0]1 [b0]2 (cid:1)(cid:1)(cid:1) [b0]h(cid:0)1 (cid:12)(cid:12)
h (cid:12) (cid:12)
(b0 +(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )! (cid:12) 1 [b1]1 [b1]2 (cid:1)(cid:1)(cid:1) [b1]h(cid:0)1 (cid:12)
= b0!b1!(cid:1)(cid:1)(cid:1)bh(cid:0)1! (cid:12)(cid:12)(cid:12) ... (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) 1 [bh(cid:0)1]1 [bh(cid:0)1]2 (cid:1)(cid:1)(cid:1) [bh(cid:0)1]h(cid:0)1 (cid:12)
(cid:16) (cid:17)
h
(b0 +(cid:1)(cid:1)(cid:1)+bh(cid:0)1 (cid:0) 2 )! Y
= (bj (cid:0)bi):
b0!b1!(cid:1)(cid:1)(cid:1)bh(cid:0)1! 0(cid:20)i<j(cid:20)h(cid:0)1
This completes the proof.
We state the following corollary with the notation that is used later in the proof of the
Erd}os-Heilbronn conjecture.
{ 9 {
Corollary 1 Let h (cid:21) 2, let p be a prime number, and let i0;i1;:::;ih(cid:0)1 be integers such that
0 (cid:20) i0 < i1 < (cid:1)(cid:1)(cid:1) < ih(cid:0)1 < p
and !
h
i0+i1+(cid:1)(cid:1)(cid:1)ih(cid:0)1 < +p:
2
Then
^
B(i0;i1;:::;ih(cid:0)1) 6(cid:17) 0 (mod p):
Proof. This follows immediately from the Theorem.
3 A review of linear algebra
Let V be a (cid:12)nite-dimensional vector space over a (cid:12)eld F, and let T : V ! V be a linear
operator. Let I : V ! V be the identityoperator. For everynonnegative integer i, we de(cid:12)ne
i
T : V ! V by
0
T (v) = I(v) = v;
(cid:16) (cid:17)
i i(cid:0)1
T (v) = T T (v)
for all v 2 V. To every polynomial
n n(cid:0)1
p(x) = cnx +cn(cid:0)1x +(cid:1)(cid:1)(cid:1)+c1x+c0 2 F[x]
we associate the linear operator p(T) : V ! V de(cid:12)ned by
n n(cid:0)1
p(T) = cnT +cn(cid:0)1T +(cid:1)(cid:1)(cid:1)+c1T +c0I:
The set of all polynomials p(x) such that p(T) = 0 forms a nonzero, proper ideal J in the
polynomial ring F[x]. Since every ideal in F[x] is principal, there exists a unique monic
polynomial pT(x) = pT;V(x) 2 J such that pT(x) divides every other polynomial in J. This
polynomial is called the minimal polynomial of T over the vector space V.
A subspace W of V is called invariant with respect to T if T(W) (cid:18) W, that is, if
T(w) 2 W for all w 2 W. Then T restricted to the subspace W is a linear operator on
W with minimal polynomial pT;W(x). Since pT;V(T)(w) = 0 for all w 2 W, it follows that
pT;W(x) divides pT;V(x), and so
deg(pT;W) (cid:20) deg(pT;V); (2)
where deg(p) denotes the degree of the polynomial p.
For v 2 V, the cyclic subspace with respect to T generated by v is the smallest subspace
of V that contains v and is invariant under the operator T. We denote this subspace by