IAI+IBI - 1
Y- A\X-leEA:B(e)-B)
3.3 Multidimensional ballot numbers
Lemma 3.3 Let A be a nonempty, finite subset of a field F, and let I A I - k. For every m > 0 there exists a polynomial gm(x) E F[x] of degree at most k - 1 such
that
gm(a) - am for all a E A.
Proof. Let A - lao, a,, ... , ak_ 1). We must show that there exists a polynomial E F[x) such that
u(ai)-uO+ulai+U2a2+...+u&_Iak-1 -a"
for i - 0, 1, ... , k - 1. This is a system of k linear equations in the k unknowns
UO, I ,-- . , uk_1, and it has a solution if the determinant of the coefficients of the unknowns is nonzero. The lemma follows immediately from the observation that this determinant is the Vandermonde determinant
1 ao a2 ... ak-I
1 al ai .. a?-1
1 ak_I ak_I ... akk-I-1
- fl
(aj - ai) 710.O<i<j<k-I
82 3. Sums of distinct congruence classes
for j = 1, ... , m. Let vj_ 1, vl be successive points on a path. We call this a step in the direction e, if
vj = vj-1 + e1.
The vector a is called nonnegative if a; > 0 for i = 0, 1,
... , h - 1. We write
a<b
if b - 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)
for all a, b, c E Zh. In particular,P(a, b) - P(0, b - a).
The path function satisfies the boundary conditions
P(a,a)- 1,
and
P(a, b) > 0 if and only if a < b.
Ifa-vo,v1,...,vm = b is a path with m > 1, then
v,,,-1 = b - e1for some i = 1, ..., h, and there is a unique path from b - e; to b. It follows that the path counting function P(a, b) also satisfies the difference equation
h
P(a, b) P(a, b - e1).
Let a < b. For i = 0, 1, ... , k - 1, every path from a to b contains exactly
b; - a; steps in the direction e;+1. Leth-1
m = E(b1 - a;).
r-o
Every path from a to b has exactly m steps, and the number of different paths is the multinomial coefficient
h-1
1:
(b1 - ai) 1 m,P(a, b) = -1 h-1
R,
(3.4) -0(b, - ai )!
i r[,
-0(b; - a,)!Let h > 2. Suppose that there are h candidates in an election. The candidates will be labeled by the integers 0, 1, ..., h - 1. If mo votes have already been cast, and if candidate i has received a; votes, then
mo - ao+al + +ah_l.
We shall call
vo-a-(ao,a,,...,ah-I)
the initial ballot vector. Suppose that there are m remaining voters, each of whom has one vote, and these votes will be cast sequentially. Let Va.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 - (VO.k, V1.k, ... , Vh-l.k) Then
fork-0,1,...,m. Let
vo.k + V1.k +...+vh-I.k - k+mo
vm -b-(bo,bl,...,bh-1)
be the final ballot vector. It follows immediately from the definition of the ballot vectors that
vk - vk-I E (el,.... eh)
for k - I .... , m, and so
a - v0,vl,...,Vm - b
is a path in Zh from a to b. Therefore, the number of distinct sequences of m votes that can lead from the initial ballot vector a to the final ballot vector b is the multinomial coefficient
\E
-0 (b; -a,))! m!fl o
f (b1 -a;)lnh f
o (bi - a,)'Let v-(vl,...,uh)and w-(wl,...,wh)be vectors in Rh.The vector vwill
be called increasing if
u1 < U2 < < Vh
and strictly increasing if
V1 <V2 <... <Vh.
Now suppose that the initial ballot vector is
a-(0,0,0,...,0)
84 3. Sums of distinct congruence classes and that the final ballot vector is
b = (bo, b, , ... , b,,_, ).
Let
m =bo+b, +...+bh_,.
Let B(bo, b, , ..., bh_ I) denote the number of ways that m votes can be cast so that all of the kth ballot vectors are nonnegative and increasing. This is the classical h-dimensional ballot number. Observe that
B(0,0,...,0)= 1,
and that
B(bo,b,,...,bh_,)>0
if and only if (bo, b, , ... , bh_ i) is a nonnegative, increasing vector. These boundary conditions and the difference equation
h- I
B(bo,b,,...,bh-i)=1: B(bo,...,bi-1,bi - 1,bi+i,...,bh-1)
i-o
completely determine the function B(bo, b, , .... bh_, ).
There is an equivalent combinatorial problem. Suppose that the initial ballot vector is
a'=(0,1,2,...,h-1)
and that the final ballot vector is
Let
b=(bo,bi,....b,_1).
m=J(bi-i)=Ebi- (2)
hLet B(bo, b, , ..., bh_,) denote the number of ways that m votes can be cast so that all of the ballot vectors vA are nonnegative and strictly increasing. We shall call this the strict h-dimensional ballot number.
A path vo, vi, .... vin Z" will be called a strictly increasing path if every
lattice point vk on the path is strictly increasing. Then B(bo, b,, ... , bh_i) is the number of strictly increasing paths from a* to b = ( b 0 . .... b,, _ i ).The strict h-dimensional ballot numbers satisfy the boundary conditions
8(0,1,...,h-1)=1
and
B(bo, bi... b,,_,) > 0
if and only if (bo, b1, ... , bh_ I) is a nonnegative, strictly increasing vector. These boundary conditions and the difference equation
h-1
B(bo,b1,...,bh-1)-EB(bo,...,bi-1,bi - l,bi+l,...,bh-I)
i-o completely determine B(bo, b1, ... , bh-I ).
There is a simple relationship between the numbers B(bo, b1... bh_I) and B(bo, b1... bh_I). The lattice point
v-(vo,VI,...,vh-I)
is nonnegative and strictly increasing if and only if the lattice point
W-v-(0, 1,2,...,h- 1)-v-a'
is nonnegative and increasing. It follows that
a' -v0,v1,v2,...,um -b
is a path of strictly increasing vectors from a* to b if and only if
0, V1 -a",v2-a ..,b-a'
is a path of increasing vectors from 0 to b - a'. Thus,
B(bo,b1....,bh-I)-B(bo,b1 - 1,b2-2....,bh_I - (h - 1)).
For 1 < i < j < h, let Hij be the hyperplane in R' consisting of all vectors (x1, ... , xh) such that xi - xJ. There are (Z) such hyperplanes. A path
a-vo,v1,v2,...,vm-b
will be called intersecting if there exists at least one vector vk on the path such that vk E Hij for some hyperplane H1 1.
The symmetric group Sh acts on Rh as follows. For a E Sh and v = (vo, vI, .. . uh_1) E Rh, let
av - (vo(0), un(I), ... , vo(h-I))
A path is intersecting if and only if there is a transposition r - (i, j) E Sh such that rvk - 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).
(3.5) Lemma 3.4 Let a be a lattice point in Zh, and let b - (b0... bh_I) be a strictly increasing lattice point in Z'. A path from a to b is strictly increasing if and only if it intersects none of the hyperplanes H1 , andB(bo,...,bh_1)-J(a',b).
86 3. Sums of distinct congruence classes
Proof. Let a path, and let
Vk - (VO.k,---,
VIA, Uh - l.k)f o r k - 0, 1, ... , m. If the path is strictly increasing, then every vector 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_I is not strictly increasing. Then I < k < m, and
Vj.k-1 < Vj-I,k-I
f o r some j - 1, ... , h - 1. Since the vector vk is strictly increasing, we have Vj-1.k : Vj.k - 1.
Since vk_I and vk are successive vectors in a path, we have
Vj-I.k-I : Vj-l.k
and
Vj.k - I < Vj.k-1
Combining these inequalities, we obtainvj.k-1 : vj-l.k-I : vj-I.k < vj.k - 1 C Vj.k-1
This implies thatVj.k-1 - Vj-l.k-I
and so the vector vk_I lies on the hyperplane Hj_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. It follows that J(a, b) is equal to the number of strictly
increasing paths from a to b, and so J(a*, b) is equal to the strict ballot number B(bo,..., bh-I ).
Lemma 3.5 Let a and b be strictly increasing vectors. Then P(aa, b) - I(aa, b)
for every a E Sh, a ' id.
Proof. If a is strictly increasing and a E Sh, a 7( id, then as is not strictly increasing, and so every path from as to b must intersect at least one of the hyperplanes Hi .j. and so P(aa, b) < 1(aa, b). On the other hand, we have 1(a a, b) < P(aa, b) by (3.5).
Lemma 3.6 Let a and b be strictly increasing lattice points. Then
E sign(a)l(aa, b) - 0.
oESA
Proof. Since a is strictly increasing, it follows that there are h! distinct lattice points of the form aa, where a E Sh, and none of these lattice points lies on a hyperplane H1 j. Let £2 be the set of all intersecting paths that start at any one of the h! lattice points as and end at b. We shall construct an involution from the set 12 to itself.
Let a E Sh, and let
aa-v0,v1,...,v.-b
be a path that intersects at least one of the hyperplanes. Let k be the least integer such that Vk E Hi,j for some i < j. Then k > I since a is strictly increasing, and the hyperplane Hi. j is uniquely determined since Vk lies on a path. Consider the transposition r - (i, j) E Sh. Then
TVk - Vk E Hi, j and
Moreover,
raa (an.
Taa - Tvo, rv1,... , TVk - Vk, Vk+I... Vm - b
is an intersecting path in n f r o m Taa t o b. For i - 0, 1, ... , k - 1, noneof the vectors Tvo, rv1, ... , TVk_1 lies on any of the hyperplanes, and Hi,j is still the unique hyperplane containing Vk. Since T2 is the identity permutation for every transposition r, it follows that if we apply the same mapping to this path from raa to b, we recover the original path from as to b. Thus, this mapping is an involution on the set S2 of intersecting paths from the h! lattice points as to b. Moreover, if a is an even (resp. odd) permutation, then an intersecting path from as is sent to an intersecting path from ran, where r is a transposition and so ra is an odd (resp. even) permutation. Therefore, the number of intersecting paths that start at even permutations of a is equal to the number of intersecting paths that start at odd
permutations of a, and so
F 1(oa, b) - E 1(a a, b).
mesh oesh
sign(.., sign(.,,
This statement is equivalent to Lemma 3.6.
Recall that [x]r denotes the polynomial x(x - 1) (x - r + 1). If bi and a(i) are nonnegative integers, then
[bilo(i)
- bi(bi - 1)(bi -2)...(b1 -a(i)+1)
b.
(b, -aWY 0
if a(i) < bi if a(i) > bi.
Theorem 3.1 Let h > 2, and let bo, b1... bh_1 be integers such that
0<bo<bl<...<bh_1.
88 3. Sums of distinct congruence classes Then
(bo+b, +...+bh_, _ (h))1
B(bo,bl,...,bh-1) -
borb,l...bh_,1
2 fl (bj - b1)
0<i<j<h-1
ProoL Let a' _ (0, 1, 2, ..., h - 1) and b - (bo, bl,
..., bh-1) E
Zh. Applying the preceding lemmas, we obtainB(bo,bl,...,bh-1) J(a', b)
P(a*, b) - 1(a*, b)
P(a', b) + sign(a)1(aa-, b)
..Sb
eyed
P(a', b) + E sign(a)P(aa*, b)
..Sb ayid
1: sign(a)P(aa', b)
aES,
sign(a)(b°+ti_;+bh-1
- (i))!
.:.
ni-o (b; - a(i))!
a '<b
(bo+...+bh_1 - (?))!
sign(a)[bolaco>[bl]ap)
bo!b,!...bh_1!
.15
b h
( 0
h-I - ())1
sign(a)[bola(o,[bl]oc,, [bh-Mah-,,
b lb l...b
0 1 h-11 aES,I [boll [bolt [bo]h-1
(bo + + bh_I - (Z))! 1 [bill [b112 ... [bllh-1 bo!b,!...bh_,!
I (bh-111 [bh-112 ... [bh-llh-1
(bo+...+bh_1 - (z))!
0<i<j<h-1
This completes the proof.
The following result will be used later in the proof of the Erdo"s-Heilbronn conjecture.
Theorem 3.2 Let h > 2, let p be a prime number, and let io, il, ... , ih_) be
integers such that0<io<il <...<ih-1 <p
and
(h)
Then
B(io,ii,...,ih-1)00
(mod p).Proof. This follows immediately from Theorem 3.1.