http://www.ictp.trieste.it/~pub_offIC/97/180

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE ENUMERATION OF LATTICE HOMOMORPHISMSOF BOOLEAN ALGEBRAS

Mamadou S. Bah1

Universite de Kankan, Kankan, Guinee2

andInternational Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

In these notes we find closed formulae for the numbers of lattice homomorphisms ofa finite Boolean algebra into another subject to various conditions of " distinguishability"and "undistinguishability" of their atoms. This is related to the more general problem ofcounting all order preserving mappings of a finite poset into another (cf. [3], [4], [6], [7]and [8]). In the case where the first poset is a chain a lot of work has already been done.A general theory of this case applicable to a large class of posets (the so-called differentialposets) is developed in [7]. Here we consider the related problem of counting certain orderpreserving mappings for Boolean algebras. To carry out the counting we first establish aone-to-one correspondence between these mappings and other configurations (families ofsubsets and binary relations) for which the real enumeration is done.

MIRAMARE - TRIESTE

November 1997

1 Regular Associate of the ICTP.2Permanent address.

1 Introduction

The problem of counting order preserving mappings of a partially ordered set into an-other is the subject of many investigations (cf. [3], [4], [6], [7] and [8]). According to R. P.Stanley "the counting of chains in partially ordered sets is a well-developed subject withmany applications both within and outside combinatorics" (cf. [6], p. 919). The currentreference on the topic is certainly [8] (chapter 3). Here we consider the related problemof counting lattice homomorphisms of Boolean algebras. To carry out the counting weestablish bijections between these mappings and families of subsets or equivalently withbinary relations for which the real enumeration is done. This enumeration is made underdifferent assumptions of "distinguishability" or "undistinguishability" on the atoms of theBoolean algebras. In other words we count not only homomorphisms but also "symme-try classes of homomorphisms", that is to say their orbits under the action of certainpermutation groups.

We use certain known facts without explicit explanations referring to [3] for grouptheory, [1] for combinatorics and [5] for Boolean algebras. For the notation of classicalcombinatorial numbers we follow [2].

2 Lattice homomorphisms, families of subsets, and

binary relations

First of all let's note that by Stone's theorem (cf. [5], p. 31), up to isomorphism, there isonly one finite Boolean algebra with a given cardinality 2n: the poset P(N) of all subsetsof a finite set N of cardinality n. Hence the lattice homomorphisms are just the mappings

such that4>(i n J ) = 4>(i) n <f>(j), <f>(i u j) = <f>{i) u <f>(j),

for any subsets I and J of N. More precisely we are interested in U-homomorphisms, i.e.mappings satisfying the first condition:

φ(I U J ) = φ(I) U φ(J) f o r a n y I , J C N

Let's note that these are not all order preserving mappings which can be defined by theweaker condition:

φ(I) U φ(J) C φ(I U J ) f o r a n y I , J C N

Remark. Pi-homomorphisms are defined by the dual condition:

φ(I n J ) = φ(I) n φ(J) f o r a n y I , J C N

But there is a one-to-one correspondence between U-homomorphisms and fi-homomorphisms.Indeed if φ is a U-homomorphism, then by putting

= φ(I) where I is the complementary of I,

we get a n-homomorphism and conversely. So for our purpose it is enough to study U-homomorphisms calling them homomorphisms (by an abuse of language) for the sake ofsimplicity.

We say that a Boolean algebra homomorphism φ is join-unitary (resp. meet-unitary)if

0(0) = 0 (resp. φ(N) = X )

A homomorphism that is meet- and join unitary is simply said to be unitary.These homomorphisms are uniquely defined by their "values" on the single element-

subsets, i.e. by a family of subsets F = (Fi)ieN of X indexed by N, where Fi = φ({i}).Moreover giving such a family of subsets of X is the same as defining a binary relation

or correspondence R between the sets N and N:

(i,a) E R -<=>• a E Fi

Let B(N,X), Fn(X) and Hom(P(N),P(X)) denote respectively the sets of all binaryrelations between N and X, of all n-families of X (n = |N|) and lattice homomorphisms.Then we have the following:

Proposition 1. There are bijections between any two of these setsProof. We have already mappings Ψ of Hom(P(N),P(X)) into Fn(X) and R of

Fn(X) intoB(N,X).Conversely we define mappings F and Φ in the opposite directions by setting

F(R) = (Fi)ieN w i t h Fi = R(i) = {a E X : (i,a) E R},

= \J Fi,

for any i E N and I C N.Now it is easy to check that the mappings F and R on the one hand and Φ and Ψ on

the other hand are inverse of each other:

RF(R) = R , FR(F) = F,

= F,

for any R E B(N,X), F E Fn(X) and φ E Hom(P(N),P(X). D.We call a family of subsets a covering of X if:

t = X (1)

The family F is said to be disjunctive if

F t n F 3 = f) for i^j (2)

We said that the family F is strict if all subsets Fi are nonempty:

Fi ^ 0, for any iE N (3)

We say that that F is a small family if no subset Fi has more than one element:

|Fi| < 1, for any iE N (4)

Following Comtet (cf. [1]) we call a disjunctive covering a division of X (we do not say"partition" because empty or repeated blocks are allowed). A covering F whose blocksare all nonempty is called an hypergraph. Finally we call a small family whose blocks areall nonempty a single-element family.

In terms of binary relations conditions (1) - (4) are equivalent to the following:

For any a E X, there is at least one i E N such that (i,a) E R (5)

For any a E X, there is at most one i E N such that (i,a) E R (6)

For any i E N, there is at least one a E X such that (i,a) E R (7)

For any i E N, there is at most one a E X such that (i,a) E R (8)

Relations R satisfying (5) (resp. (6)) are said to be right injective (resp. right surjec-tive). Relations satisfying (7) and (8) are similarly called left injective and left surjective.Injective and surjective relations on one side are called bijective on this side. Left bijectiverelations are just mappings.

By the above one-to-one correspondence we have bijections between the followingtriple of sets:

i) Right injective relations, disjunctive families of X and join-unitary homomorphims:

ii) Right surjective relations, coverings of X and meet-unitary homomorphisms φ:

φ(N) = X

iii) Left injective relations, small families of X and homomorphisms φ such that:

φ(I) = X ==> I = N

iv) Left surjective relations, strict families of X and homomorphisms φ such that:

φ(I) = 0 ==> I = 0

vi) Mappings, single-element families (words) of X and homomorphisms φ such that:

φ(I) = 0 = ^ 7 = 0,

φ(I) = X ^ I = N

vii) Right bijective relations (inverse mappings), divisions of X and Boolean algebrahomomorphisms (i.e. meet and join-unitary homomorphisms):

0(0) = 0 ,φ(N) = X

viii) Right and left injective relations, small disjunctive families and homomorphismswith values increasing "slower" than arguments:

|φ( < \I\, for any I C N

ix) Right and left surjective relations, hypergraphs and homomorphisms with valuesincreasing "faster" than the argument:

\ I \ < W ) \ , f o r a n y I C N

The cases (vi) and (vii) are well known (cf. [1] or [3] or [7]). The following remarkshows that we can further reduce the number of cases and consider the above propertiesonly on the right.

Remark. The mapping R —R R~l = {(a, i) : (i, a) G R} define bijections between thefollowing sets of binary relations:

i) Binary relations between N and X with binary relations between X and N;ii) Right injective relations between N and X with left injective relations

between X and Niii) Right surjective relations between N and X with left surjective relations

between X and N.Similarly we have the following bijection between Fn(X) and Fx(N):

w h e r eF* = {ieN:ae Fi}.

We call this family the dual of F.The " dualization" mapping establishes a one-to-one correspondence between the fol-

lowing sets of families:i) disjunctive n-families of X and small x-families of N.ii) n-coverings of X and strict x-families of N.Our first objective is to find the cardinalities of the following sets in terms of the

cardinalities n and x of N and X:B(N,X): the set of all binary relations between N and X(or equivalently the set of alln-families of X);Ir(N, X): the set of all right injective relations (the set of all disjunctive n-families of X);Sr(N, X): the set of all right surjective relations (or the set of all n-coverings of X);Ilr(N, X): the set of all left and right injective relations (or the set of all small disjunctiven-families of X: permutations!)Slr(N, X): the set of all left and right surjective relations (or the set of all n-hypergraphsofX).

We'll do the enumeration under different assumptions of "distinguishability" or "undis-tinguishability" of the elements of N and X. This amounts to counting the orbits of thenatural actions of the symmetric groups SN, SX and their direct product SN X SX on theprevious sets:

= {(ir(i),a(a)):(i,a)eR}, for any R G B(N, X) and (π, σ) G SN x SN;

(π, a)((Fi)i€N) = (a(Fn-i{i)))teN for any (Ft)teN G Fn(X);

( ( π , σ ) ( φ ) ) ( I ) = a ( 0 ( 7 r - 1 ( / ) ) ) for a n y ICN a n d a n y φ G Hom(P(N),P(X).

W i t h t h e a b o v e n o t a t i o n we h a v e :

Proposition 2. i) For any sets the following diagram is commutative:

- 1B(N,X) ^ 4 B(X,N)

i i

i iHom(P(N),P(X)) - U Hom(P(N),P(X))

ii) The actions of SN X SX on the various sets are permutable with the arrows of theabove diagram:

if R G B(N,X), F G Fn(X) and φ G Hom(P(N),P(X)), then for every (π,σ) GSN X SX we have:a)

= {{T,a){4>))\

where (f>*(A) = Uaex0*({a}) = {i & N : a E φ(i)}.Proof. The three points of the proposition are proved in a similar way. As an example

let's check point (ii) - b) for the first equation let F = {Fi)ie^ and G = (Gi)ieN be twofamilies such that

G = (π, σ)(F) or Gi = a(F7r-i^i-)), for any i G N

We must show that

G* = (σ, vr)(F*) or G*a = TT(F*), for anya G X

This follows from this chain of implications:

i G G*a <^=^ a G Gi <^=^ a G a(F7T-i^) <^=^ a~ (a) G Fn-i^

^^ K~l{i) G K-^a) ^ = ^ * G (-P -Ha))

For the second equation we have:

(7r,<j)(U Fi) = (π,σ)(Φ(F))(I), for any I C Niei

Therefore

Q.E.D. •

By Proposition 2 and the remark before it we get:Corollary. For any of the groups SN, SX and SN X SX there is a one-to-one corre-

spondence betweeni) the orbits on n-families of X and the orbits on x-families of N;ii) the orbits on n-coverings of X and the orbits on strict x-families of N;iii) the orbits on disjunctive n-families and the orbits on small x-families of N.The corresponding results are true for lattice homomorphisms and binary relations.Now we are in a position to state the main results.Theorem A. The numbers of orbits of the symmetric groups SX, SX and their direct

product SN X SX on the homomorphims of the Boolean algebra P(N) into P(X) are givenin the following table:

Type ofHomomorphisms

AllHomomorphisms

Join-unitaryHomomorphisms

Meet-unitaryHomomorphisms

Slow-growingHomomorphisms

Fast growingHomomorphisms

Unitary

Acting GroupsNone

2nx

(1 + n)x

(2 n - 1)x

£LOO:K

ELoC-1)*^2"-*-1)*

nx

SN

2^fc = 0(- L) \k)( n )

V™ (x\k k = 0 k

srx (-i\k(x\(2*~k+n-2\Z^fc = O(- L> \k)\ n )

V ^ " rx\2^ik = 0^ki

SX

/2 r l + a:-2\V x )

kk = 0 k

y-« (_l)k(n\(2<-k+x-2\Z^fc = O(- L> \k)\ x )

ln + x-l\V x )

SN x SX

ELoELo^W

1 + min{n, x}

ELoPfcW

where n is the binomial coefficient, {nk} the Stirling number of the second kind and

xk = x(x — 1) • • • (x — k + 1) the falling factorial

The following theorem shows that the numbers corresponding to the empty boxes inthe table are equally difficult to compute.

Theorem B. Let α(n,x), β(n,x) andγ(n,x) be the numbers of orbits of SN X SX onall homomorphisms , join-unitary homomorphisms and on fast growing homomorphisms.Then

(n, x) = α(n, x) — α(n, x — 1)

γ(n, x) = β(n, x) - (n -l,x)

Remark. The number α(n, x) can actually be computed using the following formula:

α(n,x) = ]T (a1 +2a2 + nan =n,b

where m = max{n, x} (we take ai = 0 (resp. bj = 0) for i greater than n (resp. for jgreater than x)).

We obtain these two results as reformulation of the results on counting binary relationsin the language of lattice homomorphisms.

3 Elementary enumeration

Let N and X be two finite sets of cardinalities n and x. Then the following results areeasy to prove.

Proposition 3. If N and X are finite sets of cardinalities n and x, then:a)

|B(N,X)| = 2nx; (9)

b)

|Ir(N,X)| = (1 + n)x; (10)

c)

d)

min{n,x} / \ / \ min{n,x} / \

I,(NX)\- V IHr l f c i - V \ \xk

k=0 k k k=0 kwhere xk = x(x — 1) • • • (x — k + 1);

e)

.l)M'M(2-*_i)s (13)

Proof. a) Since |N x X| = nx, N x X has 2nx subsets.b) A relation R C N x X is right injective, if and only if its inverse is a mapping of

X into A'U {0}. There are (1 + n)x such mappings.c) A relation R C N x X is right surjective, if and only if its inverse is a mapping of

X into the set of all nonempty sets of N. There are (2n — 1)x such mappings.d) It is easy to see that there is a one-to-one correspondence between left and right

injective relations R C N x X and the triples (A, B, f), where A is a subset of N, B asubset of X, and f a bijection of A onto B. The number of such triples is obviously equalto the sum on the right side of (d).

e) The set Sr(N, X) can be written as a disjoint union in the following manner:

Sr(N,X)= U Slr(A,X).ACN

Thus|Sr(N,X)|= E |Slr(A,X)|.

ACN

Using the Mobius inversion principle we get

|Slr(N,X)|= E \Sr(A,X)\»(A,N) =ACN

y ^ j j(2fc — i)x(—i)ra~fc = V^(—i) fcI \(2n~k — i)xofc=o V"v fc=o Vv

Remark. By symmetry we get the following formulae

|Il(N,X)| = (1 + x)n; (14)

|Sl(N,X)| = (2x-l)n. (15)

Using the enumeration of left injective relations we get another proof of the binomialformula:

Corollary. For any positive integers n and x the following formula holds:

Proof. It straightforwardly follows from the below disjoint decomposition of Il(N,X):

Il(N,X)= U XA,A<ZN

where XA denotes the set of all mappings of A into X.O

By the one-to-one correspondence between binary relations and families of subsets wecan reformulate the above results as follows.

Proposition 3'. If X is a finite set of cardinality x,thena) the number of n-families of subsets of X is 2nx;b) the number of disjunctive n-families of X is (1 + n)x;c) the number of n-coverings of X is (2n — 1)x;

d) the number of disjunctive small n-families of X is Y^=o nkxk;

e) the number of strict n-coverings of X is YJk=o(~l)fcu)(2ra~fc — 1)x.

4 "Distinguishability" and "undistinguishability"

Now we count relations between N and X assuming the elements of N (resp. X, resp.both N and X) are undistinguishable. As we noted before this is the same as countingthe orbits of the above sets under the natural actions of the symmetric groups SN, SX ortheir direct product SN X SX.

Let's denote the wanted numbers using subscripts and superscripts to indicate onwhich side we consider the relevent property (subscripts) or the action of the groups(superscripts). For example ir(n, x), irl(n, x), ir

r(n, x), and irlr(n, x) will respectively denotethe cardinality of the set Ir(N, X) and the numbers of its orbits under the actions of thegroups SN, SX, and SN X SX; in the same way blr(n,x), irlr(n,x), slrr(n,x), ilrlr(n,x), andslrlr(n, x) will be the numbers of orbits of all the sets under the action of SN X SX.

To enumerate the above orbits we use the following lemma.Lemma (Cauchy-Frobenius-Burnside). If G is a finite group acting on a finite

set S, then the number tS(G) of orbits of G on S is equal to the average number of fixedpoints of the elements ofG:

gee

Proof. See any book on finite permutation groups(cf. [3] for example). •Therefore we must find the number of fixed points of a given permutation π G SN

(rep. σ £ SX, resp. (Π,Σ) G SN X SX) on each of the previous sets. We collect all theneeded results in the following lemma.

9

Lemma 1. Let π G SN and σ E SX be two permutations of cyclic types (1a12a2 ... nan)and (1b12b2 ... xbx). Then the number of fixed points of π, σ, and (Π,Σ) on the setsB(N,X), Ir(N,X), Sr(N,X), Ilr(N,X), andSlr(N,X) are the following:

(17)

(18)

ii) a)

nT^ • • aibjgcd(i,j)= 2 i j a y , (19)

(20)

c)

|FixIr(σ)| = (21)

iii) a)

b)

iv) a)

b)

c)

k=1 d|k

+bx)

|FixIlr(π)| =min{a1,x} / \

k=0

|FixIlr(σ)| =k = 0

min{n,x} /min{ak,bk}

n E

(22)

(23)

(24)

(25)

(26)

(27)

10

v)a)

% - 1)x, (28)

k=0where r = a1 + a2 + • • • + an;

(29)k=0

where s = b1 + b2 + • • • + bx;Proof. i) a) A relation R C N x X is fixed by π if and only if it is a relation between

the cycles of π and X.b) is like (a).c) A relation R C N x X fixed by (π, σ) is just a subset of cycles of (π, σ). But any

cycles (c1c2 ... cr) of π and (u1u2 ... us) of σ define gcd(r, s) cycles of length lcm(r, s) in(Π, σ). Therefore the number of cycles in R is J2ij aibjgcd(i,j). Consequently the number

of the above relations R is 2i,jaz ]9C .ii) c) For a relation R e FixB(π,σ) to be right injective it should associate to any

cycle of length k in σ at most one cycle of length dividing k in π. This can be done in

a) Put σ = 1X (the identity of X) in (c)(b1 = x and bi = 0 for i = 2,.. . ,x.)b) Put π = 1N in (c)(a1 = n and ai = 0 for i = 2,... , n.)iii) a) A relation R e Sr(N, X) is fixed by π if and only if it is a right surjective

relation between the cycles of π and X. There are (2( a i + a 2 H '""") — 1)x such relations.b) is like (a).iv) a) For a relation R e FixB(N,R) (Π, Σ) to be left and right injective it should associate

to selected cycles of π an equal number of cycles of the same length in a. This can bedone in YYk=i(J2iHo ("M ( ^)^-k%) ways.

b) Put σ = 1X in (c).b) Put π = 1N in (c).v) a) A relation R e Slr(N,X) is fixed by π if and only if it is a left and right

surjective relations between the cycles of vr and X. The number of such relations isE L o ( - 1 ) f c (l) (2 r" f c - 1)x, where r = a1 + a2 + • • • + an.

n.b) is like (a).DWe also need the following fact.Remark. The number of permutations π G SN which have ai cycles of length i, i =

1,2,... ,n is equal to

By the Cauchy-Frobenius lemma, Lemma 1 and the remark we have the followingresults.

Theorem 1.Ifn and x are positive integers, then:

bl(n,x)= J2 (laial\2a2a2\---nanan\yl2(-ai+a2+-+an)x; (30)a1 +2a2 + nan =n

11

b)

br(n,x)= 1 b1b1!2b2b2! • • • xbxbx\) 2n[-bl+b2+-+bx>- (31)b1 +2b2 +•• -+xbx=x

c)

blr(n,x)= E IlK^lI^AU 2^° i tww; (32)

ii) a)

n xlr

ai=n,jjbj=x i=1

] l + «!)-; (33)a1 +2a2 + nan =n

f(nr)- V V1b^2hbo^---rt'xb n (1 + riYbl+b2+'"+bx> • (^4)lr{ll,X) — 2_^i \ L ul-^b2

u2- X UX.J [l-\-U) , [O^)61+262 H 1- 63;=x

YJiia.i=n^j.jbj=x i = 1 j= i k=1 d |k

iii) a)

J (ri r\ V ^ f 1 a i / 7 , l 9 a 2 / i a l . . . T i n ]\~l (o(al+a2-\ H«n) _ ->\x. (If.)

srr(n,x)= E (r16i!2b262!---^6JJ (2ra - i)^+^+-+b-)- (37)

6 i + 2 6 2 H ha;6a;=a:

( min{a1,x} / \ \

E x K ; (38)fc=o \ / I

flr(n,x)= E (Ibl6i!2b262!---^6J)-M E , K I 5 (39)6i+262+---+a:6;I;=a: V fc=O V K /

ra x \ n /min{ak,bk}

n E(40)

12

v)a)

sl

lr(n,x)= J2 a1a1

a1+2a2 +nan=n k=0

where s = a1 + a2 + • • • + an;

I n\ , ,\

sr

lr(n,x) = Y, (l1bi\2b2b2\---xbxbj)~1 X X " 1 ) ( 2 ~ \)^+h^+-+h*)b1+2b2+-+xbx=x 12xk=0 k (42)

The corresponding results are true for families of subsets and homomorphisms ofBoolean algebras.

5 Bijections for injective relations

The above formulae are not easy to use for practical computations. In this section we usebijections to find more handy formulae for disjunctive families (i.e. injective relations).

Lemma 2. There is is a one-to-one correspondence between the orbits of disjunctiven-families of X under the action of SX and the non negative integer solutions of thefollowing inequality:

a.\ + α2 + • • • + αn < x,

where x is the cardinality of X.Proof. Indeed two disjunctive n-families (F1, F2,... , Fn) and (G1, G2,... , Gn) are in

the same orbit under SN if and only if |Fi| = |Gi|, i = 1,2,... ,n. Since the subsets Fiare pairwise disjoint these numbers satisfy the inequality:

Q.E.D. •Theorem 2. The number of orbits of disjunctive n-families of X under the action of

SX is equal to ("+*).Proof. For any integer k with 0 < k < x the number of non negative integer

solutions of the equation

α1 + α2 + • • • + αn = k

is \+

k~ ) (cf. [1],pp. 15-16). Therefore the number of solutions of the previous inequalityVis given by the formula:

fn - 1\ fn\ fn + x - l\ fn + x\

By the correspondence between binary relations and families we obtain the followingresult.

Theorem 2'. For any positive integers n and x we have:

fr(n,x)=(n+

x

X) (43)

13

When "indexes" i G N are undistinguishable we have the following characterizationof symmetry classes.

Lemma 3. There is a one-to-one correspondence between the orbits of SN on disjunc-tive n-families of X and all pairs (A, A), where A is a subset of X and A is a class ofn-divisions of A under SN.

Proof. Indeed if π(F1, F2,... , Fn) = (G1, G2, . . . , Gn), then

G i = F 7 r -i W , for i = 1,2,... , n.

Since TT~1(Z) ranges over N with i we have

U GI = u ^ - i W = U Fi.i=1 i=1 i=1

Put A = U™=i Fi. Then (Fi)ieN and (Gi)ieN are n-divisions of A belonging to the sameorbit of SN. This orbit will be our A.

Conversely any orbit of n-divisions of a subset of X is an orbit of disjunctive n-familiesofX.D

Theorem 3. The number of orbits of disjunctive n-families of X under the action of

SN is equal to E L o S " = o xk{sk}, where {ks} is the Stirling number of the second kind.

Theorem 3'.Ifn and x are positive integers, then

s} (44)k=0s=0 k

Proof. By lemma 3 and the twelvefold way (cf. [8], pp. 33-36) the number of orbitsof disjunctive n-families is equal to:

|{A : orbit of divisions of A}| =AQX

x I n \ x n / \

E E E(f) E E *) W

E(f) =EE*kl-k=0 \ACA,\A\=k l=0 ) k=0l=0 k

Q.E.D. •When both groups are acting we get the following result.Lemma 4. There is a one-to-one correspondence between the orbits on disjunctive n-

families of X under SN X SX and the pairs (k, On,k) where k is a positive integer suchthat 0 < k < x and On,k is the set of orbits of n-divisions of any k-element set Ak underthe action of SN X SAk .

Proof. We prove two facts from which the lemma will follow.i) For any integer k such that 0 < k < n the disjunctive families whose union is a

k-element subset of X constitute an invariant subset under the action of SN X SX. Let'sdenote it by Dk.

ii) If we fix a subset Ak of cardinality k, then the orbits of SN X SX contained in Dk

are in a one-to-one carrespondence with the orbits of n-divisions of Ak under SN X SAK .

i) If (Fi)ieN € Dk and (Gi)ieN = (Π, cr)((Fi)ieN), with ( Π , Σ ) G SN X SX, then sincen~l(i) ranges over N with i we have:

U Gil = | U ^(^r-i(o)l = W(\JF*-HO)\ = I U ^ W l = I U F*\ = k.

14

Therefore (Gi)ieN lies also in Dk. This proves (i).ii) If F and G both belong to Dk then there are bijections

i£N

and

i£N

which naturally transform F and G into n-divisions P = (Pi)ieN and Q = (Qi)ieN of Ak.If F and G are in the same SN X SX-orbit, then we can find permutations π G SN anda & SX such that

Gi = cr(F7I—i(j)), for any i E N.

We have alsoτ1(Fi) = Pi and τ2(Gi) = Qi, for any i G N.

Denoting by a' the restriction of σ to \JieN Fi we get

Qi = r2cr /r1"1(P7 r-iW), for any i G A , where r 2 a'rf 1 G 5 ^

orQ = (π, r 2 a / r f 1 ) ( P ) with (π, T2CJT^1 e SN X SAk.

This shows that P and Q are in the same SN X SAk-orbit.Conversely any orbit of n-divisions of Ak under SN X SAk defines an orbit of disjunctive

n-families of X under SN X SX. This proves (ii).Now we derive the lemma from the following disjoint decomposition of the set Sn(X)/SN><

SX of orbits of disjunctive n-families of X:

Sn(X)/SN x SX = U Dk/SN x SX

k=0

Q.E.D. •Theorem 4. The number of orbits of disjunctive n-families of X under the action of

Sn x SX is equal to J2l=o J2f=oPi(k), where pl(k) is the number of partitions ofk in l parts.Theorem 4!.If n and x are positive integers, then

k=0 l=o

Proof. By lemma 4 and the twelvefold way (cf. [8], pp. 33-36) we have

fc-0 k=0 l=0

For relations that are injective on both sides (or equivalently for small disjunctivefamilies) we have the the following results.

Lemma 5. There are one-to-one correspondences between:(i) the orbits of small disjunctive n-families of X under the action of SN and the

subsets of X with at most n elements;

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(ii) the orbits of small disjunctive n-families of X under the action of SX and thesequences of n symbols 0's and 1's with no more than x symbols 1's;

(iii) the orbits of small disjunctive n-families of subsets of X with at most one elementunder the action of SN X SX and the set {0,1, 2,... ,min{n,x}}.

Proof. It is a slight modification of the proofs of Lemmas 3 and 4.D

Theorem 5. i) The number of orbits of small disjunctive n-families of

X under the action of SN is Y^k=o kx ;

ii) the number of the similar orbits under the action of SX is equal to Y^=o nk

iii) the number of orbits under the action of SN X SX is 1 + min{n,x}.Theorem 5'. If n and x are positive integers, then

i)

(46)k=0 k

ii)

X / r l \

- (47)k=0

ilrlr(n,x) = 1+ min{n,x}. (48)

Proof. Points (i) and (iii) are immediate consequences of the corresponding pointsof Lemma 5. Point (ii) follows from the fact that any of the sequences in Lemma 5-point(ii) is totally defined by the k positions of the 1's (0 < k < x) out of n positions.•

6 Bijections for surjective relations

We use bijections to find closed formulae for orbits of surjective relations (coverings). Aswe shall see it is quite natural to handle the case of general families at the same time.Let's first consider the action on the set N of "indexes" as the easiest. We have thefollowing result.

Lemma 6. There is a one-to-one correspondence betweeni) the orbits of n-families of X under the action of SN and the non negative integer

solutions of the equation

α1 + α2 + • • • + α2x = n,

where x = |X|;ii) the orbits of strict n-families of X under the action of SN and the non negative

integer solutions of the equation

α1 + α2 + 2 % _ ! = n.

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Proof. i) Let F and G be two n-families of X which belong to the same orbit of

SN. Then any subset A C N appears among the Fi's as often as it does among the Gi's.

Indeed we have

Fi = A, if and only if Gπ(i) = A,

where π G SN is the permutation sending F to G. So if αA(F) denotes the "multiplicity"

of A in F then we have:

αA(F) = αA(G) for every ACX.

Conversely if this relation is satisfied for every subset of X, then the Fi's and the Gi’s

are formed by the same different subsets A1,A2,... ,Ar with the same multiplicities

α1, α2, . . . ,αr (the only difference being the positions they occupy in the two sequences

(F1, F2,... , Fn) and (G1, G2, . . . , Gn)). Let ki1, ki2, . . . , kiαi denote the positions occu-

pied by Ai in F (i = 1, 2,... , r) and li1,li2, . . . , liαi be the corresponding positions in G.

Define a permutation π of N by setting

π(kij) = lij for i = 1, 2,... , r and j = 1, 2,... , αi.

Then it is easy to check that π satisfies the equation:

d = F^-i(i) for i G N.

Therefore F and G are in the same SN-orbit.

This shows that there is a one-to-one correspondence between the SN-orbits of n-

families of X and the number of non negative integer solutions of the equation:

]T αA = n.ACX

ii) We apply the same argument for nonempty subsets to get the correspondence between

the orbits on strict families and the solutions of the equation:

Q.E.D. •

Theorem 6. i) The number of orbits of n-families of subsets of X under the action

of SN is2x + n - 1\

{ n )'(ii) the number of orbits of strict n-families of X under the action of SN is equal to

iii) the number of orbits of n-coverings of X for the same action is equal to

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iv) the number of orbits of hypergraphs of X under this action is

to ( W V n

Theorem 6'. If n and x are positive integers, then

i)

(49)

s\(n,x)=^ ^ ; - ^ ; (50)

iii)

iv)

- _ 1 ) f c M ^ - f c + ^ - 2 \ ( 5 2 )

Proof. (i) and (ii) follow immediately from the lemma and from the fact that theequation

o.\ + α2 + • • • + αm = n

has (m+™"1) solutions (cf. [1], pp. 15-16).

iii) We use Mobius inversion formula to substract orbits of coverings from generalorbits of families. By taking all families we take once each class of coverings of any subsetofX. Thus

n~l)=12 |Covn(A)/SN|,

where COVN(A)/SN denotes the set of orbits of coverings of A under the action of SN.Using Mobius inversion formula we get:

AQX \ U J fc=0 \KJ \ U J

iv) We apply the argument of (iii) to nonempty subsets. •For the actions of SX we use Proposition 2 and Theorem 6.

Theorem 7. i) The number of orbits of n-families of X under the action of the

group SX is equal to \ +x~l\ ;

ii) the number of orbits ofn-coverings of X under the action of SX is equal to 2n+a;-2x

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iii) the number of orbits of SX on strict n-families is equal to

A , sk(n\(T-k + x - l \/ (-1) , ;

fro WV * yiv) the number of orbits of SX on strict n-coverings of X is equal to

Proof. The three points are proved in the same way. We prove point (ii) as anexample. By the point (ii) of the corollary to Proposition 2 the number of orbits of SXof SX on n-coverings of X is equal to the number of its orbits on strict x-families of N.Exchanging the roles of N and X and using Theorem 6 - point (ii) we find that thisnumber is equal to

(2n + x - 2\

Q.E.D. •The reformulation of this result in the language of binary relations gives the following

result.Theorem 7'.For any positive integers n and x we have

i)

¥(n,x)=^n+^-^ (53)

ii)

srr(n,x)=(2n + *~2^ (54)

iii)

iv)

r(n,x) = E(-l)fc(Y)f2ra k + X~2) (56)fc=0 \kJ V X J

For the actions of SN X SX we do not obtain closed formulae. Instead we provesome useful inductive relations between the numbers br(n,x), blr(n,x), sr

r(n,x), srlr(n,x),sl

lr(n,x) and slrlr(n,x).Theorem 8. If n and x are positive integers, then

i)

srr(n,x) = br(n,x)-br(n,x-l); (57)

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ii)

sl

r

r(n, x) = blr(n, x) - blr(n, x - 1); (58)

iii)

sr

lr(n,x) = sr

r(n,x)-sr

r(n-l,x); (59)

iv)

slrlr(n,x) = sl;(n,x)-sl;(n-l,x); (60)

v)

4 = br(n, x) - [br(n, x - 1) + br(n - 1, x)] + br(n - 1, x - 1); (61)

vi)

slrlr(n, x) = blr(n, x) - [blr(n, x - 1) + blr(n - 1, x)] + blr(n - 1, x - 1) (62)

Similar properties hold for left surjective relations.Proof. Points (i), (ii), (iii) and (iv) are proved in the same way. As an example we'll

prove (i). To do so we'll prove the following two facts from which the relation (i) can bededuced:

For any non negative integer k with 0 < k < x let's fix a subset Ak C X. Then(a) the set Ck of all n-families whose union has cardinality k is invariant under the

action of SX.(b) There is a one-to-one correspondence between the orbits of n-families contained

in Ck and the orbits of n-coverings of Ak (under the action of SAk!).a) If (F1, F 2 , . . . , Fn) and (G1, G 2 , . . . , Gn) are two families of subsets of X such that

Gi = σ(Fi), i = 1,2,... ,n, with σ & SX, then

ji=1 i=1 i=1 i=1

Therefore if one of the families is in Ck, the same holds for the other.(b) Two n-coverings R1 and R2 such that R2 = Τ(R2) with τ G SAk are obviously in

the same orbit of n-families of X. Conversely let (Fi)ni=1 and (Gi)in=1 be two n-familiesof X such that Gi = σ(Fi), i = 1,2,... ,n and | U™=i Fi| = k. Let's construct bijectionsτ1 '• ni=1 Fi -^ Ak and Τ2 : U?=i Gi —> Ak. These mappings transform the families (Fi)ni=1

and Gi)ni=1 into n-coverings of Ak, say T1 and T2. Denoting by σ1 the restriction of σ toU?=i ^ we get T2 = T2<TiTi1(T1), with raairf 1 e SAk. This proves (b).

Now from (a) and (b) and the fact that the sets Ck, k = 0,1, 2,... ,x constitute apartition of the set of n-families of X we deduce the following formula:

br(n, x) = sr

r(n, 0) + sr

r(n, 1) + • • • + sr

r(n, x,)

which is equivalent to (i).(v) and (vi) follow immediately from (i) and (iii) on the one hand and (ii) and (iv) on

the other hand.D

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Now we can directly read out the proof of our main results from Theorems 1- 8.Proof of Theorems A and B. The formulae in the first five rows of the table in

Theorem A follow from Proposition 2 and Theorems 2-7. They are obtained by reformu-lating the results of these theorems in the language of lattice homomorphisms. The lastrow is derived from the twelvefold way (cf. [8], pp. 33-36).

Theorem B is the reformulation of formulae (58) and (60) of Theorem 8. •

7 Acknowledgements

This work was done within the framework of the Associateship Scheme of the Interna-tional Centre for Theoretical Physics, Trieste, Italy. Financial support from the SwedishInternational Development Cooperation Agency is acknowledged. A part of this work wasdone during a stay at Freiburg University under the auspices of the German Associationfor Academic Exchange (DAAD). It was the subject of talks at Professor Kegel's algebraicseminar at Freiburg and at the Pedagogical Institute of Erfurt-Mulhausen. The authorwould like to thank Professors Kegel from Freiburg and Rosenbaum from Erfurt for theirhelp and hospitality.

8 References

1. L. Comtet. Advanced Combinatorics: The Art of Finite and Infinite Expansions. D.Reidel, 1974.2. R. Graham, D. E. Knuth, O. Patashnik. Concrete Mathematics. Adison-Wesley, 1989.3. A. Kerber. Algebraic Combinatorics Via Finite Group Actions. BI-Wissenschaftsverlag,1991.4. G. Kreweras. Sur une classe de problemes lies au treillis des partitions des entiers,cahiers du BURO, vol. 6, 1965.5. J. D. Monk, R. Bonnet (ed.). Handbook of Boolean Algebras. Vol. 1 (by S. Kopel-berg). North-Holland, 1989.6. R. P. Stanley. Ordered Structures and Partitions. Mem. Amer. Math. Soc., N. 119,Amer. Math. Soc., 1971.7. R. P. Stanley. Differential Posets. Journal Amer. Math. Soc., Vol. 1, N. 4, 1988.8. R. P. Stanley. Enumerative Combinatorics. Cambridge University Press, 1997.

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