heitzig reinhold 2002 counting finite lattices

Algebra univers. 48 (2002) 43–530002-5240/02/010043 – 11c©Birkhauser Verlag, Basel, 2002 Algebra Universalis

Counting Finite Lattices

Jobst Heitzig and Jurgen Reinhold

Abstract. The correct values for the number of all unlabeled lattices on n elements areknown for n ≤ 11. We present a fast orderly algorithm generating all unlabeled latticesup to a given size n. Using this algorithm, we have computed the number of all unlabeledlattices as well as that of all labeled lattices on an n-element set for each n ≤ 18.

1. Introduction

An efficient way for the mathematician to find and test new hypotheses is to con-sider examples. When working with finite structures, it can be helpful to generatesmall objects by a computer program. For example, the search for a counter-example to a new hypothesis can be done by such a program.

A mathematical structure is often called a labeled structure if one wants toexpress the difference between the structure itself and its isomorphism type whichis then called an unlabeled structure. Since mathematicians usually study structuresonly up to isomorphism, it is often more useful to consider unlabeled structures.

The principle of an orderly algorithm for the generation of all unlabeled struc-tures of a given type up to an appropriate finite size n was introduced by R.C. Read [14] in 1976. Considering structures like posets or graphs, Read’s idea canbe sketched as follows: Turn the class of all labeled structures of size at most n intoa finite set by assuming that the underlying set of the structures is always a naturalnumber m = {0, . . . , m − 1}, m ≤ n. Given two structures S, T with underlyingsets m and m + 1, respectively, call S the mother of T and T a child of S if S isa substructure of T . It should be possible to find an algorithmic description of allchildren of S. But, many of these children may be isomorphic. To avoid this ob-stacle, one introduces an appropriate linear ordering on the finite set of all labeledstructures and calls a structure S canonical, if it is the smallest structure with re-spect to this ordering compared with all other structures of the same isomorphismtype. The canonical structures form a set of representatives for the isomorphism

Presented by R. Freese.Received April 4, 2000; accepted in final form November 2, 2001.2000 Mathematics Subject Classification: 05A15, 06B99, 05–04.Key words and phrases: Orderly algorithm, (un-)labeled lattice, tree, canonical, levelized.

43

44 J. Heitzig and J. Reinhold Algebra univers.

types and, therefore, may be identified with unlabeled structures. The chosen lin-ear ordering should have the property that the mother of a canonical structure isalso canonical. Now, one can generate all canonical (unlabeled) structures of sizeat most n by starting with the trivial (empty) structure and always branching fromone structure S of size m < n to all canonical children of S.

Diverse authors have introduced different orderly algorithms which generate andcount various types of structures (see, e.g. [1,3,4,6,11,13,15,16,17]). In this paper,we introduce a fast orderly algorithm that generates lattices, i.e. posets (= partiallyordered sets) P in which any two elements a, b ∈ P have a meet (= greatest lowerbound) a ∧P b and a join (= least upper bound) a ∨P b. It is well known and easyto see that a finite poset that has a greatest element and in which any two elementshave a meet is already a lattice.

We denote by L(n) and l(n) the numbers of all labeled and unlabeled lattices onan n-element set, respectively. In [12], Koda presented the numbers l(11), l(12) andl(13). It seems that he used an orderly algorithm which generated all unlabeledposets and then counted only those posets which actually are lattices. We aresure that Koda’s values for l(12) and l(13) are wrong. Using our algorithm, wecalculated the numbers l(n) as well as L(n) for n ≤ 18. Besides presenting thealgorithm and the results we shall also sketch a couple of methods that we used toverify our results.

2. A tree of lattices

In this paper it will be convenient to consider n-lattices, i.e. lattices with under-lying set n = {0, . . . , n−1} whose smallest element is 0 and whose greatest elementis 1. We will only consider the case where n ≥ 2 and, consequently, 0 �= 1. To avoidconfusion, we shall use the symbol � for the order relation of posets and latticesand reserve the symbol ≤ for the usual order relation on the natural numbers. LetP be a poset and A ⊆ P . The set ↓A = {p ∈ P | p � a for some a ∈ A} is calledthe lower set generated by A and the set ↑A = {p ∈ P | a � p for some a ∈ A}is called the upper set generated by A. If an interval [a, b] = {p ∈ P | a � p � b}contains exactly the two different elements a, b then we write a ≺ b and call b an(upper) cover of a and a a lower cover of b. Covers of the smallest element 0 ina lattice are called atoms. Each atom is join-irreducible, i.e. it is different from0 and not the join of two strictly smaller elements. Coatoms and meet-irreducibleelements are defined dually. While every subset Q of a poset P is again a poset P |Qcarrying the induced order relation � |Q, this smaller poset need not be a latticeeven if P is. If �|Q is the identity relation on Q then Q is called an antichain inP . The following lemma is probably known to lattice-theorists, and its proof is aneasy exercise.

Vol. 48, 2002 Counting Finite Lattices 45

Lemma 1. Let a �= 0 be an element of a finite lattice L. Then L′ = L|L\{a} is alattice if and only if a is join- or meet-irreducible in L.

By Lemma 1, a successive removal of atoms from a given finite lattice will producesmaller lattices. Conversely, we wish to generate larger lattices by adding newatoms. Actually, it is possible to describe for a given n-lattice L all (n + 1)-latticesM containing the element n as an atom such that the n-lattice M |n = M \ {n}equals L. To this end, call a nonempty antichain A ⊆ L \ {0} of an n-lattice L alattice-antichain if a ∧L b ∈ {0}∪ ↑A holds for any two elements a, b ∈ ↑A.

Lemma 2. A subset A ⊆ L\{0} of an n-lattice L is a lattice-antichain if and onlyif L is a subposet of an (n + 1)-lattice LA in which the element n is an atom andA is the set of its covers.

Proof. Let M be an (n + 1)-lattice such that n ∈ M is an atom covered exactly bythe elements of the subset A ⊆ L \ {0}, where L = M |n. Hence M = LA. Since A

consists of the minimal elements strictly above n in LA, it is an antichain, and forc, d ∈ ↑A = ↑{n} \ {n} one has c ∧LA d ∈ {n}∪ ↑A and thus c ∧L d ∈ {0}∪ ↑A.Thus A is a lattice-antichain in L.

Conversely, let A be a lattice-antichain in an n-lattice L = (n,�). Define LA =(n + 1,�A) by

�A =� ∪ {(0, n), (n, n)} ∪ (n×↑A).

It is now straightforward to show that LA is a lattice. �

Consider the unique 2-lattice 2 and define

S2 = {2}

Now, recursively define a set S3 of 3-lattices, a set S4 of 4-lattices etc. by setting

Sn+1 = {LA | L ∈ Sn, A is a lattice-antichain in L}.

Finally, let

S =∞⋃

n=2

Sn.

For L = (l,�L), M = (m,�M ) ∈ S set L �S M if and only if l ≤ m and �L =�M |l. Obviously, �S is an order relation on S such that (S,�S) is a tree, i.e. forevery element L ∈ S, the lower set ↓{L} is linearly ordered. A straightforwardinduction using Lemma 2 shows that an n-lattice L = (n,�L) is an element of thetree S if and only if for each number m = 2, . . . , n− 1 the element m is an atom inthe (m + 1)-lattice (m + 1, �L|m+1).


3. The orderly algorithm

We define the weight w(L) = (w2(L), . . . , wn−1(L)) of an n-lattice L by settingwi(L) =

∑i≺j 2j . Since a finite poset is uniquely determined by its covering relation,

the map L → w(L) is injective.Let L, M be n-lattices. Then we write w(L) <lex w(M) and say that w(L) is

(lexicographically) smaller than w(M) if there is an i ≤ n − 1 such that wi(L) <

wi(M) and wk(L) = wk(M) for all k = 2, . . . , i − 1. We call an n-lattice C acanonical lattice if there is no n-lattice isomorphic to C that has a smaller weight.To decide whether an n-lattice L is canonical, one has to check whether there is apermutation of the labels of L that yields an isomorphic copy of L with a smallerweight.

If A is a lattice-antichain in an n-lattice L such that LA is canonical then L iscanonical, too. Indeed, if ϕ is an isomorphism between L and the canonical latticeC ∼= L then Cϕ[A] ∼= LA. But the assumption L �= C, i.e. w(L) > w(C), impliesw(LA) > w(Cϕ[A]), a contradiction.

Let T2 = {2} and define

Tn+1 = {LA | L ∈ Tn, A is a lattice-antichain in L such that LA is canonical}.Ordering the set T =

⋃∞n=2 Tn in the same way as the set S in the preceding sec-

tion yields the tree (T ,�T ). As we shall see in the next section, in every canonical(n + 1)-lattice the element n is an atom. Since for each canonical (n + 1)-lattice L,the n-lattice L|n and thus all the lattices L|2, . . . , L|n are again canonical lattices,T contains exactly all canonical lattices.

Now, it is easy to formulate a recursive algorithm that generates for a givennatural number n ≥ 2 exactly all canonical lattices on up to n elements by a depth-first search through the tree Tn. It consists of the following procedure next lattice

which is initially called with the arguments m = 2 and L = 2 (the two-elementlattice).

next lattice (integer m, canonical m-lattice L)

begin

if m < n

then for each lattice-antichain A of L do

if LA is a canonical lattice

then next lattice (m + 1, LA)

if m = n then output L

end.


4. Levelized lattices

The set of all maximal elements in a finite poset P is called the first level of P

and will be denoted by lev1(P ). One recursively defines the (m + 1)-th level of P

by

levm+1(P ) = lev1(P \m⋃

i=1

levi(P )).

Moreover, for p∈P let depP (p) denote the depth of p in P , i.e. the number k suchthat p∈ levk(P ).

It is easy to see that an element p ∈ P is contained in levm(P ) if m is themaximal cardinality of a chain in P with least element p. In the case of a lattice,for example, the second level is the set of all coatoms, and the second to the lastnonempty level contains atoms only, but there may also be atoms in higher levels.Notice that p � q implies depP (p) > depP (q).

We call an n-lattice L levelized if depL(i) ≤ depL(j) for all i, j ∈ L \ {0}with i ≤ j. It is easy to see that L is levelized if and only if

⋃kh=1 levh(L) =

{1, . . . , |⋃k

h=1 levh(L)|} for k = 1, . . . , depL(n − 1).Let L be a levelized n-lattice. Then for each m = 2, . . . , n−1 the (m+1)-lattice

L|m+1 is also levelized and thus m is an atom in L|m+1, in particular, L ∈ S.Moreover, if A is a lattice-antichain in L then depL(m) = depLA(m) for eachm = 1, . . . , n−1, and thus LA is levelized if and only if A∩(levk−1(L)∪levk(L)) �= ∅,where k = depL(n − 1).

Theorem 1. Every canonical lattice L is levelized.

Proof. Assume that L is not levelized. Then there is a smallest number m such thatthe set Mm =

⋃mh=1 levh(L) is different from {1, . . . , |Mm|}. Let i be the smallest

number in {1, . . . , |Mm|} \Mm and choose any j ∈ Mm \{1, . . . , |Mm|}. Then i < j

but depL(i) > depL(j). Define L′ to be the isomorphic copy of L obtained byexchanging the labels i and j.

(i) wk(L′) = wk(L) for 2 ≤ k < i: By the choice of i, one has k ∈ Mm andthus depL(k) ≤ m. Let c be any cover of k in L. Then depL(c) ≤ m − 1. SinceMm−1 =

⋃m−1h=1 levh(L) coincides with {1, . . . , |Mm−1|} by the choice of m, it follows

from c ∈ Mm−1 and i /∈ Mm−1 ⊆ Mm that c < i < j. Therefore, the covers of k inL are different from i and j and thus coincide with those in L′.

(ii) wi(L′) = wj(L): From depL′(j) > depL′(i) it follows that j is not a cover ofi in L′. Therefore, the covers of i in L′ coincide with those of j in L.

(iii) wj(L) < wi(L): Since j ∈ Mm, every cover of j is contained in Mm−1 ={1, . . . , |Mm−1|} and, therefore, wj(L) < 2|Mm−1|+1. Since depL(i) > m = depL(j),there is a cover c of i in L with depL(c) ≥ m, whence c /∈ Mm−1 = {1, . . . , |Mm−1|}and thus c ≥ |Mm−1| + 1 and wi(L) ≥ 2c ≥ 2|Mm−1|+1 > wj(L).


In all, it follows that w(L′) < w(L), a contradiction. �

Dealing with levelized n-lattices, one has to consider fewer permutations in orderto check their canonicity. Indeed, by the above theorem, it suffices to considerpermutations that leave the levels of L fixed. This idea makes it possible to improvethe performance of the algorithm considerably.

5. Speeding up the algorithm

In this section, we present a few further ideas to make the algorithm run faster.As we shall see, special properties of canonical lattices make it possible to considerfewer lattice-antichains A on the given n-lattice L and to decide more quicklywhether the lattice LA is canonical. As demonstrated in the preceding section, itwill suffice to consider levelized lattices only, so the lattice-antichains A should havethe property A∩

(levk−1(L)∪ levk(L)

)�= ∅ for k = depL(n−1). The next theorem

yields another way to reduce the number of lattice-antichains A to be considered,namely to deal only with those antichains A which satisfy wn(LA) =

∑a∈A 2a ≥

wn−1(L).

Theorem 2. Let L be a canonical n-lattice. Then w2(L) ≤ · · · ≤ wn−1(L).

Proof. Let 2 ≤ i < j. If depL(i) = depL(j) then consider the lattice L′ obtainedfrom L by exchanging the labels i and j. Since L is levelized by Theorem 3, we havewk(L′) = wk(L) for k = 1, . . . , i− 1. Hence wi(L) ≤ wi(L′) by the canonicity of L.Since the covers of i in L′ coincide with those of j in L, we have wi(L′) = wj(L).It follows that wi(L) ≤ wj(L).

If depL(i) < depL(j) then there is a cover c of j such that depL(c) = depL(j) −1 ≥ depL(i), and since L is levelized, there is no m ∈ n with m ≥ c and i ≺ m.Hence wi(L) < 2c ≤ wj(L). �

When the algorithm tries to decide for a given canonical lattice L and a givenlattice-antichain A on L whether the (n + 1)-lattice LA is canonical, it searchesfor a permutation ϕ of the labels of LA such that the weight of the lattice withthe resulting labeling is smaller than that of LA. As mentioned at the end of thelast section, we only have to consider permutations ϕ that leave the levels of LA

fixed. Another possible method to reduce the number of permutations consideredis based on the following observation: Let k = depLA(n). Since L is levelized,m = {0} ∪

⋃k−1i=1 levi(L) is a natural number. Since L is canonical, so is L|m.

Therefore, we only need to consider permutations ϕ such that the restriction ϕ|mis an automorphism of L|m.

Recall that a lattice L is vertically decomposable if it contains an element which isneither the greatest nor the least element of L but comparable with every element


of L. Let lv(n) be the number of unlabeled vertically indecomposable lattices.Applying the simple recursion formula

l(n) =n∑

k=2

lv(k)l(n − k + 1), n ≥ 2, (1)

where the running index k may be regarded as the cardinality of the upmost verticalcomponent of the lattices, one can compute the numbers l(1), . . . , l(n) from thevalues of lv(1), . . . , lv(n).

Theorem 3. Let L = (n,�) be a canonical n-lattice containing an element k ∈{3, . . . , n − 1} comparable with the elements 2, . . . , k − 1. Then L is verticallydecomposable.

Proof. Let A be the set of covers of k. Since L is levelized by Theorem 3, we have↑A = {1, . . . , k − 1}. Assume that there is a smallest number p ∈ {k + 1, . . . , n}such that p �� a for some a ∈ A. By Theorem 4, we have wp(L) ≥ wk(L). Ifwp(L) > wk(L) then p ≺ c for some c ∈ {k, . . . , p − 1} and thus p � c � a by theminimality of p. If wp(L) = wk(L) =

∑b∈A 2b then A is also the set of covers of

p. Thus in both cases we have p � a, a contradiction. It follows that p � a holdsfor every a ∈ A and every p ∈ {k, . . . , n − 1}. Suppose that k is not comparablewith every element in L. Then there is a q ∈ {k + 1, . . . , n − 1} which is maximalwith respect to � such that q �� k. By the maximality of q, the antichain A isalso the set of covers of q. Since L is a lattice, it follows that |A| = 1, and since↑A = {1, . . . , k − 1}, it follows that A = {k − 1}. Hence the element k − 1 iscomparable with every element in L. �

Theorem 5 makes it easy to modify the algorithm so that it only generates andcounts vertically indecomposable lattices. Unfortunately, the proportion of verti-cally decomposable n-element lattices becomes small when n grows. For example,for n = 18 less than 1/5 of all unlabeled n-lattices are vertically decomposable.

6. The labeled case

Although our algorithm generates only one member of each isomorphism classof finite lattices, it also provides us with the numbers L(n) of all labeled latticeson an n-element set. Indeed, an n-lattice is recognized to be canonical if the searchthrough the permutations of the labels of L does not yield an isomorphic copy witha smaller weight. Therefore, the algorithm may count by the way the number ofthose permutations which yield an isomorphic copy with the same weight, i.e. itmay determine the cardinality of the automorphism group Aut(L) of L. Now, the


numbers L(n) can be computed by

L(n) =∑

L canonicaln-lattice

n!|Aut(L)| . (2)

Similar to the extrapolation formulae of Erne [5] for posets, the following theo-rem shows that it is possible to compute the numbers L(n) by just generating allunlabeled (n − 1)-lattices L and then determining for every lattice-antichain A ofL the number a(LA) of atoms in LA.

Theorem 4.

L(n) =∑

L canonical(n − 1)-lattice

n!|Aut(L)|

∑A lattice-

antichain in L

1a(LA)

Proof. Let Lk(n) be the number of all labeled lattices on the set n with exactlyk atoms, and let Lk(n) be the number of all labeled lattices on the set n withexactly k atoms such that the element n − 1 is one of these atoms. Moreover, letLj

k(n) be the number of all labeled lattices L on the set n which contain exactly j

lattice-antichains A such that LA has exactly k atoms. Then

Lk(n) =2n∑

j=1

j · Ljk(n − 1).

The fractions Lk(n)/(nk

)and Lk(n)/

(n−1k−1

)have the same value since both can be

interpreted as the number of labeled lattices on the set n in which exactly theelements n − k, . . . , n − 1 are atoms. Hence

Lk(n) = Lk(n) ·(nk

)(n−1k−1

) = Lk(n) · n

k.

In all, we get

L(n) =n∑

k=1

Lk(n) =n∑

k=1

Lk(n) · n

k=

n∑k=1

2n∑j=1

j · Ljk(n − 1) · n

k,

and applying (2), it follows that

L(n) =∑

L canonical(n − 1)-lattice

(n − 1)!|Aut(L)|

∑A lattice-

antichain in L

n

a(LA). �

An algorithmic realization of Theorem 6 computes the numbers L(n) much fasterthan that of (2). But since the search for all lattice-antichains and the determina-tion of the numbers a(LA) needs some additional time, we did not compute L(19)although we generated all unlabeled lattices with up to 18 elements.


7. Concluding remarks

An implementation of our algorithm on a single personal computer yields theresults presented in the table in a reasonable time only for n ≤ 17. Indeed, aPentium III, 500 Mhz, needs about two days to determine the numbers l(n) andL(n) for n ≤ 16 and another three weeks for the values of l(17) and L(17). Inorder to determine the numbers l(18) and L(18), with the kind support of theRRZN Hannover, we developed a parallelized version of the algorithm which ranon the Cray T3e ‘BERTE’ at the Conrad Zuse Zentrum in Berlin. We worked with50 of BERTE’s 450 Mhz processors at once. Divided into several partial jobs, theprogram ran about 6 days. Some details of the parallelization have been presentedin [7].

When working with computer programs containing hundreds of commands, itis almost impossible to prove that the results are correct. However, using severaldifferent ways, we were able to partially verify our results. For example, we countedthe number of vertically indecomposable lattices as well as the number of all latticesand then checked that (1) holds. Moreover, we implemented the extrapolationformula given in Theorem 6 and compared the resulting values for the numbersL(n) with the ones obtained by a direct counting.

The asymptotics given in [9,10] are not suitable for a verification of our numbersnor of their rough size. The asymptotic behaviour of the sequence (Ln)n is notrecognizable for the small values n ≤ 18.

Our algorithm can easily be modified in order to generate arbitrary posets insteadof lattices. The fact that this modification has verified all known values for posets(see [2]) gives us even more evidence that the algorithm works properly. Moreover,we were able to compute the number of all unlabeled posets on 14 elements (see [7]).The fact that this new result needed even less effort than Chaunier’s and N. Lygeros’computation [2] of the last known value for 13 elements demonstrates the superiorityof our algorithm. We plan to use other modifications to generate and count differenttypes of lattices, for example, distributive, modular or complemented lattices. Theadditional structural properties will make the resulting trees considerably smallerso that we expect to obtain results for underlying sets with much more than 18elements.


The numbers of lattices on up to 18 elements.

elements labeled lattices unlabeled latticesn L(n) l(n)

1 1 12 2 13 6 14 36 25 380 56 6 390 157 157 962 538 5 396 888 2229 243 179 064 1 078

10 13 938 711 210 5 99411 987 858 368 750 37 62212 84 613 071 940 452 262 77613 8 597 251 494 954 564 2 018 30514 1 020 353 444 641 839 854 16 873 36415 139 627 532 137 612 581 090 152 233 51816 21 788 453 795 572 514 675 760 1 471 613 38717 3 840 596 246 648 027 262 079 472 15 150 569 44618 758 435 490 711 709 577 216 754 642 165 269 824 761

References

[1] G. Brinkmann and E. Steffen, Chromatic-index-critical graphs of orders 11 and 12,European J. Combin. 19 (1998), no. 8, 889–900.

[2] C. Chaunier and N. Lygeros, The number of orders with thirteen elements, Order 9 (1992),no. 3, 203–204.

[3] C. J. Colbourn and R. C. Read, Orderly algorithms for graph generation, Internat. J.Comput. Math. 7 (1979), no. 3, 167–172.

[4] C. J. Colbourn and R. C. Read, Orderly algorithms for generating restricted classes ofgraphs, J. Graph Theory 3 (1979), no. 2, 187–195.

[5] M. Erne, On the cardinalities of finite topologies and the number of antichains in partiallyordered sets, Discrete Math. 35 (1981), 119–133.

[6] D. K. Garnick and J. H. Dinitz, On the number of one-factorizations of the complete graphon 12 points, in: Proceedings of the Twenty-fourth Southeastern International Conferenceon Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1993), Congr. Numer.

94 (1993), 159–167.[7] J. Heitzig and J. Reinhold, The number of unlabeled orders on fourteen elements, Order 17

(2000), no. 4, 333–341.[8] D. J. Kleitman and B. L. Rothschild, Asymptotic enumeration of partial orders on a finite

set, Trans. Amer. Math. Soc. 205 (1975), 205–220.


[9] D. J. Kleitman and K. J. Winston, The asymptotic number of lattices, in: Combinatorialmathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. andOptimal Design, Colorado State Univ., Fort Collins, Colo., 1978), Ann. Discrete Math. 6(1980), 243–249.

[10] W. Klotz and L. Lucht, Endliche Verbande, J. Reine Angew. Math. 247 (1971) 58–68.[11] Y. Koda, Orderly algorithms for generating k-colored graphs, in: Proceedings of the

Twenty-third Southeastern International Conference on Combinatorics, Graph Theory, andComputing (Boca Raton, FL, 1992), Congr. Numer. 91 (1992), 223–238.

[12] Y. Koda, The numbers of finite lattices and finite topologies, Bull. Inst. Combin. Appl. 10(1994), 83–89.

[13] B. D. McKay, Isomorph-free exhaustive generation, J. Algorithms 26 (1998), no. 2, 306–324.[14] R. C. Read, Every one a winner or how to avoid isomorphism search when cataloguing

combinatorial configurations, in: Algorithmic aspects of combinatorics (Conf., VancouverIsland, B. C., 1976), Ann. Discrete Math. 2 (1978), 107–120.

[15] G. F. Royle, An orderly algorithm and some applications in finite geometry, Discrete Math.185 (1998), no. 1–3, 105–115.

[16] P. Schultz, Enumerations of rooted trees with an application to group presentations,Discrete Math. 41 (1982), no. 2, 199–214.

[17] E. Seah and D. R. Stinson, On the enumeration of one-factorizations of complete graphscontaining prescribed automorphism groups, Math. Comp. 50 (1988), no. 182, 607–618.

Jobst HeitzigUniversitat Hannover, Institut fur Mathematik, Welfengarten 1, 30167 Hannover, Germany

e-mail : [email protected]

Jurgen ReinholdUniversitat Hannover, Institut fur Mathematik, Welfengarten 1, 30167 Hannover, Germanye-mail : [email protected]

To access this journal online:

http://www.birkhauser.ch

heitzig reinhold 2002 counting finite lattices

Documents