CCARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM
JUKKA KOHONEN
Abstract.
A vertical 2-sum of a two-coatom lattice L and a two-atom lat-tice U is obtained by removing the top of L and the bottom of U , and identi-fying the coatoms of L with the atoms of U . This operation creates one or twononisomorphic lattices depending on the symmetry case. Here the symmetrycases are analyzed, and a recurrence relation is presented that expresses thenumber of such vertical 2-sums in some family of interest, up to isomorphism.Nonisomorphic, vertically indecomposable modular and distributive latticesare counted and classified up to 35 and 60 elements respectively. Asymptoti-cally their numbers are shown to be at least Ω(2 . n ) and Ω(1 . n ), where n is the number of elements. The number of semimodular lattices is shown togrow faster than any exponential in n . Introduction
Let L and U be finite lattices. Their vertical sum is obtained by identifying thetop of L with the bottom of U . If L has two coatoms and U has two atoms, their vertical 2-sum is obtained by removing the top of L and the bottom of U , andidentifying the coatoms of L with the atoms of U .The vertical sum leads to a simple and well-known recurrence relation. A latticeis a vi-lattice (short for vertically indecomposable ) if it is not a vertical sum oftwo non-singleton lattices. If f ( n ) and f vi ( n ) are the numbers of nonisomorphic n -element lattices and vi-lattices, respectively, then(1) f ( n ) = n (cid:88) k =2 f vi ( k ) f ( n − k + 1) , for n ≥ Cartesian counting because each term expresses the cardinality ofa Cartesian product, namely, of the set of k -element vi-lattices and the set of( n − k +1)-element lattices. One does not need to list the members of a Cartesianproduct to find its cardinality. Recurrence (1) has been used in counting smalllattices [4, 5, 7, 8, 12] and in proving lower bounds [4, 13].Many vi-lattices can be further decomposed as vertical 2-sums of smaller lattices.So let us pursue a kind of Cartesian counting of vi-lattices. Now we must observethat from two given lattices, one obtains two vertical 2-sums, because there are twoways to match the coatoms and the atoms. Whether the results are isomorphicdepends on the symmetries of L and U .Our main result, Theorem 1, is a recurrence relation that distinguishes the sym-metry cases, and expresses the exact number of nonisomorphic lattices obtainableas vertical 2-sums. To apply the recurrence, we need to classify and count thecomponent lattices by symmetry type. Key words and phrases.
Counting, vertical 2-sum, modular lattices, distributive lattices. a r X i v : . [ m a t h . C O ] J u l JUKKA KOHONEN
The motivations of this study are threefold. First, our recurrence provides a newway of counting small lattices. Only the component lattices are generated explicitly;their vertical 2-sums are then counted with the recurrence in the Cartesian fashion.This is faster, so we can count further. We count modular and distributive vi-lattices of at most 35 and 60 elements, respectively. This also provides a verificationof previous countings (of at most 33 and 49 elements, respectively), because themethod is different.The second motivation is a more compact lattice listing. A full listing of dis-tributive vi-lattices of at most 60 elements would contain about 4 . × lattices.We can shrink the list to less than 1 /
200 of that size, to 2 . × lattices, by leav-ing out all vertical 2-sums. A smaller listing is more practical to store and to study,and one can still recover the left-out lattices by performing the vertical 2-sums.The third motivation is in improving lower bounds. A simple recurrence forvertical 2-sums was derived in [13], but it is only a loose lower bound as it does notconsider the symmetry cases. The new recurrence gives tighter bounds because ofan extra factor of 2 in the asymmetric cases. It may not sound much, but the factorcompounds when vertical 2-sum is applied repeatedly. Some further improvementcomes from counting small lattices larger than before. For nonisomorphic modularvi-lattices, we improve the lower bound from Ω(2 . n ) [13] to Ω(2 . n ). Fornonisomorphic distributive vi-lattices, we improve from Ω(1 . n ) [4] to Ω(1 . n ),which is close to the empirical growth rate.2. Vertical 2-sum and symmetry
In order to understand how the vertical 2-sum operates on lattices, we classifythem by the number and symmetry of their atoms and coatoms. Our aim is inmodular and distributive vi-lattices, but we state the results more generally whenconvenient. All lattices considered in this work are finite. If L is a lattice, we write a ( L ) and c ( L ) for the numbers of its atoms and coatoms, and 0 L and 1 L for itsbottom and top. Definition 1.
Let L and U be disjoint lattices of length 3 or greater, L with twocoatoms c , c and U with two atoms a , a . Then their vertical 2-sums are thetwo lattices obtained by removing 1 L and 0 U , and identifying ( c , c ) with either( a , a ) or ( a , a ). L and U are the summands of the vertical 2-sum.Note that vertical 2-sums are indeed lattices, those of graded lattices are graded,and those of vi-lattices are vi-lattices [13]. We do not consider summands of length 2as that would be essentially an identity operation. If S is a vertical 2-sum of L and U , then | S | = | L | + | U | − Definition 2.
If a lattice has two coatoms [atoms], they are symmetric if the latticehas an automorphism that swaps them, and fixed otherwise.
Lemma 1.
Let L and U be lattices with vertical 2-sums S and S . Then S and S are nonisomorphic if and only if L has fixed coatoms and U has fixed atoms.Proof. If L has an automorphism that swaps the coatoms, then extending it withthe identity mapping on U yields an isomorphism S → S . If U has symmetricatoms, the case is similar. Finally, if there is an isomorphism S → S , it musteither fix the coatoms of L and swap the atoms of U , or vice versa; but this isimpossible if L has fixed coatoms and U has fixed atoms. (cid:3) ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 3
From here on we confine our attention to graded vi-lattices. We divide theminto three kinds as follows.
Definition 3. If L is a graded vi-lattice, then its k th level , denoted L k , is the setof elements that have rank k . A neck is a two-element level other than the atomsand the coatoms. We say that L is(1) a composition , if it contains a neck;(2) a piece , if does not contain a neck, has rank 3 or greater, and at least oneof a ( L ) and c ( L ) equals two;(3) special otherwise.A composition has necessarily at least 8 elements, and a piece has at least 6.All compositions ensue from pieces by repeated application of the vertical 2-sum.Specials are not vertical 2-sums, but also cannot act as their summands, becausethey are too short (rank two or smaller) or contain too many atoms and coatoms. Definition 4.
A piece L is:(1) a middle piece , if a ( L ) = c ( L ) = 2;(2) a bottom piece , if a ( L ) ≥ c ( L ) = 2;(3) a top piece , if a ( L ) = 2 and c ( L ) ≥ Definition 5.
A middle piece is of symmetry type:(1) MF, if its atoms and coatoms are fixed;(2) MA, if its atoms are symmetric and coatoms are fixed;(3) MC, if its atoms are fixed and coatoms are symmetric;(4) MX, if it has an automorphism that swaps the atoms but fixes the coatoms,and another automorphism that swaps the coatoms but fixes the atoms;(5) MH, if it is not MX, but has an automorphism that swaps both the atomsand the coatoms.
Definition 6.
A bottom piece is of symmetry type:(1) BF, if its coatoms are fixed;(2) BS, if its coatoms are symmetric.
Definition 7.
A top piece is of symmetry type:(1) TF, if its atoms are fixed;(2) TS, if its atoms are symmetric.
Definition 8.
A composition is of symmetry type:(1) CF, if it has two coatoms and they are fixed;(2) CS, if it has two coatoms and they are symmetric;(3) CN (“composition-nonextensible”), if it has three or more coatoms.The symmetry types are illustrated in Figure 1. If we take the BF example asthe lower summand, and the MC example as the upper summand, we obtain twononisomorphic vertical 2-sums (both of type CS) as shown in Figure 2.Note that in an MX piece atoms and coatoms can be swapped independently,but in an MH piece only simultaneously; the shapes of the letters X and H aremeant as mnemonics for this.
JUKKA KOHONEN
MF MA MC MX MHBF BS TF TS special
Figure 1.
Example pieces of each symmetry type (and a special).
Figure 2.
The two vertical 2-sums of a BF piece and an MC piece(elements common to the summands are shown as hollow circles).We will form compositions “bottom up”, adding new pieces over pieces or smallercompositions. The following lemma characterizes how many and what kinds ofcompositions are formed in different cases.
Lemma 2. If L is a piece or a composition, and U is a piece, then the number andtype of their nonisomorphic vertical 2-sums are as follows. U type L type, any of MF MA MC MX MH TF TSCF/BF/MF/MA 2 × CF 1 × CF 2 × CS 1 × CS 1 × CF 2 × CN 1 × CNCS/BS/MC/MX/MH 1 × CF 1 × CF 1 × CS 1 × CS 1 × CS 1 × CN 1 × CNCN/TF/TS none none none none none none none
ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 5
Proof.
Let us first prove the numbers. We proceed by the rows.Row 1: L has two fixed coatoms. By Lemma 1, if U has fixed atoms (MF, MCor TF), then there are two nonisomorphic vertical 2-sums; otherwise there is one.Row 2: L has two symmetric coatoms. By Lemma 1 there is one vertical 2-sumup to isomorphism.Row 3: L has three or more coatoms, so no vertical 2-sums are formed.Next we deduce the symmetry type of each vertical 2-sum S i ( i = 1 , U is MF or MA, it has fixed coatoms. Then S i cannot have an automorphismthat swaps the coatoms of S i , because its restriction to U would be an automor-phism that swaps the coatoms of U . Thus S i has fixed coatoms.If U is MC or MX, it has an automorphism that swaps its coatoms and fixes itsatoms; extending with the identity mapping in L gives, in each S i , an automorphismthat swaps the coatoms. Thus S i has symmetric coatoms.If U is MH, then an automorphism of S i that swaps the coatoms also swaps theatoms of U , which are also the coatoms of L . Thus S i has symmetric coatoms ifand only if L has symmetric coatoms.If U is TF or TS, then S i has three or more coatoms and is of type CN. (cid:3) Cartesian counting in a lattice family
In this section we present our main result, a recurrence relation that counts allnonisomorphic compositions in some desired lattice family, provided that the familyhas suitable form. We also give examples of such families.
Definition 9.
A family F of graded vi-lattices is (vertically) if thefollowing conditions hold:(C1) If L, U ∈ F and S is one of their vertical 2-sums, then S ∈ F .(C2) If S ∈ F is a vertical 2-sum of L and U , then L, U ∈ F .The first condition ensures that vertical 2-sums stay in the family, and the secondensures that all compositions in F are indeed obtained as vertical 2-sums of smallerlattices in F . Theorem 1.
Let F be a 2-summable family, and let XX n denote the number ofnonisomorphic n -element lattices in F having symmetry type XX. Then for n < we have CF n = CS n = CN n = 0 , and for n ≥ the following recurrences hold: CF n = n − (cid:88) j =6 (cid:16) LF j · (2 · MF k + MA k + MH k ) + LS j · (MF k + MA k ) (cid:17) CS n = n − (cid:88) j =6 (cid:16) LF j · (2 · MC k + MX k ) + LS j · (MC k + MX k + MH k ) (cid:17) CN n = n − (cid:88) j =6 (cid:16) LF j · (2 · TF k + TS k ) + LS j · (TF k + TS k ) where k = n − j + 4 , and LF j = CF j + BF j + MF j + MA j LS j = CS j + BS j + MC j + MX j + MH j . JUKKA KOHONEN
Proof.
For n < n ≥ n -element CF-type composition S ∈ F . Thereis exactly one way of expressing S as a vertical 2-sum of two lattices L, U suchthat U is a piece. This U contains the elements of S above and including itshighest-ranked neck, plus an augmented bottom element. By condition (C2) wehave L, U ∈ F . Furthermore, because | U | ≥ | L | + | U | − n , it followsthat | L | ≤ n − L and U , where U is apiece, lead to nonisomorphic results. To be more precise: If L (cid:29) L (cid:48) or U (cid:29) U (cid:48) , and U and U (cid:48) are pieces, then the vertical 2-sums of L and U are not isomorphic to thevertical 2-sums of L (cid:48) and U (cid:48) .All nonisomorphic n -element CF-type compositions in F can be counted byconsidering (for all j = 6 , . . . , n −
2) first the choices of a j -element lower summand L ∈ F , and then the choices of an upper summand U ∈ F such that U is a piecewith k = n − j + 4 elements, subject to the requirement that the resulting vertical2-sums are of type CF.Now LF j is the number of nonisomorphic lower summands that have fixedcoatoms. For each such lower summand, by collecting the CF-type results fromthe first row of the table in Lemma 2, we get 2 · MF k + MA k + MH k nonisomorphicvertical 2-sums, which are in F by condition (C1).Similarly, LS j is the number of nonisomorphic lower summands that have sym-metric coatoms. For each such lower summand, by collecting the CF-type resultsfrom the second row of the table in Lemma 2, we get MF k + MA k nonisomorphicvertical 2-sums, which are in F by condition (C1).Adding up the cases we obtain the stated expression for CF n . The expressionsfor CS n and CN n follow in the same manner. (cid:3) Not all families of graded vi-lattices are 2-summable. For a simple example,finite graded rank-four vi-lattices fail both conditions (C1) and (C2). Interestingly,finite geometric lattices are 2-summable but in a vacuous way.
Theorem 2.
The only finite geometric lattice that has a two-element level is M .Proof. Let L be a finite geometric lattice that has a two-element level { a, b } . Be-cause L is atomistic, neither a or b has any join-irreducible covers, thus a and b are covered by exactly one element c . Further, because L is necessarily verticallyindecomposable, it follows that c = 1 L , and a, b are the coatoms.The numbers of atoms and coatoms in a finite geometric lattice of rank r areknown as the Whitney numbers W and W r − . It is known that W ≤ W r − ; seee.g. Dowling and Wilson [3]. This implies that our L has two atoms. But we haveshown that L cannot have two-element levels other than the coatoms; thus theatoms are the coatoms, and L = M . (cid:3) In other words, Theorem 2 says that all finite geometric lattices are special; thereare no pieces and no compositions, so no use for the vertical 2-sum. Modular anddistributive lattices will be more interesting for our purposes. We first prove anauxiliary result by elementary means.
ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 7
Lemma 3.
In the Hasse diagram of a finite semimodular lattice, the subgraphinduced by two consecutive levels is connected.Proof.
Let L be a finite semimodular lattice, L k its k th level, and H k the subgraphinduced by L k and L k +1 . We use induction on k . Clearly H is connected. Assumethen that H k − is connected. If L k is a singleton, then obviously H k is connected.Otherwise, let ( U, V ) be any partition of L k into two nonempty subsets. Because H k − is connected, there is an element in L k − that is covered by some two elements u ∈ U and v ∈ V . Then by semimodularity u and v are covered by some w ∈ L k +1 ,so there is a path from U to V in H k . Finally, from every element in L k +1 there isan edge to L k . Thus H k is connected. (cid:3) Note that Lemma 3 also follows from previously known, more general results:Bj¨orner [1] proved that finite semimodular lattices are lexicographically shellable,and Collins [2] proved that graded lexicographically shellable lattices are rank-connected (i.e. the subgraph induced by two consecutive levels is connected).
Lemma 4.
The family of finite semimodular vi-lattices is 2-summable.Proof.
We use subscripted symbols ∧ L , ∨ L and ≺ L to denote meet, join andcovered-by in lattice L .(C1) Let L and U be finite semimodular vi-lattices, S their vertical 2-sum, and N the two identified elements of L and U . Clearly S is a vi-lattice. We showthat S is semimodular by using Birkhoff’s condition [6, p. 331]. Let a, b ∈ S such that they cover a ∧ S b . Then a and b have the same rank, and areeither below N , in N or above N . If a, b are below N , then because L issemimodular, a, b ≺ S a ∨ L b = a ∨ S b . If a, b are in or above N , then because U is semimodular, a, b ≺ S a ∨ U b = a ∨ S b .(C2) Let S be a finite semimodular vi-lattice that is a vertical 2-sum of L and U .Clearly L and U are vi-lattices. We show that they are semimodular, againusing Birkhoff’s condition.First let a, b ∈ L such that they cover a ∧ L b . If a, b are not the coatomsof L , then a ∨ L b = a ∨ S b . If a, b are the coatoms of L , then a ∨ L b = 1 L . Inboth cases a, b ≺ L a ∨ L b . Thus L is semimodular.Let then a, b ∈ U such that they cover a ∧ U b . If a, b are not the atoms of U ,then a ∧ U b = a ∧ S b , and because S is semimodular, a, b ≺ U a ∨ U b = a ∨ S b . If a, b are the atoms of U , then they are a neck of S . Because S is semimodular,it follows from Lemma 3 that a, b have a common upper cover c in S . Thenalso a, b ≺ U c . Thus U is semimodular. (cid:3) Lemma 5.
The family of finite modular vi-lattices is 2-summable.Proof.
Follows from Lemma 4 by duality. (cid:3)
Lemma 6.
The family of finite distributive vi-lattices is 2-summable.Proof.
We recall that a finite modular lattice is distributive if and only if it doesnot contain a cover-preserving diamond [6, p. 109], that is, five distinct elements o, a, b, c, i such that o ≺ a, b, c ≺ i .(C1) Let L and U be finite distributive vi-lattices and S their vertical 2-sum.By Lemma 5 S is modular. Since L and U do not contain a cover-preservingdiamond, the only possibility for S to contain one would be with o ∈ L and JUKKA KOHONEN i ∈ U , but the neck consisting of the two identified elements of L and U cannot contain three distinct elements a, b, c . Thus S is distributive.(C2) Let S be a finite distributive vi-lattice that is a vertical 2-sum of L and U . ByLemma 5 L and U are modular vi-lattices. Since S does not contain a cover-preserving diamond, the only possibility for L to contain one would be with i = 1 L , but this is impossible because L has only two coatoms. Thus L isdistributive. The case of U is similar. (cid:3) Computations
Method of classifying a lattice.
Given a graded vi-lattice represented byits covering graph, a short piece of program code classifies the lattice into the typesdescribed in Section 2. Calculating lattice length, counting atoms and coatoms,and finding possible necks is straightforward. For analyzing the symmetry type weuse the Nauty library, version 2.7r1 [14, 15].Nauty returns the automorphism group of a given directed graph as a list ofgenerators ( γ , . . . , γ k ). To classify a bottom piece we check if any generator swapsthe coatoms; in that case the coatoms are symmetric, otherwise fixed. With a toppiece we check if any generator swaps the atoms.To classify a middle piece some more cases are required. For each generator, wecheck if it:(A) swaps the atoms and fixes the coatoms; or(B) swaps both the atoms and the coatoms; or(C) swaps the coatoms and fixes the coatoms.Generators that touch neither atoms nor coatoms are ignored. Then:(1) If there are no generators of types A/B/C, the piece is MF.(2) If there are generators of type A, but none of B/C, the piece is MA.(3) If there are generators of type C, but none of A/B, the piece is MC.(4) If there are generators of type B, but none of A/C, the piece is MH.(5) If there are generators of at least two of the types A/B/C, the piece is MX.It is easily seen that this procedure produces the correct classification. Note thatfor an MX piece, Nauty does not necessarily return generators of types A and C.It can instead return, for example, a generator γ i of type A, and a generator γ j oftype B. But then γ i ◦ γ j is an automorphism that swaps the coatoms and fixes theatoms, and then we know that the piece is indeed MX.4.2. Modular lattices.
Modular vi-lattices were previously generated and countedup to 30 elements in [12], and up to 33 elements in unpublished work [16]. That wasdone with a program that starts from length-two seed lattices, and then adds newlevels of elements recursively. The program lists exactly one representative latticefrom each isomorphism class.We use here essentially the same lattice-generating program, modified so thatit skips all compositions, and generates only the pieces and the specials. Themodification is simply that two-element levels are not allowed between coatom andatom levels, because such a level would form a neck.With this program, all modular pieces and specials of n ≤
35 elements weregenerated (up to isomorphism), and classified as described in § n ≤
35 elements are then calculated using Theorem 1.
ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 9
The results of the exact counting are shown in Table 1. Rows MF–TF and “spe-cial” are from direct counting with the lattice-generating program. Rows CF–CNare calculated with the recurrence in Theorem 1. Row “vi-latt.” contains the num-bers of all modular vi-lattices: this is the sum of specials, pieces and compositions.Finally, row “all” has the numbers of all modular lattices (including vertical sumsof vi-lattices), calculated with the vertical sum recurrence (1).An exponential lower bound is derived as follows. Using Theorem 1 with theknown numbers of modular pieces of up to 35 elements, and plugging in zeros forlarger pieces (whose numbers we do not know), we obtain lower bounds on CF n ,CS n and CN n for n arbitrarily large. We observe that the growth ratios (from n to n + 1) of all three lower bounds settle a little above 2.3122 for n large enough.To obtain rigorous lower bounds, we choose a convenient starting point n = 50,convenient constant coefficients in front, and apply induction. Theorem 3.
The numbers of nonisomorphic modular compositions of types CF,CS and CN have the following lower bounds, when n ≥ : CF n ≥ . × . n CS n ≥ . × . n CN n ≥ . × . n Proof.
For 50 ≤ n ≤
85 the claim follows by direct calculation with Theorem 1,using the numbers of pieces from Table 1, and zeros when the number of pieces isnot known.For n >
85 the claim follows by induction on n . Let n >
85 be arbitrary,and assume that the claimed lower bounds hold on CF m , CS m and CN m when n − ≤ m ≤ n −
1. Then applying Theorem 1 gives the claimed lower bounds onCF n , CS n and CN n , which completes the induction. (cid:3) Corollary 1.
There are at least . × . n nonisomorphic modular vi-lattices of n elements when n ≥ .Proof. Add up the three lower bounds from Theorem 3. (cid:3)
Corollary 2.
There are at least . × . n nonisomorphic modular latticesof n elements when n ≥ .Proof. For 100 ≤ n ≤
400 the claim follows by direct calculation with recur-rence (1), using as input the lower bounds on vi-lattices of up to 400 elementscomputed using Theorem 3.For the induction step, let n >
400 be arbitrary, and assume that the claimedlower bound holds for the previous 300 values. Applying (1) then gives the claimedlower bound for the number of n -element modular lattices. This completes theinduction. (cid:3) Distributive lattices.
Distributive vi-lattices were previously counted up to49 elements by Ern´e et al. [4, 17]. To count n -element distributive lattices, theyactually generated posets that have n antichains; these are in a bijective correspon-dence with the distributive lattices.Our approach is more direct. We generate the distributive lattices directly, usingthe same program that we used for modular lattices, with some modifications. The first modification is a condition that ensures that we generate only the distributivelattices. Since the original program generates modular lattices, we only need tocheck that whenever a new element is created, it does not create a cover-preservingdiamond [6, p. 109]. This ensures that we generate the distributive lattices butno others.We also employ an important optimization that cuts short search branches thatcannot lead to distributive lattices. Our lattice-generating program (see [12] formore details) builds lattices levelwise, top down, until the number of elementsreaches a preset maximum. When creating a new level, it adds new elements indecreasing order of updegree. The last step on each level is thus to create meet-irreducible elements. In the original algorithm, this step can create a large numberof meet-irreducible elements, limited only by the maximum lattice size. But in adistributive lattice we can limit their number as follows. We recall (see Corollary 112in [6]) that the number of meet-irreducible elements in a distributive lattice equalsthe lattice length. As we build a lattice, we keep track of the meet-irreducibleelements created so far, and at each level we compute an updated upper bound R on the lattice length (based on the current length and the budget of remainingelements). The number of meet-irreducible elements, including the ones alreadycreated, is then limited to be at most R .With this program, all distributive pieces and specials of n ≤
60 elements weregenerated (up to isomorphism), and classified with the method described in § n , CS n andCN n whose growth ratios settle a little above 1.7250 for n large enough. To obtainrigorous lower bounds, we choose a convenient starting point n = 100, convenientconstant coefficients in front, and apply induction. Theorem 4.
The numbers of nonisomorphic distributive compositions of types CF,CS and CN have the following lower bounds, when n ≥ : CF n ≥ . × . n CS n ≥ . × . n CN n ≥ . × . n Proof.
For 100 ≤ n ≤
161 the claim follows by direct calculation with Theorem 1,using the numbers of pieces from Table 2, and zeros when the number of pieces isnot known.For n >
161 the claim follows by induction on n . Let n >
161 be arbitrary,and assume that the claimed lower bounds hold on CF m , CS m and CN m when n − ≤ m ≤ n −
1. Then applying Theorem 1 gives the claimed lower bounds onCF n , CS n and CN n , which completes the induction. (cid:3) Corollary 3.
There are at least . × . n nonisomorphic distributivevi-lattices of n elements when n ≥ .Proof. Add up the three lower bounds from Theorem 4. (cid:3)
ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 11
Corollary 4.
There are at least . × . n nonisomorphic distributive latticesof n elements when n ≥ .Proof. For 100 ≤ n ≤
400 the claim follows by direct calculation with recur-rence (1), using as input the lower bounds on vi-lattices of up to 400 elementscomputed using Theorem 4.For the induction step, let n >
400 be arbitrary, and assume that the claimedlower bound holds for the previous 300 values. Applying (1) then gives the claimedlower bound for the number of n -element distributive lattices. This completes theinduction. (cid:3) Semimodular lattices.
Although semimodular vi-lattices are 2-summable,vertical 2-sum is not very useful with them. For example, of the 1 753 185 150semimodular vi-lattices of 25 elements, only about 23% are compositions. This isin stark contrast with modular and distributive lattices. Basically this is becausesemimodular lattices are short and wide (cf. Figures 4–5 in [12]). For this reasonwe do not include tables of semimodular lattices here, but such tables can be easilycomputed using the accompanying program code.We could apply the same techniques as above to obtain an exponential lowerbound. But an asymptotically stronger lower bound is obtained by constructingsemimodular lattices from Steiner triple systems. A
Steiner triple system is a setof k elements ( points ) and a collection of their 3-sets ( triples ), such that each pairof distinct points occurs in exactly one triple. By counting the pairs it is easilyseen that the number of triples must be k ( k − /
6. It is known that Steiner triplesystems on k points exist if and only if k ≡ k arecalled admissible .Given a Steiner triple system on k ≥ k + k ( k − / Theorem 5.
For any n ≥ , the number of nonisomorphic semimodular rank-three vi-lattices containing n elements is at least (cid:0) . n / (cid:1) n . Proof.
Let n ≥
100 be given. Choose the largest admissible k such that k + k ( k − / ≤ n. Then k ≥
21, and because admissible values are at most 4 units apart, we have(2) n < ( k + 4) + ( k + 4)( k + 3) / < k / . Let N be the number of nonisomorphic Steiner triple systems on k points. By Wil-son’s Theorem 2 [18], we have N ≥ ( e − k ) k / , and using the bound k > √ n from (2) we obtain N ≥ (cid:0) e − · √ n (cid:1) n/ = (cid:0) e − / · / · n / (cid:1) n ≥ (cid:0) . n / (cid:1) n . From these N Steiner triple systems on k points, we can form N nonisomorphicsemimodular rank-three vi-lattices that have n (cid:48) = k + k ( k − / our choice of k , we have n (cid:48) ≤ n . To each lattice, add n − n (cid:48) extra coatoms covering anarbitrarily chosen atom of the highest updegree. This operation makes the latticeshave exactly n elements, and preserves semimodularity and nonisomorphism, so theclaim follows. (cid:3) The lower bound in Theorem 5 is very loose (it does not even exceed 1 until n ≈ n . The bound might be improved in several ways, for example, by usingKeevash’s recent improvement on Wilson’s lower bound [9].4.5. Notes on computation.
The main computational load was in generatingthe pieces and specials. For the largest sizes this was parallelized by running thelattice-generating program until a predefined number of elements had been added.The search state at those points was saved to a file, and the remaining work wasdivided among worker processes. For modular lattices of 33, 34 and 35 elements,this computation took 8 .
9, 23 . . . n grows byone. For distributive lattices of 58, 59 and 60 elements, the computation took 6 . . . . § Partial verification.
We describe here some of the methods that were usedto partially verify the correctness of the computational results.The pieces and specials were generated and classified twice, on different com-puters. The counts and the actual lattice listings were verified to be identical bycomparing their MD5 checksums. This would help against transient hardware andoperational errors, but not against systematic errors in the program the.The number of MA pieces equals the number of MC pieces in each column ofTables 1 and 2. This is as it should, because such pieces are duals of each other.The same holds between BF and TF pieces, and between BS and TS pieces.We also performed a more thorough duality check. The rank sequence of a gradedlattice is the sequence of its level sizes from bottom to top. The rank sequences ofa lattice and its dual are reverses of each other. In Figure 1 the BF example piecehas rank sequence (1 , , , , , , ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 13 sequence is (1 , , , , , , , , , , , , , , , , , including composi-tions. In each class, the count thus obtained matches the calculated count in Ta-ble 2. Generating these 50-element lattices took 12 . n = 1, because we count the singleton as a vi-lattice). The previouscountings did not employ the vertical 2-sum.5. Concluding remarks
One of our stated goals was to create more compact lattice listings by leaving outall compositions (vertical 2-sums of smaller lattices). As seen in Tables 1 and 2, thiswas more successful with distributive lattices than with modulars. Compositionsmake up 79% of the modular vi-lattices of 35 elements, and 99 .
6% of the distributivevi-lattices of 60 elements.The observed growths of the numbers of modular and distributive vi-lattices,to the extent that they are now known, are illustrated in Figure 3. In modularvi-lattices our lower bound Ω(2 . n ) seems loose; the observed ratios keep in-creasing, hinting perhaps of a (very slightly) superexponential growth. We notethat no exponential upper bound is currently known on the number of modularlattices. In distributive vi-lattices the observed ratios seem to be converging, andour lower bound Ω(1 . n ) seems pretty good.Ern´e et al. have shown an upper bound of O (2 . n ) on nonisomorphic distributivevi-lattices [4]. Our improved lower bound Ω(2 . n ) on nonisomorphic modularvi-lattices is still not enough to separate the growth rates of these two families.To close the gap there are different options. We could count modular pieces further.Empirically, adding one element increases the base in our lower bound by 0 . n grows). Counting the pieces up to 40 elementswould probably raise the lower bound above Ω(2 . n ). But this would take about10 000 cpu-core-days with the current lattice-generating program, and was deemednot worth the effort. Improving the algorithm or the lower bound techniques mightbe a better idea. Another option is to improve the upper bound on distributivevi-lattices. Indeed, Ern´e et al. note that “with more effort” it might be improvedconsiderably, at least to 2 . n . Combined with our lower bound, this would sufficeto separate the growth rates.Although our lower bounds are based on large computations, we must point outthat proper analysis of symmetry is the key to good lower bounds. Indeed, using allour data on distributive lattices ( n ≤ . n ) for distributivevi-lattices. In contrast, using just the distributive middle pieces of n ≤
21 elements(a truly modest collection of 134 lattices), our symmetry-distinguishing methodalready gives Ω(1 . n ).
20 25 30 352.252.32.352.4 modular vi-lattices with 2-sumswithout 2-sums
45 50 55 601.551.61.651.71.751.8 distributive vi-lattices
Figure 3.
Ratio between the numbers of nonisomorphic vi-lattices of ( n −
1) and n elements, with and without vertical 2-sums.Accompanying program code is available in Bitbucket [10]. This includes C pro-grams to generate the pieces and the specials, to classify and count them by symme-try type, and to perform the Cartesian counting. Also included is SageMath codefor verifying the exponential lower bounds. The lattice listings (pieces and specialsonly) were stored in digraph6 format and compressed with xz. The compressedlistings take about 167 GB of disk space, and will be available in [11]. Acknowledgments.
Computational resources were provided by CSC – IT Centerfor Science and by the Aalto Science-IT project.
References [1] Anders Bj¨orner. Shellable and Cohen-Macaulay partially ordered sets.
Transactions of theAmerican Mathematical Society , 260:159–183, 1980.[2] Karen L. Collins. Planar lattices are lexicographically shellable.
Order , 8:375–381, 1992.[3] Thomas A. Dowling and Richard M. Wilson. Whitney number inequalities for geometriclattices.
Proceedings of the American Mathematical Society , 47:504–512, 1975.[4] Marcel Ern´e, Jobst Heitzig, and J¨urgen Reinhold. On the number of distributive lattices.
TheElectronic Journal of Combinatorics , 9:Article
Journal of Algebra ,545:213–236, 2020.[6] George Gr¨atzer.
Lattice Theory: Foundation . Birkh¨auser, Basel, 2011.[7] Jobst Heitzig and J¨urgen Reinhold. Counting finite lattices.
Algebra universalis , 48:43–53,2002.[8] Peter Jipsen and Nathan Lawless. Generating all finite modular lattices of a given size.
Algebrauniversalis , 74:253–264, 2015.[9] Peter Keevash. Counting Steiner triple systems. In
European Congress of Mathematics , pages459–481, 2018.[10] Jukka Kohonen. Cartesian lattice counting. https://bitbucket.org/jkohonen/cartesian-lattice-counting/ .[11] Jukka Kohonen. Modular and distributive lattices without vertical sums and 2-sums. https://b2share.eudat.eu . In preparation.
ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 15
Table 1.
Number of modular lattice up to isomorphism. n type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16MF 0 0 0 0 0 1 0 0 2 3 3 13 24 48 105 242MA 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 7MC 0 0 0 0 0 0 0 0 0 0 0 1 1 2 5 7MX 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1MH 0 0 0 0 0 0 0 0 1 0 1 0 1 0 2 2BF 0 0 0 0 0 0 1 1 1 4 6 11 25 56 113 257BS 0 0 0 0 0 0 0 0 0 1 1 2 4 5 9 15TF 0 0 0 0 0 0 1 1 1 4 6 11 25 56 113 257TS 0 0 0 0 0 0 0 0 0 1 1 2 4 5 9 15CF 0 0 0 0 0 0 0 2 2 6 16 38 80 208 464 1115CS 0 0 0 0 0 0 0 0 0 0 0 0 0 3 5 15CN 0 0 0 0 0 0 0 0 2 4 10 28 66 154 375 884special 1 1 0 1 1 1 1 3 3 5 10 20 35 75 151 317pieces 0 0 0 0 0 1 2 2 5 13 18 41 85 174 361 803compos. 0 0 0 0 0 0 0 2 4 10 26 66 146 365 844 2014vi-latt. 1 1 0 1 1 2 3 7 12 28 54 127 266 614 1356 3134all 1 1 1 2 4 8 16 34 72 157 343 766 1718 3899 8898 20475 n type 17 18 19 20 21 22 23 24MF 518 1185 2664 6092 13849 31932 73458 170112MA 15 28 61 122 270 570 1259 2729MC 15 28 61 122 270 570 1259 2729MX 1 4 5 11 18 35 63 124MH 5 8 11 19 22 43 51 105BF 557 1250 2763 6267 14125 32225 73561 169304BS 30 52 109 207 422 835 1721 3544TF 557 1250 2763 6267 14125 32225 73561 169304TS 30 52 109 207 422 835 1721 3544CF 2580 6156 14382 34236 80703 192141 455548 1086269CS 35 84 191 457 1054 2482 5795 13601CN 2091 4959 11736 27832 66009 156845 372956 888193special 657 1426 3074 6783 15006 33707 75944 172893pieces 1728 3857 8546 19314 43523 99270 226654 521495compos. 4706 11199 26309 62525 147766 351468 834299 1988063vi-latt. 7091 16482 37929 88622 206295 484445 1136897 2682451all 47321 110024 256791 601991 1415768 3340847 7904700 18752943 Table 1 (continued). Number of modular lattice up to isomorphism. n type 25 26 27 28 29 30MF 394356 918597 2142885 5016593 11766661 27673169MA 6054 13395 29981 67308 152290 345897MC 6054 13395 29981 67308 152290 345897MX 239 474 945 1911 3917 8094MH 148 290 454 826 1359 2352BF 390258 904769 2102583 4905597 11472236 26908706BS 7475 15902 34379 75030 165752 369140TF 390258 904769 2102583 4905597 11472236 26908706TS 7475 15902 34379 75030 165752 369140CF 2586652 6179943 14763845 35347971 84670699 203133686CS 31931 75120 176999 417863 988002 2340245CN 2117276 5054559 12078748 28902161 69228582 166012187special 395073 908830 2098043 4866320 11320574 26427788pieces 1202317 2787493 6478170 15115200 35352493 82931101compos. 4735859 11309622 27019592 64667995 154887283 371486118vi-latt. 6333249 15005945 35595805 84649515 201560350 480845007all 44588803 106247120 253644319 606603025 1453029516 3485707007 n type 31 32 33 34 35MF 65203834 153963391 364151886 862779754 2047145114MA 790496 1813615 4180886 9673363 22467366MC 790496 1813615 4180886 9673363 22467366MX 16975 35876 76749 165615 360878MH 3958 6696 11466 19465 33807BF 63245392 148991342 351620380 831365583 1968780807BS 829576 1877307 4277558 9800078 22571155TF 63245392 148991342 351620380 831365583 1968780807TS 829576 1877307 4277558 9800078 22571155CF 487682310 1172237243 2819860668 6789965627 16361898245CS 5551716 13191092 31388574 74798062 178482514CN 398494238 957517799 2302844911 5543373958 13354884177special 61853133 145160950 341431589 804878006 1901058538pieces 194955695 459370491 1084397749 2564642882 6075178455compos. 891728264 2142946134 5154094153 12408137647 29895264936vi-latt. 1148537092 2747477575 6579923491 15777658535 37871501929all 8373273835 20139498217 48496079939 116905715114 282098869730 ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 17
Table 2.
Number of distributive lattices up to isomorphism. n type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17MF 0 0 0 0 0 1 0 0 0 0 0 3 0 0 4 5 4MA 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0MC 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0MX 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0MH 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0BF 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 3 1BS 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1TF 0 0 0 0 0 0 0 0 0 1 0 0 0 2 1 3 1TS 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1CF 0 0 0 0 0 0 0 2 0 4 2 10 6 32 18 83 74CS 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2CN 0 0 0 0 0 0 0 0 0 0 0 2 0 4 2 14 8special 1 1 0 1 0 0 0 1 0 0 0 1 0 1 0 3 1pieces 0 0 0 0 0 1 0 0 1 2 0 3 2 4 8 12 8compos. 0 0 0 0 0 0 0 2 0 4 2 12 6 37 20 97 84vi-latt. 1 1 0 1 0 1 0 3 1 6 2 16 8 42 28 112 93all 1 1 1 2 3 5 8 15 26 47 82 151 269 494 891 1639 2978 n type 18 19 20 21 22 23 24 25 26MF 16 10 29 49 63 94 213 219 459MA 0 1 1 2 1 3 3 10 8MC 0 1 1 2 1 3 3 10 8MX 0 0 0 0 0 0 1 0 1MH 1 0 0 0 1 1 0 3 2BF 8 7 16 16 40 38 102 116 229BS 0 1 1 2 1 5 2 8 4TF 8 7 16 16 40 38 102 116 229TS 0 1 1 2 1 5 2 8 4CF 230 233 672 726 1928 2342 5516 7280 16178CS 1 5 2 14 9 37 27 99 95CN 41 27 120 104 343 347 1005 1119 2953special 6 2 10 6 26 18 56 48 131pieces 33 28 65 89 148 187 428 490 944compos. 272 265 794 844 2280 2726 6548 8498 19226vi-latt. 311 295 869 939 2454 2931 7032 9036 20301all 5483 10006 18428 33749 62162 114083 210189 386292 711811 Table 2 (continued). Number of distributive lattices up to isomorphism. n type 27 28 29 30 31 32 33MF 726 1099 1691 3112 4176 7573 11728MA 15 14 50 47 87 121 227MC 15 14 50 47 87 121 227MX 0 2 0 4 3 5 4MH 1 2 2 8 2 10 6BF 303 596 749 1513 2033 3647 5316BS 16 11 32 25 59 62 151TF 303 596 749 1513 2033 3647 5316TS 16 11 32 25 59 62 151CF 22302 47348 68582 138752 208961 409676 632745CS 281 301 789 926 2307 2865 6611CN 3594 8607 11348 25363 35198 74935 108658special 129 328 339 769 914 1913 2371pieces 1395 2345 3355 6294 8539 15248 23126compos. 26177 56256 80719 165041 246466 487476 748014vi-latt. 27701 58929 84413 172104 255919 504637 773511all 1309475 2413144 4442221 8186962 15077454 27789108 51193086 n type 34 35 36 37 38 39MF 18593 29332 49894 73906 125464 196346MA 279 584 732 1333 1963 3362MC 279 584 732 1333 1963 3362MX 18 15 20 22 71 68MH 16 9 16 26 31 40BF 9431 13450 24024 35267 60195 91542BS 147 317 369 755 927 1833TF 9431 13450 24024 35267 60195 91542TS 147 317 369 755 927 1833CF 1211099 1914417 3583636 5772993 10632469 17361550CS 8863 19257 27094 56362 82534 165301CN 221306 333256 655975 1014990 1947706 3081099special 4783 6192 11888 16279 29902 42083pieces 38341 58058 100180 148664 251736 389928compos. 1441268 2266930 4266705 6844345 12662709 20607950vi-latt. 1484392 2331180 4378773 7009288 12944347 21039961all 94357143 173859936 320462062 590555664 1088548290 2006193418 ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 19
Table 2 (continued). Number of distributive lattices up to isomorphism. n type 40 41 42 43 44MF 316251 501232 824706 1277065 2104201MA 4763 8706 12369 21206 32381MC 4763 8706 12369 21206 32381MX 137 132 314 389 739MH 39 72 82 100 131BF 154501 234005 395667 606372 1005172BS 2286 4424 5957 10857 15117TF 154501 234005 395667 606372 1005172TS 2286 4424 5957 10857 15117CF 31576370 52152822 93836823 156430892 279134095CS 250864 486894 758311 1439247 2287530CN 5786183 9323041 17211623 28123776 51240833special 75946 109160 191940 283583 488243pieces 639527 995706 1653088 2554424 4210411compos. 37613417 61962757 111806757 185993915 332662458vi-latt. 38328890 63067623 113651785 188831922 337361112all 3697997558 6815841849 12563729268 23157428823 42686759863 n type 45 46 47 48MF 3324318 5359557 8530466 13845649MA 54061 81642 139008 210288MC 54061 81642 139008 210288MX 858 1710 2298 4073MH 157 263 252 415BF 1566178 2579549 4019531 6615167BS 27265 38591 68176 99857TF 1566178 2579549 4019531 6615167TS 27265 38591 68176 99857CF 468610361 830767951 1402696804 2473422299CS 4259645 6888356 12629975 20697992CN 84668721 152604200 254469592 454703002special 731917 1246418 1890209 3178981pieces 6620341 10761094 16986446 27700761compos. 557538727 990260507 1669796371 2948823293vi-latt. 564890985 1002268019 1688673026 2979703035all 78682454720 145038561665 267348052028 492815778109 Table 2 (continued). Number of distributive lattices up to isomorphism. n type 49 50 51 52MF 21848698 35484402 56423044 90846703MA 349946 545640 894103 1392365MC 349946 545640 894103 1392365MX 5652 10074 14075 24970MH 440 697 770 1034BF 10365640 16917992 26705669 43421020BS 171474 256172 436542 657562TF 10365640 16917992 26705669 43421020TS 171474 256172 436542 657562CF 4195545640 7366918781 12541052681 21947209314CS 37503756 62106875 111453001 186162978CN 763638695 1355285715 2289013709 4040233169special 4883596 8125938 12584095 20810796pieces 43628910 70934781 112510517 181814601compos. 4996688091 8784311371 14941519391 26173605461vi-latt. 5045200597 8863372090 15066614003 26376230858all 908414736485 1674530991462 3086717505436 5689930182502 n type 53 54 55 56MF 144993779 233835914 372140014 600341635MA 2298377 3582354 5856047 9243133MC 2298377 3582354 5856047 9243133MX 35929 61578 91665 155593MH 1401 1762 2365 2920BF 68622251 111467609 176619879 285832100BS 1108440 1696359 2822398 4362272TF 68622251 111467609 176619879 285832100TS 1108440 1696359 2822398 4362272CF 37469608053 65395110178 111905017483 194884875094CS 331472539 557422364 986467033 1667764379CN 6854924656 12046207362 20511977357 35920244327special 32424737 53285185 83549296 136565579pieces 289089245 467391898 742830692 1199375158compos. 44656005248 77998739904 133403461873 232472883800vi-latt. 44977519230 78519416987 134229841861 233808824537all 10488501786986 19334113091637 35639590512519 65696773057331 ARTESIAN LATTICE COUNTING BY THE VERTICAL 2-SUM 21
Table 2 (continued). Number of distributive lattices up to isomorphism. n type 57 58 59 60MF 958148836 1540236160 2462775718 3959945640MA 15005164 23705048 38546064 60946820MC 15005164 23705048 38546064 60946820MX 231975 392608 596474 990499MH 3980 5148 6470 8675BF 454454916 733959291 1168085737 1885053587BS 7218116 11214722 18454173 28883114TF 454454916 733959291 1168085737 1885053587TS 7218116 11214722 18454173 28883114CF 334097123844 580839511384 997199063829 1731270488614CS 2936919383 4986729668 8746804291 14902405273CN 61338845389 107114456244 183330850349 319426226966special 215048026 350313997 553415624 898644768pieces 1911741183 3078392038 4913550610 7910711856compos. 398372888616 692940697296 1189276718469 2065599120853vi-latt. 400499677825 696369403331 1194743684703 2074408477477all 121102696325898 223236665889804 411506035223499 758556959660012 [12] Jukka Kohonen. Generating modular lattices up to 30 elements. Order , 36:423–435, 2018.[13] Jukka Kohonen. Exponential lower bounds of lattice counts by vertical sum and 2-sum.
Algebra universalis , 80, 2019.[14] Brendan D. McKay and Adolfo Piperno. Nauty and Traces home page (version 2.7r1). http://pallini.di.uniroma1.it/ .[15] Brendan D. McKay and Adolfo Piperno. Practical graph isomorphism, II.
Journal of SymbolicComputation , 60:94–112, 2014.[16] OEIS, the on-line encyclopedia of integer sequences. https://oeis.org/A006981 . Number ofunlabeled modular lattices with n elements.[17] OEIS, the on-line encyclopedia of integer sequences. https://oeis.org/A006982 . Number ofunlabeled distributive lattices with n elements.[18] Richard M. Wilson. Nonisomorphic Steiner triple systems.
Mathematische Zeitschrift ,135:303–313, 1974.
Department of Mathematics and Systems Analysis, Aalto University, Finland
E-mail address ::