Coprime Ehrhart theory and counting free segments
CCOPRIME EHRHART THEORY AND COUNTING FREE SEGMENTS
SEBASTIAN MANECKE AND RAMAN SANYAL
Abstract.
A lattice polytope is free (or empty ) if its vertices are the only lattice points it contains. In thecontext of valuation theory, Klain (1999) proposed to study the functions α i ( P ; n ) that count the numberof free polytopes in nP with i vertices. For i = 1 , this is the famous Ehrhart polynomial. For i > , thecomputation is likely impossible and for i = 2 , computationally challenging.In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relativelyprime coordinates, and use it to compute α ( P ; n ) for unimodular simplices. We show that the coprimeEhrhart function can be explicitly determined from the Ehrhart polynomial and we give some applicationsto combinatorial counting. Introduction
In this paper, we are exclusively concerned with lattice polytopes , that is, convex polytopes P withvertices in Z d , for some d . A lattice polytope P is called free (or empty ) if P ∩ Z d are precisely thevertices of P . Free polytopes have been studied in relation to integer programming; see, for example, [20,19, 22, 14, 18] and they are related to hollow polytopes, whose interior do not contain lattice points. Ourinterest in free polytopes comes from valuation theory and geometric combinatorics. For a set S ⊂ R d ,denote by [ S ] : R d → { , } its indicator function and let α ( S ) := | S ∩ Z d | . Klain [16] basically provedthe following identity for lattice polytopes P :(1) ( − dim P [relint( P )] = − (cid:88) Q ( − α ( Q ) [ Q ] , where the sum is over all free polytopes Q ⊆ P . Applying the Euler characteristic to both sides of (1)then yields − (cid:88) Q ( − α ( Q ) = (cid:88) i ≥ ( − i α i ( P ) , where α i ( P ) to be the number of free polytopes Q ⊆ P with α ( Q ) = i . Klain [16, Sect. 10] proposed tostudy the functions α i ( P, n ) := α i ( n · P ) where n · P = { np : p ∈ P } is the n -th integer dilate of P . This is motivated by the fact α ( P, n ) = | n · P ∩ Z d | is the famous Ehrhart polynomial; see, for example, [4]. For i > , the function α i is not a valuation onpolytopes. Moreover, in dimensions ≥ , there are infinitely many free polytopes up to unimodularequivalence, which probably renders task of computing α i ( P ; n ) hopeless in general. However, any twofree segments are unimodularly equivalent and there is hope for the computation of α ( P ; n ) . Here, α ( P ) is the number of pairs in P ∩ Z d that are visible from each other. If P is a dilate of the unit cube, then thisis related to digital lines [9]. For the unit square the sequence α ([0 , ; n ) is given in [1] but an explicitdescription does not seem to be known.The main goal of our endeavor was to find explicit description of α ( P ; n ) where P is a unimodularsimplex; see Theorem 3 below. In our investigation, it turned out that we need the following number-theoretic variation of Ehrhart theory: The coprime Ehrhart function of a lattice polytope P ⊂ R d is Date : August 19, 2020.1991
Mathematics Subject Classification.
Key words and phrases.
Jordan totient function, coprime Ehrhart functions, free polytopes, empty polytopes. a r X i v : . [ m a t h . C O ] A ug SEBASTIAN MANECKE AND RAMAN SANYAL the function(2)
CE( P ; n ) := |{ ( a , . . . , a d ) ∈ nP ∩ Z d : gcd( a , . . . , a d , n ) = 1 }| . If P is a half-open free segment (that is, one endpoint removed), then CE( P ; n ) = φ ( n ) . Theorem 1 belowgives an explicitly computable description for general lattice polytopes.Recall that the Ehrhart function of P is E ( P ; n ) := | nP ∩ Z d | for n ∈ Z ≥ . Ehrhart [10] famously provedthat the Ehrhart function agrees with polynomial of degree r = dim P : there are numbers e i ( P ) ∈ R for i = 0 , . . . , r such that(3) E ( P ; n ) = e r ( P ) n r + e r − ( P ) n r − + · · · + e ( P ) n for all n ≥ . For k ≥ , the Jordan totient function is given by J k ( n ) := (cid:12)(cid:12)(cid:12)(cid:12)(cid:110) ( a , . . . , a k ) ∈ Z k : 1 ≤ a i ≤ n for i gcd( a , . . . , a k , n ) = 1 (cid:111)(cid:12)(cid:12)(cid:12)(cid:12) . For k = 0 , we have J ( n ) = 1 if n = 1 and = 0 otherwise. For k = 1 , J ( n ) = φ ( n ) is the Euler totientfunction. See [21] for more on properties of J k ( n ) . Theorem 1.
Let P be an r -dimensional lattice polytope. Then CE( P ; n ) = e r ( P ) J r ( n ) + e r − ( P ) J r − ( n ) + · · · + e ( P ) J ( n ) , for all n ≥ . The Jordan totient function can be computed as J k ( n ) = n k (cid:89) p | n (cid:18) − p k (cid:19) , where p ranges over all prime factors of n . This prompts us to define J k ( − n ) := ( − k J k ( n ) for all n ≥ .The following is the counterpart to Ehrhart–Macdonald reciprocity (see [4]), which states that the numberof lattice points in the relative interior of nP is given by ( − dim P E ( P ; − n ) . Theorem 2.
Let P ⊂ R d be an r -dimensional lattice polytope and n ≥ . Then CE(relint( P ); n ) = ( − r CE( P ; − n ) . Theorem 1 allows us to give an easily computable description of α ( P ; n ) , where P is a unimodularsimplex. Let us first note that α ( P ; n ) is invariant under unimodular transformations , that is, lineartransformations T ( x ) = Ax + b with A ∈ SL( Z d ) and b ∈ Z d . It is therefore sufficient restrict to the d -dimensional standard simplex ∆ d := conv( e , . . . , e d +1 ) = { x ∈ R d +1 : x ≥ , x + · · · + x d +1 = 1 } . We also define the polytope ∇ d := ∆ d + ( − ∆ d ) , the difference body [23, Sect. 10.1] of ∆ d . Theorem 3.
For d ≥ , we have α (∆ d ; n ) = 12 n (cid:88) (cid:96) =0 (cid:18) n − (cid:96) + dd (cid:19) CE( ∂ ∇ d ; (cid:96) ) . The Ehrhart polynomial of ∇ d is given in (9) in Section 3. Together with Ehrhart–Macdonald reciprocitythis gives E ( ∂ ∇ d ; n ) = E ( ∇ d ; n ) − ( − d E ( ∇ d ; − n ) and we can use Theorem 1 to compute α (∆ d ; n ) . For OPRIME EHRHART THEORY AND COUNTING FREE SEGMENTS 3 ≤ d ≤ this yields the following list:(4) CE( ∂ ∇ ; n ) = 6 J ( n )CE( ∂ ∇ ; n ) = 10 J ( n ) + 2 J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n ) + 2 J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n ) + J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n ) + J ( n ) + 2 J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n ) + J ( n ) + J ( n )CE( ∂ ∇ ; n ) = J ( n ) + J ( n ) + J ( n ) + J ( n ) + 2 J ( n ) In the plane, there are exactly four free polytopes up to unimodular equivalence. In particular, there areno free polygons with more than four vertices. Using the relations given in [16, Cor. 7.4], this allows us tocompletely determine all functions α i ( P ; n ) for unimodular triangles. Corollary 4.
Let P ⊂ R be a unimodular triangle. α ( P ; n ) = (cid:18) n + 12 (cid:19) , α ( P ; n ) = 6 n (cid:88) (cid:96) =0 (cid:18) n − (cid:96) + 22 (cid:19) J ( (cid:96) ) − n − nα ( P ; n ) = 3 n (cid:88) (cid:96) =0 (cid:18) n − (cid:96) + 22 (cid:19) J ( (cid:96) ) , α ( P ; n ) = n (cid:88) (cid:96) =0 (cid:18) n − (cid:96) + 22 (cid:19) J ( (cid:96) ) −
32 ( n + n ) and α i ( P ) = 0 for i > . The paper is organized as follows. In Section 2, we briefly develop coprime Ehrhart theory and proveTheorems 1 and 2. Section 3 is devoted to the study of α (∆ d ; n ) . We close with afterthoughts and openquestions in Section 4. Acknowledgements.
This project grew out of discussions in the seminar
Point configurations, valuations,and anti-matroids at the Goethe University Frankfurt in the (unusual) summer 2020. We thank theparticipants for creating a wonderful albeit virtual atmosphere. Our research was driven by experimentsconducted with
Sage [26] and
The On-Line Encyclopedia of Integer Sequences [24].2.
Coprime Ehrhart functions A valuation on lattice polytopes is a function ϕ satisfying ϕ ( ∅ ) = 0 and ϕ ( P ∪ Q ) = ϕ ( P ) + ϕ ( Q ) − ϕ ( P ∩ Q ) for all lattice polytopes P, Q such that P ∪ Q and P ∩ Q are also lattice polytopes. We call ϕ lattice-invariant if ϕ ( T ( P )) = ϕ ( P ) for all unimodular transformations T ( x ) . Lemma 5.
Let n ≥ be fixed. Then the map P (cid:55)→ CE( P ; n ) is a lattice-invariant valuation.Proof. It is straightforward to see that P (cid:55)→ CE( P ; n ) is a valuation. To see lattice invariance, let a ∈ n · T ( P ) = A ( nP ) + nb . That is a = Aa (cid:48) + nb for some lattice point a (cid:48) ∈ nP . It is now clear that gcd( a, n ) = gcd( Aa (cid:48) + nb, n ) = gcd( Aa (cid:48) , n ) = gcd( a (cid:48) , n ) . Hence CE( T ( P ); n ) = CE( P ; n ) . (cid:3) Proof of Theorem 1.
Betke–Kneser [6] showed that a lattice-invariant valuation is uniquely determined byits values on the unimodular simplices S k = conv(0 , e , . . . , e k ) for k = 0 , . . . , d . This implies that thevaluations { e i ( P ) : i = 0 , . . . , d } form a basis for the space of (real-valued) lattice-invariant valuations.Thus, for n ≥ fixed, there are c n,i ∈ R such that(5) CE( P ; n ) = c n,d e d ( P ) + c n,d − e d − ( P ) + · · · + c n, e ( P ) SEBASTIAN MANECKE AND RAMAN SANYAL
To determine the coefficients c n,i it suffices evaluate (5) at sufficiently many lattice polytopes or, in fact, half open polytopes; see the methods used in [15]. The k -dimensional half-open cube is H k := (0 , k . ItsEhrhart polynomial is readily given by E ( H k ; n ) = n k and hence e j ( H k ) = 1 if j = k and = 0 otherwise.To complete the proof, we simply note that CE( H k ; n ) = J k ( n ) . (cid:3) Proof of Theorem 2.
In order to prove Theorem 2, we recall the following implication of a classical resultdue to McMullen [17]. If ϕ is a lattice-invariant valuation and P an r -dimensional lattice polytope, thenthe function ϕ P ( k ) := ϕ ( kP ) agrees with a polynomial of degree at most r . Moreover ( − r ϕ P ( −
1) = ϕ (relint( − P )) .If we set ϕ ( P ) := CE( P ; n ) for n ≥ fixed, then we obtain ϕ P ( k ) = e r ( P ) J r ( n ) k r + e r − J r − ( n ) k r − + · · · + e ( P ) J ( n ) k and hence CE(relint( P ); n ) = CE(relint( − P ); n ) = ( − r ϕ P ( −
1) = ( − r CE( P ; − n ) (cid:3) To conclude this section, let us briefly remark that both results can also be proved with the use of number-theoretic Möbius inversion. For this, we note that E ( P ; n ) = (cid:88) d | n CE( P ; nd ) and hence CE( P ; n ) = (cid:88) d | n µ ( d ) E ( P ; nd ) . using linearity, we only need to consider (cid:88) d | n µ ( d ) (cid:16) nd (cid:17) k = n k (cid:88) d | n µ ( d ) 1 d k = n k (cid:89) p | n (cid:18) − p k (cid:19) = J k ( n ) . The Jordan totient functions J k ( n ) take the role of the monomial basis n k . In Ehrhart theory andcombinatorics, there are two more important bases. For d ≥ , let S d = { x ∈ R d : x ≥ , x + · · · + x d ≤ } ,O d = { x ∈ R d : 0 ≤ x ≤ x ≤ · · · ≤ x d ≤ } . Both polytopes are unimodular simplices with Ehrhart polynomial E ( S d ; n ) = E ( O d ; n ) = (cid:0) n + dd (cid:1) . Ehrhart–Macdonald reciprocity now states (cid:0) n − d (cid:1) = E (relint( O d ); n ) = ( − d E ( O d , − n ) = ( − k (cid:0) − n + dd (cid:1) . In partic-ular E ( S d ; n ) = d (cid:88) k =0 c ( n, k ) n ! n k , where c ( n, k ) are the unsigned Stirling numbers of the first kind [25, Prop. 1.3.7]. The correspondingcoprime Ehrhart function has a nice interpretation. Proposition 6.
For k ≥ , B k ( n ) := CE( S k ; n ) is the number of compositions µ = ( µ , . . . , µ l ) with µ i ≥ and µ + · · · + µ l = n of length l ≤ k + 1 with gcd( µ , . . . , µ l ) = 1 . The function ( − k B k ( − ( n + 1)) was studied by Gould [13] under the name R k ( n ) as the number ofcompositions of n with exactly k relatively prime parts. The functions B k ( n ) and R k ( n ) take the role ofthe binomial coefficients in Coprime Ehrhart theory.We can also consider the fraction of lattice points in nP that get counted by CE( P ; n ) as n goes to infinity. OPRIME EHRHART THEORY AND COUNTING FREE SEGMENTS 5
Corollary 7.
Let P ⊂ R d be an r -dimensional lattice polytope. Then lim sup n →∞ CE( P ; n ) E ( P ; n ) = 1 ζ ( r ) , where ζ is the Riemann zeta function.Proof. lim sup n →∞ CE( P ; n ) E ( P ; n ) = lim sup n →∞ J r ( n ) n r = lim sup n →∞ (cid:89) p | n (cid:18) − p r (cid:19) = 1 ζ ( r ) . (cid:3) In the case that P is the unit cube, this seems to be related to [12].3. Counting free segments in unimodular simplices
In this section, we will determine α ( P ; n ) , the number of free segments contained in nP , where P is aunimodular d -simplex. We start by some considerations that apply to general lattice polytopes.Let P ⊂ R d be a lattice polytope and S = [ a, b ] a free segment. For n ≥ , we write(6) E S ( P ; n ) = |{ t ∈ Z d : t + S ⊆ nP }| = | ( nP − a ) ∩ ( nP − b ) ∩ Z d | . Note that E S ( P ; n ) is invariant under translation of S and we may assume that a = 0 . We call a vector b ∈ Z d primitive if gcd( b ) = 1 and we write E b ( P ; n ) = E [0 ,b ] ( P ; n ) . This gives us the representation(7) α ( P ; n ) = 12 (cid:88) b primitive E [0 ,b ] ( P ; n ) . The factor stems from the fact that [0 , − b ] = [0 , b ] − b .The functions E S ( P ; n ) are vector partition functions [27] and related to multivariate Ehrhart functions.If P is a unimodular simplex, then the next result states that E S ( P ; n ) is in fact an Ehrhart polynomial.We now consider the standard unimodular simplex ∆ d ⊂ R d +1 and define P d := { b ∈ Z d +1 : gcd( b , . . . , b d +1 ) = 1 , b + · · · + b d +1 = 0 } . For b ∈ P d , there are unique b + , b − ∈ Z d ≥ such that b = b + − b − and b + i b − i = 0 for i = 1 , . . . , d . We furtherdefine (cid:96) ( b ) := (cid:88) i b + i = (cid:88) i b − i . Lemma 8.
Let b ∈ P d and n ≥ . Then n ∆ d ∩ ( n ∆ d − b ) = ∅ if and only if (cid:96) ( b ) > n . If (cid:96) ( b ) ≤ n , then n ∆ d ∩ ( n ∆ d − b ) = b − + ( n − (cid:96) ( b ))∆ d . In particular, E b ( n ) = (cid:0) n − (cid:96) ( b )+ dd (cid:1) for (cid:96) ( b ) ≤ n and E b ( n ) = 0 otherwise.Proof. The polytope n ∆ d ∩ ( n ∆ d − b ) is given by all points x ∈ R d +1 such that x + · · · + x d +1 = n and x i ≥ min(0 , − b i ) for all i = 1 , . . . , d + 1 . Summing the inequalities yields x + · · · + x d +1 ≥ (cid:96) ( b ) . Thus, there is no solution if and only if (cid:96) ( b ) > n .Otherwise, the solutions are given by points of the form x = b − + x (cid:48) with (cid:80) i x (cid:48) i = n − (cid:96) ( b ) and x (cid:48) ≥ .That is, n ∆ d ∩ ( n ∆ d − b ) = b − +( n − (cid:96) ( b ))∆ d . The second statement follows from the fact that E b (∆ d ; n ) = E (∆ d ; n − (cid:96) ( b )) = (cid:0) n − (cid:96) ( b )+ dd (cid:1) . (cid:3) SEBASTIAN MANECKE AND RAMAN SANYAL
Combining Lemma 8 with (7), yields α (∆ d ; n ) = 12 (cid:88) b ∈P d E b (∆ d ; n ) = 12 (cid:88) p ∈P d (cid:96) ( b ) ≤ n (cid:18) n − (cid:96) ( b ) + dd (cid:19) = 12 n (cid:88) (cid:96) =0 (cid:18) n − (cid:96) + dd (cid:19) c (cid:96),d , where c (cid:96),d := { b ∈ P d : (cid:96) ( S ) = (cid:96) } .Let ∇ d := ∆ d + ( − ∆ d ) . This is a convex polytope with vertices e i − e j for i (cid:54) = j . The combinatorial andarithmetic structure of ∇ d is easy to understand; see, for example, [8, Sect. 3] and below. Proposition 9.
Let d, (cid:96) ≥ . Then c (cid:96),d = CE( ∂ (∆ d − ∆ d ); (cid:96) ) . Proof.
Let b ∈ Z d +1 with (cid:80) i b i = 0 and recall that b = b + − b − where b + , b − ∈ Z d +1 ≥ with disjoint supports.It follows that b ∈ ∂ ( k ∇ d ) if and only if (cid:96) ( b ) = k . Adding the condition gcd( b, n ) = 1 now proves theclaim. (cid:3) For c ∈ R d +1 , let ∇ cd := { x ∈ ∇ d : (cid:104) c, x (cid:105) ≥ (cid:104) c, y (cid:105) for all y ∈ ∇ d } be the face in direction c . Let I + := { i ∈ [ d + 1] : c i = max( c ) } and I − := { i ∈ [ d + 1] : c i = min( c ) } . Then ∇ cd = conv( e i − e j : i ∈ I + , j ∈ I − ) . If c is not a multiple of (1 , . . . , , then I + ∩ I − = ∅ and ∇ cd is unimodularly isomorphic to ∆ | I + |− × ∆ | I − |− .The number of faces that are isomorphic to ∆ k − × ∆ l − is (cid:0) d +1 k,l (cid:1) . The boundary of ∇ d is the disjointunion of the relative interiors of proper faces. This gives us the following expression(8) c (cid:96),d = (cid:88) k,l ≥ k + l ≤ d +1 (cid:18) d + 1 k, l (cid:19) CE(relint(∆ k − × ∆ l − ); (cid:96) ) . Note that relint(∆ k − × ∆ l − ) = relint(∆ k − ) × relint(∆ l − ) . Hence E (relint(∆ k − × ∆ l − ); n ) = E (relint(∆ k − ); n ) · E (relint(∆ l − ); n ) = (cid:18) n − k − (cid:19)(cid:18) n − l − (cid:19) and using Theorem 1 yields explicit expressions for c (cid:96),d and subsequently for α (∆ d ; n ) .A different representation of c (cid:96),d is as follows. The polytope ∇ d is reflexive (cf. [3]). This implies that E (relint( ∇ d ); n ) = E ( ∇ d ; n − and hence E ( ∂ ∇ d ; n ) = E ( ∇ d ; n ) − E ( ∇ d ; n − . Using [8, Cor. 3.16],an explicit description of E ( ∇ d ; n ) is given by(9) E ( ∇ d ; n ) = d (cid:88) j =0 (cid:18) dj (cid:19)(cid:18) nj (cid:19)(cid:18) n + d − jd − j (cid:19) . Afterthoughts and questions
Geometric combinatorics and coprime chromatic functions.
There are a number of countingfunctions that can be expressed in terms of Ehrhart polynomials; see [4]. Perhaps most prominent isthe chromatic polynomial of a graph. Let G = ( V, E ) be a simple graph. An n -coloring is a map c : V → { , . . . , n } with c ( u ) (cid:54) = c ( v ) for all edges uv ∈ E . George Birkhoff [7] introduced the function χ G ( n ) counting the number of n -colorings of a graph. Birkhoff and Whitney [28] proved that χ G ( n ) agreeswith a polynomial in n of degree d = | V | . This is known as the chromatic polynomial of G . Beck andZaslavsky [5] realized χ G ( n ) as an Ehrhart polynomial of an inside-out polytope. An explicit formula isgiven by χ G ( n ) = d (cid:88) k =0 w k ( G ) n d − k , OPRIME EHRHART THEORY AND COUNTING FREE SEGMENTS 7 where w k ( G ) are the Whitney numbers of the first kind of G ; [2].Now, a coprime coloring is an n -coloring c with the additional constraint that the set c ( V ) ∪ { n } iscoprime. If we denote by χ cG ( n ) the number of coprime n -colorings, then Theorem 1 readily gives us χ cG ( n ) = d (cid:88) k =0 w k ( G ) J d − k ( n ) . Similarly, we may define coprime order functions on posets; see [4, Ch. 1].4.2.
Rational coprime Ehrhart theory and coprime Π -partitions. If P ⊂ R d is a polytope withvertices in Q d , then E ( P ; n ) agrees with a quasi-polynomial . That is, there are periodic functions c i ( n ) such that E ( P ; n ) = c d ( n ) n d + · · · + c ( n ) n for all n ≥ . Question 1.
Can the coprime Ehrhart function
CE( P ; n ) of a rational polytope be related to its Ehrhartfunction? Our construction of coprime Ehrhart functions is in line with the usual approach to Ehrhart theory.For a rational polytope P ⊂ R d , its homogenization is the pointed polyhedral cone C ( P ) = { ( x, t ) : t ≥ , x ∈ tP } = cone( P × { } ) . The set M ( P ) = C ( P ) ∩ Z d +1 is a finitely generated monoid and E ( P ; n ) = |{ ( a, t ) ∈ M ( P ) : t = n }| . If we denote by Z d +1prim := { a ∈ Z d +1 , gcd( a ) = 1 } , then CE( P ; n ) = |{ ( a, n ) ∈ Z d +1prim : ( a, n ) ∈ M ( P ) }| . A grading of M ( P ) is a linear function (cid:96) : Z d +1 → Z such that (cid:96) ( p ) > for all p ∈ M ( P ) \ . The associated Hilbert function H (cid:96) ( P ; n ) = |{ p ∈ M ( P ) : (cid:96) ( p ) = n }| isa quasipolynomial for all rational polytopes P [4, Sect. 4.7]. This rests on the rationality of the integerpoint transform (cid:80) p ∈ M ( P ) z p ∈ Z [[ z ± , . . . , z ± d +1 ]] . Question 2.
Is there a coprime version of the rational integer point transform?
A nice combinatorial consequence would be a coprime theory of Π -partitions. Let Π be a finite set partiallyordered by (cid:22) . A Π -partition of n ≥ is a map f : Π → Z ≥ such that (cid:80) a ∈ Π f ( a ) = n and f ( a ) ≤ f ( b ) whenever a (cid:22) b . This setup was introduced by Stanley (see [25]) as a generalization of usual partitionsand plane partitions. It can be shown that the function c Π ( n ) counting the Π -partitions of n is of theform H (cid:96) ( P ; n ) for some rational polytope P and linear function (cid:96) . A Π -partition is strict if f ( a ) > and f ( a ) < f ( b ) when a ≺ b . It would be desirable to obtain explicit formulas for counting coprime (strict) Π -partitions.Work in this direction was done by El Bachraoui [11]. A relatively prime partition of n ∈ Z ≥ is asequence of natural numbers λ > λ > · · · > λ k > such that n = λ + λ + · · · + λ k and the λ i arecoprime. The number of parts of the partition is k . We write rpp k ( n ) for the number of relatively primepartitions of n with exactly k parts. Note that rpp ( n ) is the number of coprime < a < b with n = a + b and hence rpp ( n ) = ϕ ( n ) . For the number of relatively prime partitions with parts El Bachraoui [11]showed that for n ≥ ( n ) = 112 J ( n ) . Mixed versions.
Upon closer inspection of the proof of Theorem 3, it can be seen that α (∆ d ; n ) = |{ ( t, b ) ∈ ( Z d +1 ) : b ∈ n ∇ d , t ∈ b + + ( n − (cid:96) ( b ))∆ d , gcd( b, n ) = 1 }| . This prompts the question of a mixed version of
CE( P ; n ) . For a lattice polytope P ⊂ R d and I ⊆ [ d ] ,define CE I ( P ; n ) := |{ p ∈ nP ∩ Z d : gcd( { p i : i ∈ I } ∪ { n } ) = 1 }| . It would be interesting if a reasonable expression for CE I ( P ; n ) could be found in general. SEBASTIAN MANECKE AND RAMAN SANYAL
Counting free triangles.
In every dimension ≥ , the unimodular triangle is the unique free poly-tope with vertices, up to unimodular equivalence. Question 3.
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