New upper bounds for the period of a negative orientable sequence

New upp er b ounds for the p erio d of a negativ e orien table sequence Chris J. Mitc hell and Peter R. Wild Information Securit y Group, Roy al Hollo wa y , Univ ersity of London me@c hrismitchell.net; p eterrwild@gmail.com 6th F ebruary 2026 Abstract Negativ e orien table sequences, i.e. perio dic sequences with elemen ts from a finite alphab et of size at least three in whic h an n -tuple or the negativ e of its reverse appears at most once in a p erio d of the sequence, w ere in tro duced b y Alhakim et al. in 2024. The main goal in defining them w as as a means of generating orien table sequences, whic h ha ve automatic position location applications, although they are p oten tially of in terest in their own righ t. In this pap er we dev elop new upp er b ounds on the p eriod of negativ e orientable sequences which, for n > 2, are significantly sharper than the previous known bound. The approac h used to develop the new bounds in v olves examining the nodes in the subgraph of the de Bruijn graph corresp onding to a negativ e orien table sequence, and to consider the implications of the fact that the in-degree of every vertex in this subgraph must equal the out- degree. How ever, despite impro ving the b ounds, a gap remains b etw een the largest known p eriod for a negative orientable sequence and the corresp onding b ounds for every n > 2. 1 In tro duction Orien table sequences are p eriodic sequences with elemen ts from an alphabet of size k with the property that, for some n (the or der ), an y n -tuple o ccurs at most once in a p eriod of the sequence in either direction. Clearly an orien table sequence of order n , an O S k ( n ), is also an O S k ( m ) for ev ery m ≥ n . Orien table sequences w ere introduced in the early 1990s [2, 3] in connection with position location applications. The binary case was studied b y Dai et al. [4], who presented an upper bound and a construction metho d yielding sequences with asymptotically optimal p eriod. F urther metho ds of construction for this case app eared muc h more 1 recen tly [5, 6, 8]. The general alphab et case (i.e. for k ≥ 2) w as first con- sidered by Alhakim et al. [1]. Subsequen tly , construction methods for this case ha v e b een describ ed, [7, 9, 10], and upp er bounds for the p erio d of a k -ary orientable sequence w ere giv en in [10] and impro ved in [9]. Although the construction metho d of Gabri´ c and Saw ada [7] has b een shown to yield sequences of asymptotically optimal p eriod, the problem of determining the precise v alue of the largest p erio d for an OS k ( n ) remains an op en question except for small v alues of n — for further details of the curren t state of the art see [9]. Alhakim at al. [1] introduced the notion of a ne gative orientable se quenc e , the main fo cus of this pap er, to help construct new orientable sequences. A negative orien table sequence of order n , N O S k ( m ), is a p erio dic k -ary sequence in whic h an n -tuple or the negativ e of its reverse app ears at most once in a p eriod of the sequence. Suc h sequences w ere subsequen tly studied further in [9, 10], where a range of metho ds of construction are described, and an upp er bound on the p erio d of a N OS k ( m ) is given in [9]. Much like the orientable sequence case, determining the precise v alue of the largest p eriod for an N O S k ( n ) remains unresolv ed, in this case even for small n . In this pap er w e use the same approach as emplo yed in [9] to dev elop new p eriod upp er bounds for orientable sequences to develop new and sharp er b ounds for the p erio d of a negativ e orien table sequence. The main tool is to examine nodes in the subgraph of the de Bruijn graph corresp onding to a negativ e orien table sequence, and to consider the implications of the fact that the in-degree of ev ery vertex in this subgraph must equal the out-degree. The remainder of the pap er is structured as follows. Section 2 gives fun- damen tal definitions, some elemen tary counting results for n -tuples, and a brief introduction to the de Bruijn graph. This is follow ed in Section 3 by the developmen t of the new perio d bounds. Finally , Section 4 concludes the pap er, summarising the state of kno wledge regarding the maximum p erio d for negative orien table sequences. 2 Preliminaries 2.1 Basic definitions W e need the following key definitions, largely follo wing the notational con- v entions of [10]. W e refer throughout to a k -ary n -tuple to mean a sequence of length n of symbols dra wn from Z k . Note that we assume that k > 2 throughout since the negativ e of a tuple equals the tuple when k = 2. Since we are in terested in tuples o ccurring either forwards or backw ards in a sequence, w e introduce the notion of a rev ersed tuple, so that if u = 2 ( u 0 , u 1 , . . . , u n − 1 ) is a k -ary n -tuple then u R = ( u n − 1 , u n − 2 , . . . , u 0 ) is its r everse . W e are also in terested in negating all the elements of a tuple, and hence if u = ( u 0 , u 1 , . . . , u n − 1 ) is a k -ary n -tuple, w e write − u for ( − u 0 , − u 1 , . . . , − u n − 1 ). W e write s n ( i ) for the tuple ( s i , s i +1 , . . . , s i + n − 1 ). Definition 2.1 ([1]) . A k -ary n -window se quenc e S = ( s i ) is a p eriodic sequence of elemen ts from Z k ( k > 1, n > 1) with the prop ert y that no n -tuple app ears more than once in a p eriod of the sequence, i.e. with the prop ert y that if s n ( i ) = s n ( j ) for some i, j , then i ≡ j (mo d m ) where m is the p erio d of the sequence. This pap er is concerned with a sp ecial class of n -windo w sequences: negativ e orien table sequences. Definition 2.2 ([1]) . A k -ary n -windo w sequence S = ( s i ) is said to b e a ne gative orientable se quenc e of or der n (a N OS k ( n )) if s n ( i )  = − s n ( j ) R , for an y i, j . Definition 2.3 ([10]) . Supp ose n ≥ 1 and k > 2. A k -ary n -tuple u = ( u 0 , u 1 , . . . , u n − 1 ) is said to b e ne gasymmetric if u i = − u n − 1 − i for every i , 0 ≤ i ≤ n − 1, i.e. if u = − u R . Clearly an N OS q ( n ) cannot con tain any negasymmetric n -tuples. Definition 2.4. Supp ose n ≥ 2 and k > 2. If a = ( a 0 , a 1 , . . . , a n − 1 ) is a k -ary n -tuple, then a is said to be uniform if and only if a i = a j for ev ery i, j ∈ { 0 , 1 , . . . , n − 1 } . Definition 2.5. Supp ose n ≥ 2 and k > 2. A k -ary n -tuple u = ( u 0 , u 1 , . . . , u n − 1 ) is said to b e alternating if and only if there exist c 0 and c 1 ( c 0  = c 1 ) such that u 2 i = c 0 and u 2 i +1 = c 1 for every i ≥ 0. R emark 2.1 . Since the ab ov e definition requires c 0  = c 1 , an alternating n -tuple is alw a ys non-uniform. Definition 2.6. Supp ose n ≥ 2 and k > 2. A k -ary n -tuple u = ( u 0 , u 1 , . . . , u n − 1 ) is said to b e uniform-alternating if and only if u i +1 = − u i for every i , 0 ≤ i ≤ n − 2. R emark 2.2 . A uniform-alternating n -tuple is uniform if u 0 = 0 or u 0 = k / 2 ( k even). Definition 2.7. Supp ose n ≥ 2 and k > 2. An n -tuple ( a 0 , a 1 , . . . , a n − 1 ) is said to b e left-semi-ne gasymmetric (or left-sns for short) if a i = − a n − i − 2 , 0 ≤ i ≤ n − 2. Equiv alently , ( a 0 , a 1 , . . . , a n − 1 ) is left-sns if and only if ( a 0 , a 1 , . . . , a n − 2 ) is negasymmetric. Analogously , an n -tuple ( a 0 , a 1 , . . . , a n − 1 ) is said to b e right-semi-ne gasymmetric (or right-sns for short) if a i = − a n − i , 1 ≤ i ≤ n − 1. Equiv alently , ( a 0 , a 1 , . . . , a n − 1 ) is right-sns if and only if ( a 1 , a 2 , . . . , a n − 1 ) is negasym- metric. 3 2.2 Coun ting sets of n -tuples The following simple results will b e of use b elow. Lemma 2.1. Supp ose n ≥ 1 and k > 2 . Then: i) the numb er of k -ary ne gasymmetric n -tuples is k ( n − 1) / 2 if n is o dd and k is o dd 2 k ( n − 1) / 2 if n is o dd and k is even; k n/ 2 if n is even; ii) the numb er of uniform k -ary n -tuples is k ; iii) the numb er of uniform-alternating k -ary n -tuples is k ; iv) the numb er of n -tuples that ar e b oth uniform and uniform-alternating is 1 if k is o dd and 2 if k is even; v) the numb er of uniform ne gasymmetric k -ary n -tuples is 1 if k is o dd and 2 if k is even; vi) the numb er of uniform-alternating ne gasymmetric k -ary n -tuples is 1 if n and k ar e o dd, 2 if n is o dd and k is even, and k if n is even; vii) the numb er of alternating ne gasymmetric k -ary n -tuples is zer o if n and k ar e b oth o dd, 2 if n is o dd and k is even, k − 1 if n is even and k is o dd, and k − 2 if n and k ar e b oth even (and in every c ase they ar e uniform-alternating); viii) the numb er of left-sns k -ary n -tuples is: k ( n +1) / 2 if n is o dd; k n/ 2 if n is even and k is o dd; 2 k n/ 2 if n and k ar e b oth even; ix) the numb er of non-uniform left-sns k -ary n -tuples is: k ( n +1) / 2 − 1 if n and k ar e b oth o dd; k ( n +1) / 2 − 2 if n is o dd and k is even; k n/ 2 − 1 if n is even and k is o dd; 2 k n/ 2 − 2 if n and k ar e b oth even; 4 x) the numb er of non-uniform-alternating left-sns k -ary n -tuples is: k ( n +1) / 2 − k if n is o dd; k n/ 2 − 1 if n is even and k is o dd; 2 k n/ 2 − 2 if n and k ar e b oth even. xi) the numb er of non-uniform non-alternating left-sns k -ary n -tuples is: k ( n +1) / 2 − k if n is o dd; k n/ 2 − 1 if n is even and k is o dd; 2 k n/ 2 − 4 if n and k ar e b oth even; Final ly observe that (viii), (ix), (x) and (xi) also hold if left-sns is r eplac e d with right-sns. Pr o of. (i) is Lemma 3.1 of [10]. (ii), (iii) and (iv) are immediate. (v) follows b y observing that a uniform negasymmetric n -tuple must hav e ev ery entry equal to zero or k / 2. F or (vi), if n is odd then every elemen t in the n -tuple m ust be either 0 or k / 2 and hence the n -tuple must b e uniform; the result follows from (v). If n is ev en then every uniform-alternating n -tuple is negasymmetric, and the result follows from (iii). F or (vii), if n is o dd then every elemen t in the n -tuple must b e either 0 or k / 2, and the result follo ws from (v) since an alternating n -tuple cannot b e uniform. If n is ev en then an alternating negasymmetric n -tuple m ust b e uniform-alternating, and the result follo ws from (vi) and (iv) since an alternating n -tuple cannot b e uniform. (viii) follo ws from (i) and observing that an n -tuple is left-sns if and only if its first n − 1 en tries form a negasymmetric ( n − 1)-tuple. A left-sns n -tuple can only be uniform if it is the all-zero or the all- k / 2 n -tuple, and (ix) follows from (v) and (viii). F or (x), If n is o dd, then a uniform-alternating n -tuple is left-sns and in this cas e the n umber is simply (viii) minus (iii). If n is ev en, a uniform- alternating n -tuple is only left-sns if it is also uniform, i.e. the num b er is (viii) minus (iv). Finally , for (xi), first observe that an n -tuple cannot b e b oth alternating and uniform b y definition, hence (xi) is simply (ix) less the num b er of alternating left-sns n -tuples. Also, the num b er of alternating left-sns n -tuples is the same as the num b er of alternating negasymmetric ( n − 1)-tuples, i.e., from Lemma 2.1(vii), zero if n is even and k is o dd, 2 if n and k are b oth ev en, k − 1 if n and k are b oth o dd, and k − 2 if n is o dd and k is even. 5 2.3 Negativ e orien table sequences and the de Bruijn digraph W e also need the follo wing definitions relating to the de Bruijn graph. Let B k ( n − 1) b e the de Bruijn digraph with vertices lab eled with k -ary ( n − 1)-tuples and edges lab eled with k -ary n -tuples. Definition 2.8. Supp ose k > 2 and n ≥ 2. Let B − k ( n − 1) b e the subgraph of the de Bruijn digraph B k ( n − 1) with all the edges corresp onding to negasymmetric n -tuples remo v ed. The following elemen tary result follo ws immediately from Lemma 2.1. Lemma 2.2. If k > 2 and n ≥ 2 the numb er N k ( n ) of e dges in B − k ( n − 1) is k n − k ( n − 1) / 2 if n is o dd and k is o dd k n − 2 k ( n − 1) / 2 if n is o dd and k is even k n − k n/ 2 if n is even. Definition 2.9. Supp ose k > 2 and n ≥ 2. Suppose S is an N O S k ( n ). Let B − ( S, n ) b e the subgraph of B k ( n − 1) with v ertices those of B k ( n − 1) and with edges corresp onding to those n -tuples which appear in either S or − S R . W e refer to B − ( S, n ) as the nega-sequence-subgraph. The following simple lemma is k ey . Lemma 2.3. Supp ose k > 2 and n ≥ 2 . Supp ose S is an N O S k ( n ) of p erio d m . Then: i) B − ( S, n ) c ontains 2 m e dges; ii) every vertex of B − ( S, n ) has in-de gr e e e qual to its out-de gr e e; and iii) B − ( S, n ) is a sub gr aph of B − k ( n − 1) . Pr o of. i) Since S has perio d m , a total of 2 m n -tuples app ear in S and − S R . They are all distinct since S is an N OS k ( n ). ii) S and − S R corresp ond to edge-disjoint Eulerian circuits in B − ( S, n ), and the result follo ws. iii) This is immediate since an N O S k ( n ) cannot con tain an y negasym- metric n -tuples. 6 3 The new b ounds 3.1 In-out-degree constrain ts on the nega-sequence-subgraph In this section and the next we consider prop erties of the nega-sequence- subgraph for a N OS k ( n ) S . W e first hav e the follo wing. Lemma 3.1. Supp ose k > 2 and n ≥ 3 . A vertex in B − k ( n − 1) has: i) in-de gr e e k − 1 if and only if its lab el is left-sns; otherwise it has in- de gr e e k ; ii) out-de gr e e k − 1 if and only if its lab el is right-sns; otherwise it has out-de gr e e k . Pr o of. F or (i), the in-degree of every vertex in B k ( n − 1) is k . Ho wev er, if (and only if ) an in b ound edge corresponds to a negasymmetric n -tuple, then this edge will not b e in B − k ( n − 1). Such an ev ent can o ccur if and only if the vertex is labelled with a left-sns ( n − 1)-tuple, and there can only b e one suc h negasymmetric in b ound edge. The result follows. The pro of of (ii) follo ws using an exactly analogous argumen t. By Lemma 2.3(ii), this immediately tells us that some edges in B − k ( n − 1) cannot o ccur in B − ( S, n ) if S is a N O S k ( n ). How ever, b efore describing exactly when this o ccurs, w e first need the follo wing simple result. Lemma 3.2. Supp ose k > 2 and n ≥ 4 . Supp ose the ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) is b oth left-sns and right-sns. Then i) if n is o dd and k is o dd then ( a 0 , a 1 , . . . , a n − 2 ) is the al l-zer o uniform ( n − 1) -tuple; ii) if n is o dd and k is even then ( a 0 , a 1 , . . . , a n − 2 ) is either uniform or alternating, wher e a i ∈ { 0 , k / 2 } for every i ; iii) if n is even then ( a 0 , a 1 , . . . , a n − 2 ) is uniform-alternating. F urther, al l the ( n − 1) -tuples in (i), (ii) and (iii) ar e b oth left-sns and right-sns. Pr o of. Supp ose the ( n − 1)-tuple ( a 0 , a 1 , . . . , a n − 2 ) is both left-sns and righ t- sns. Then a i = − a n − i − 3 , 0 ≤ i ≤ n − 3, and a i = − a n − i − 1 , 1 ≤ i ≤ n − 2. Since n ≥ 4 this implies that there exist constan ts c 0 and c 1 suc h that c 0 = a 2 i , 0 ≤ 2 i ≤ n − 2, and c 1 = a 2 j +1 , 0 ≤ 2 j + 1 ≤ n − 2. 7 If n is ev en then w e ha ve a ( n − 2) / 2 = − a ( n − 2) / 2 − 1 (from left-semi-negasymmetry), and hence c 0 = − c 1 , and (iii) follo ws. If n is o dd then, from left-semi-negasymmetry , a ( n − 3) / 2 = − a ( n − 3) / 2 and from right-semi-negasymmetry a ( n − 1) / 2 = − a ( n − 1) / 2 , and hence c 0 = − c 0 and c 1 = − c 1 . If k is o dd then we hav e c 0 = c 1 = 0, and (i) follo ws. if k is ev en then c 0 and c 1 are b oth either 0 or k / 2, and (ii) follo ws. The following result follows immediately from Lemmas 3.1 and 3.2. Corollary 3.3. Supp ose k > 2 and n ≥ 4 . Supp ose S is an N O S k ( n ) and c onsider a vertex in B − k ( n − 1) with lab el a = ( a 0 , a 1 , . . . , a n − 2 ) , wher e a is non-uniform. i) if n and k ar e b oth o dd, and a is left-sns, then its in-de gr e e is k − 1 and its out-de gr e e is k ; ii) if n and k ar e b oth o dd and a is right-sns, then its out-de gr e e is k − 1 and its in-de gr e e is k ; iii) if n is o dd and k is even and a is left-sns and non-alternating, then its in-de gr e e is k − 1 and its out-de gr e e is k ; iv) if n is o dd and k is even and a is right-sns and non-alternating, then its in-de gr e e is k and its out-de gr e e is k − 1 ; v) if n is even and a is left-sns and not uniform-alternating, then its in-de gr e e is k − 1 and its out-de gr e e is k ; vi) if n is even and a is right-sns and not uniform-alternating, then its out-de gr e e is k − 1 and its in-de gr e e is k . The ab o ve corollary immediately tells us that certain edges in B − k ( n − 1) cannot o ccur in B − ( S, n ), as follo ws. Corollary 3.4. Supp ose k > 2 and n ≥ 4 and S is an N O S k ( n ) . Then i) if n and k ar e b oth o dd, for every vertex in B − k ( n − 1) c orr esp onding to a non-uniform left-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( a 0 , a 1 , . . . , a n − 2 , x ) in B − k ( n − 1) , for some x , that is not in B − ( S, n ) . ii) if n and k ar e b oth o dd, for every vertex in B − k ( n − 1) c orr esp onding to a non-uniform right-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( y , a 0 , a 1 , . . . , a n − 2 ) in B − k ( n − 1) , for some y , that is not in B − ( S, n ) . 8 iii) if n is o dd and k is even, for every vertex in B − k ( n − 1) c orr esp onding to a non-uniform non-alternating left-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( a 0 , a 1 , . . . , a n − 2 , x ) in B − k ( n − 1) , for some x , that is not in B − ( S, n ) . iv) if n is o dd and k is even, for every vertex in B − k ( n − 1) c orr esp onding to a non-uniform non-alternating right-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( y , a 0 , a 1 , . . . , a n − 2 ) in B − k ( n − 1) , for some x , that is not in B − ( S, n ) . v) if n is even, for every vertex in B − k ( n − 1) c orr esp onding to a n on- uniform-alternating left-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( a 0 , a 1 , . . . , a n − 2 , x ) in B − k ( n − 1) , for some x , that is not in B − ( S, n ) . vi) if n is even, for every vertex in B − k ( n − 1) c orr esp onding to a n on- uniform-alternating right-sns ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , ther e is an e dge ( y , a 0 , a 1 , . . . , a n − 2 ) in B − k ( n − 1) , for some y , that is not in B − ( S, n ) . Pr o of. The result follows immediately from Lemma 2.3(ii) and Corollary 3.3. 3.2 Degree-parit y constrain ts on the nega-sequence-subgraph W e first giv e the follo wing simple lemma. Lemma 3.5. Supp ose k > 2 and n ≥ 2 . If an ( n − 1) -tuple is b oth left-sns and ne gasymmetric then it is uniform. Similarly, if an ( n − 1) -tuple is b oth right-sns and ne gasymmetric then it is uniform. Pr o of. Let a = ( a 0 , . . . , a n − 2 ) be an ( n − 1)-tuple whic h is both left-sns and negasymmetric. Then a i = − a n − 2 − i = − a n − 3 − i for i = 0 , 1 , . . . , n − 2. It is immediate that a j = a j +1 for j = 0 , 1 , . . . , n − 3 and th us a is uniform. The second claim follo ws by an analogous argument. The following lemma is key . Lemma 3.6. Supp ose k > 2 and n ≥ 2 . Supp ose S is an N O S k ( n ) and c onsider a vertex in B − ( S, n ) with lab el a = ( a 0 , a 1 , . . . , a n − 2 ) , wher e a is ne gasymmetric. Then a has even in-de gr e e and even out-de gr e e in B − ( S, n ) . Pr o of. Both S and − S R corresp ond to an Eulerian circuit in B − ( S, n ), and these circuits are edge-disjoint and cov er all the edges of B − ( S, n ). If a is negasymmetric then both circuits pass through this vertex equally many 9 times. It follo ws that a has ev en in-degree and ev en out-degree in B − ( S, n ). W e also need the follo wing. Lemma 3.7. Supp ose k > 2 and n ≥ 2 , and c onsider a vertex in B − k ( n − 1) with lab el a = ( a 0 , a 1 , . . . , a n − 2 ) . Then in al l the fol lowing c ases this vertex has o dd in-de gr e e and o dd out-de gr e e in B − k ( n − 1) : i) if k is o dd and a is ne gasymmetric but not uniform; ii) if n is o dd, k is even and a is either uniform or alternating, wher e a i ∈ { 0 , k / 2 } for every i ; iii) if n and k ar e b oth even and a is uniform-alternating. Pr o of. i) Since a is negasymmetric but not uniform, by Lemma 3.5 is cannot b e left-sns or righ t-sns. Th us, b y Lemma 3.1 it has in-degree and out-degree k , and since k is o dd the result follows. ii) Since a is either uniform or alternating, where a i ∈ { 0 , k / 2 } for ev ery i , b y Lemma 3.2(ii) it is both left-sns and right-sns. Th us, by Lemma 3.1 it has in-degree and out-degree k − 1, and since k is even the result follo ws. iii) Since a is uniform-alternating, by Lemma 3.2(iii) it is b oth left-sns and right-sns. Th us, b y Lemma 3.1 it has in-degree and out-degree k − 1. Since k is even the result follows. Com bining Lemmas 3.6 and 3.7 immediately giv es the follo wing imp ortan t result. Corollary 3.8. Supp ose k > 2 and n ≥ 2 and S is an N OS k ( n ) . Then in al l the fol lowing c ases ther e ar e e dges ( a 0 , a 1 , . . . , a n − 2 , x ) and ( y , a 0 , a 1 , . . . , a n − 2 ) in B − k ( n − 1) that ar e not in B − ( S, n ) . i) if k is o dd, for every vertex in B − k ( n − 1) c orr esp onding to a ne gasym- metric non-uniform ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) ; ii) if n is o dd and k is even, for every vertex in B − k ( n − 1) c orr esp onding to a uniform or alternating ne gasymmetric ( n − 1) -tuple ( a 0 , a 1 , . . . , a n − 2 ) , wher e a i ∈ { 0 , k / 2 } for every i ; iii) if n and k ar e b oth even, for every vertex in B − k ( n − 1) c orr esp onding to a uniform-alternating ne gasymmetric ( n − 1) -tuple ( a, − a, a, − a, . . . , a ) . 10 3.3 Constrain t interactions The results ab o ve indicate in what circumstances certain edges in B − k ( n − 1) cannot o ccur in B − ( S, n ) if S is an N O S k ( n ). In particular we hav e considered cases where one of the incoming edges to a v ertex corresp onding to a negasymmetric or righ t-sns ( n − 1)-tuple cannot o ccur in B − ( S, n ), and also where one of the outgoing edges from a vertex corresp onding to a negasymmetric or left-sns ( n − 1)-tuple cannot o ccur in B − ( S, n ). While we w ould like to add together the n umbers of eliminated edges for each case, w e need to ensure w e av oid ‘double coun ting’. More sp ecifically , w e need to consider when an edge can be outgoing from a negasymmetric or left-sns ( n − 1)-tuple and also incoming to a negasym- metric or righ t-sns ( n − 1)-tuple. This motiv ates the follo wing result. Lemma 3.9. Supp ose k > 2 . Supp ose a = ( a 0 , a 1 , . . . , a n − 2 ) and b = ( a 1 , a 2 , . . . , a n − 1 ) ar e k -ary ( n − 1) -tuples. i) If n ≥ 3 , and a and b ar e b oth ne gasymmetric then – if n is even then a and b ar e b oth uniform or alternating, wher e a i ∈ { 0 , k / 2 } for every i ; – if n is o dd then a and b ar e uniform-alternating. ii) If n ≥ 5 , a is ne gasymmetric and b is right-sns then ther e exist c j , 0 ≤ j ≤ 2 , such that a 3 i + j = c j , for every i and 0 ≤ j ≤ 2 . Mor e over if n ≡ 0 (mo d 3) then c 2 ∈ { 0 , k / 2 } and c 0 = − c 1 ; if n ≡ 1 (mo d 3) then c 1 ∈ { 0 , k / 2 } and c 0 = − c 2 ; and if n ≡ 2 (mo d 3) then c 0 ∈ { 0 , k / 2 } and c 1 = − c 2 . iii) If n ≥ 5 , a is left-sns and b is ne gasymmetric then ther e exist c j , 0 ≤ j ≤ 2 , such that a 3 i + j = c j , for every i and 0 ≤ j ≤ 2 . Mor e over if n ≡ 0 (mo d 3) then c 0 ∈ { 0 , k / 2 } and c 1 = − c 2 ; if n ≡ 1 (mo d 3) then c 2 ∈ { 0 , k / 2 } and c 0 = − c 1 ; and if n ≡ 2 (mo d 3) then c 1 ∈ { 0 , k / 2 } and c 0 = − c 2 . iv) If n ≥ 5 , a is left-sns and b is right-sns then ther e exist c j , 0 ≤ j ≤ 3 , such that a 4 i + j = c j , for every i and 0 ≤ j ≤ 3 . Mor e over if n ≡ 0 (mo d 4) then c 0 = − c 1 and c 2 = − c 3 ; if n ≡ 1 (mo d 4) t hen c 0 = − c 2 , c 1 ∈ { 0 , k / 2 } , and c 3 ∈ { 0 , k / 2 } ; if n ≡ 2 (mo d 4) then c 0 = − c 3 and c 1 = − c 2 ; and if n ≡ 3 (mo d 4) then c 1 = − c 3 , c 0 ∈ { 0 , k / 2 } , and c 2 ∈ { 0 , k / 2 } . Pr o of. i) By negasymmetry of a and b , resp ectiv ely , we ha ve a i = − a n − 2 − i and a i +1 = − a n − 1 − i for every i , 0 ≤ i ≤ n − 2. Hence there exist c 0 and c 1 suc h that a 2 i = c 0 and a 2 i +1 = c 1 for every i . 11 If n is ev en then from the first equation w e ha ve a n/ 2 − 1 = − a n/ 2 − 1 , i.e. a n/ 2 − 1 ∈ { 0 , k / 2 } , and from the second equation we ha ve a n/ 2 = − a n/ 2 , i.e. a n/ 2 ∈ { 0 , k / 2 } ; the result follows. If n is o dd then from the first equation we ha ve a ( n − 1) / 2 − 1 = − a ( n − 1) / 2 − 2 , i.e. c 0 = − c 1 , and the result follows. ii) By negasymmetry of a , a i = − a n − 2 − i for every i (0 ≤ i ≤ n − 2), and b y righ t-semi-negasymmetry of b , a j = − a n +1 − j for every j (2 ≤ j ≤ n − 1). Hence a i = a i +3 , 0 ≤ i ≤ n − 3. No w, since n ≥ 5, a 3 i + j = c j , for every i and 0 ≤ j ≤ 2, for some c j . If n ≡ 0 (mo d 3) then, b y negasymmetry , c 0 = a 0 = − a n − 2 = − c 1 , i.e. c 0 = − c 1 , and b y right-semi-negasymmetry we ha ve c 2 = a 2 = − a n − 1 = − c 2 , i.e. c 2 ∈ { 0 , k / 2 } . If n ≡ 1 (mo d 3) then, b y negasymmetry , c 0 = a 0 = − a n − 2 = − c 2 and so c 0 = − c 2 , and again by negasymmetry c 1 = a 1 = − a n − 3 = − c 1 , i.e. c 1 ∈ { 0 , k / 2 } . If n ≡ 2 (mo d 3) then, b y negasymmetry , c 1 = a 1 = − a n − 3 = − c 2 , i.e. c 1 = − c 2 , and again b y negasymmetry c 0 = a 0 = − a n − 2 = − c 0 , i.e. c 0 ∈ { 0 , k / 2 } . iii) By left-semi-negasymmetry of a , a i = − a n − 3 − i for ev ery i (0 ≤ i ≤ n − 3), and b y negasymmetry of b , a j = − a n − j for ev ery j (1 ≤ j ≤ n − 1). Using exactly the same argumen t as for (ii), it follo ws that a 3 i + j = c j , for every i and 0 ≤ j ≤ 2, for some c j . If n ≡ 0 (mo d 3) then, b y negasymmetry , c 1 = a 1 = − a n − 1 = − c 2 and so c 1 = − c 2 , and b y left-semi-negasymmetry c 0 = a 0 = − a n − 3 = − c 0 , i.e. c 0 ∈ { 0 , k / 2 } . If n ≡ 1 (mo d 3) then, b y negasymmetry , c 1 = a 1 = − a n − 1 = − c 0 and so c 1 = − c 0 , and again by negasymmetry c 2 = a 2 = − a n − 2 = − c 2 , i.e. c 2 ∈ { 0 , k / 2 } . If n ≡ 2 (mo d 3) then, by left-semi-negasymmetry , c 0 = a 0 = − a n − 3 = − c 2 and so c 0 = − c 2 , and again b y negasymmetry c 1 = a 1 = − a n − 1 = − c 1 , i.e. c 1 ∈ { 0 , k / 2 } . iv) By left-semi-negasymmetry of a , a i = − a n − 3 − i for ev ery i (0 ≤ i ≤ n − 3), and b y right-semi-negasymmetry of b , a j = − a n +1 − j for ev ery j (2 ≤ j ≤ n − 1). Hence a i = a i +4 , 0 ≤ i ≤ n − 3. Thus, since n ≥ 5, a 4 i + j = c j , for ev ery i and 0 ≤ j ≤ 3, for some c j . If n ≡ 0 (mo d 4) then, by left-semi-negasymmetry , c 0 = a 0 = − a n − 3 = − c 1 , and so c 0 = − c 1 , and by righ t-semi-negasymmetry , c 2 = a 2 = − a n − 1 = − c 3 , and so c 2 = − c 3 . If n ≡ 1 (mo d 4) then, by left-semi-negasymmetry , c 0 = a 0 = − a n − 3 = − c 2 , by left-semi-negasymmetry , c 1 = a 1 = − a n − 4 = − c 1 , i.e. c 1 ∈ { 0 , k / 2 } , and b y righ t-semi-negasymmetry , c 3 = a 3 = − a n − 2 = − c 3 , and so c 3 ∈ { 0 , k / 2 } . 12 If n ≡ 2 (mo d 4) then, by left-semi-negasymmetry , c 0 = a 0 = − a n − 3 = − c 3 , and so c 0 = − c 3 , and by righ t-semi-negasymmetry , c 2 = a 2 = − a n − 1 = − c 1 , and so c 2 = − c 1 . If n ≡ 3 (mo d 4) then, by left-semi-negasymmetry , c 1 = a 1 = − a n − 4 = − c 3 , by left-semi-negasymmetry , c 0 = a 0 = − a n − 3 = − c 0 , i.e. c 0 ∈ { 0 , k / 2 } , and b y righ t-semi-negasymmetry , c 2 = a 2 = − a n − 1 = − c 2 , and so c 2 ∈ { 0 , k / 2 } . 3.4 Dev eloping the b ound Before establishing our new p erio d b ound, we first outline our strategy , enabling the proof of the b ound to be described more simply . In Sections 3.1 and 3.2 we describ ed t wo wa ys of sho wing that certain edges in B − k ( n − 1) cannot occur in B − ( S, n ) when S is a N O S k ( n ). In partic- ular we show ed that in t wo cases incoming edges to certain categories of v ertex cannot o ccur, and also in t wo cases that outgoing edges from certain categories of v ertex cannot occur. This suggests a straightforw ard strategy for b ounding the p eriod of an N OS k ( n ), namely that the p eriod is at most half the maximum cardinality of B − ( S, n ) (by Lemma 2.3(i)). In turn | B − ( S, n ) | is b ounded ab o ve b y the n umber of edges in B − k ( n − 1), i.e. N k ( n ), as sp ecified in Lemma 2.2, less the n umber of edges in B − k ( n − 1) that cannot o ccur in B − ( S, n ), as sp ecified in Corollaries 3.4 and 3.8. This strategy is complicated b y the fact that, as noted in Section 3.3, there is a danger of ‘double coun ting’ certain excluded edges. F or example, Corol- lary 3.4 asserts that certain outgoing edges from a left-sns ( n − 1)-tuple cannot o ccur, and Corollary 3.8 asserts that certain edges incoming to a uniform or negasymmetric ( n − 1) tuple cannot o ccur. These t wo sets of excluded edges may ov erlap, and hence w e need to tak e this in to account; Lemma 3.9 is of key imp ortance in this resp ect. The following notation is in tended to simplify the arguments. Let U in and U out b e the sets of incoming and outgoing edges excluded b y Corollary 3.4, and P in and P out b e the sets of incoming and outgoing edges excluded b y Corollary 3.8. The ab o ve discussion leads to the following k ey lemma. 13 Lemma 3.10. If k > 2 and S is an N O S k ( n ) then: | B − ( S, n ) | ≤                      N k (2) − | P out | , if n = 2 N k (3) − | P out | − | P in | + | P out ∩ P in | , if n = 3 N k (4) − | U out | − | P out | , if n = 4 N k ( n ) − | U out | − | U in | − | P out | − | P in | + | U out ∩ P in | + | P out ∩ U in | + | U out ∩ U in | + | P out ∩ P in | , if n > 4 . wher e | X ∩ Y | denotes the maximum p ossible c ar dinality for such a set, and N k ( n ) denotes the numb er of non-ne gasymmetric k -ary n -tuples (se e L emma 2.2). Pr o of. The argumen t for n = 2 is immediate. A similar commen t applies when n = 3. The n = 4 and n ≥ 5 cases follow from observing that U in ∩ P in = U out ∩ P out = ∅ b ecause a negasymmetric ( n − 1)-tuple can neither b e non-uniform left-sns nor non-uniform right-sns, from Lemma 3.5. R emark 3.1 . The reason to restrict the sets considered when n ≤ 4 is b ecause certain sets are empty or migh t b e equal for small n . Moreov er Corollary 3.4 only applies for n ≥ 4. 3.5 The b ound W e can no w giv e the main result. Theorem 3.11. Supp ose k > 2 and S = ( s i ) is an N O S k ( n ) of p erio d m . i) If n = 2 then: m ≤ ( k 2 − k 2 , if k is o dd k 2 − k − 2 2 , if k is even ii) If n = 3 then: m ≤ ( k 3 − 2 k +1 2 , if k is o dd k 3 − 2 k − 6 2 , if k is even iii) If n = 4 then: m ≤ ( k 4 − 2 k 2 +1 2 , if k is o dd k 4 − 2 k 2 + k − 2 2 , if k is even 14 iv) If n > 4 then: m ≤            k n − 5 k ( n − 1) / 2 +4 k 2 , if n and k ar e o dd k n − 6 k ( n − 1) / 2 +3 k +2 2 , if n is o dd and k is even k n − 3 k n/ 2 − 2 k ( n − 2) / 2 + k 2 +3 k 2 , if n is even and k is o dd k n − 3 k n/ 2 + k 2 + k − 2 2 , if n and k ar e even. Pr o of. The pro of builds on, and uses the notation of, Lemma 3.10. i) n = 2. If k is o dd then P out = ∅ by Corollary 3.8(i), since there are no negasymmetric non-uniform 1-tuples. If k is even then, by Corollary 3.8(iii), | P out | = 2, since there are 2 uniform-alternating negasymmetric 1-tuples b y Lemma 2.1(vi). The result follows from Lemma 3.10. ii) n = 3. If k is o dd then, by Corollary 3.8(i), | P in | = | P out | equals the num b er of negasymmetric non-uniform 2-tuples, i.e. k − 1 by Lemma 2.1(v),(vii). In this case | P in ∩ P out | = k − 1 b y Lemma 3.9(i) and Lemma 2.1(ii),(iv). If k is even then, b y Corollary 3.8(ii), | P in | = | P out | equals the num b er of uniform or alternating negasymmetric 2-tuples where ev ery element is either 0 or k/ 2, i.e. 4. Next observe that when ev aluating | P in ∩ P out | w e need only consider alternating 2-tuples in whic h every element is either 0 or k / 2, since an edge can only b e outgoing from a uniform negasymmetric ( n − 1)-tuple and incoming to a uniform negasymmetric ( n − 1)-tuple if it corresponds to a uniform negasymmetric n -tuple, and suc h an edge cannot o ccur in B − k ( n − 1). Then | P in ∩ P out | is equal to the num b er of alternating negasymmetric n -tuples in whic h ev ery elemen t is either 0 or k / 2, i.e. 2. The result follo ws from Lemma 3.10. iii) n = 4. By Corollary 3.4(v), | U out | equals the num b er of non-uniform- alternating left-sns 3-tuples, i.e. k 2 − k b y Lemma 2.1(x). If k is odd then, by Corollary 3.8(i), | P out | equals the num b er of negasymmetric non-uniform 3-tuples, i.e. k − 1 b y Lemma 2.1(i),(v). If k is even then, b y Corollary 3.8(iii), | P out | equals the n umber of uniform-alternating negasymmetric 3-tuples, i.e. 2 b y Lemma 2.1(vi). The result follo ws from Lemma 3.10. iv)a) n > 4 ; n and k o dd . By Corollary 3.4(i),(ii), | U out | = | U in | equals the num b er of non-uniform left-sns ( n − 1)-tuples, i.e. k ( n − 1) / 2 − 1 by Lemma 2.1(ix). Also, b y Lemma 3.9(iv), | U out ∩ U in | = k − 1 since if n ≡ 1 (mo d 4) there are k choices for c 0 , and one c hoice eac h for c 1 and c 3 , giving 15 k p ossibilities, one of whic h is uniform (when c 0 = c 1 = c 3 = 0). An analogous argument applies if n ≡ 3 (mo d 4). By Corollary 3.8 (i), | P out | = | P in | equals the num b er of negasymmetric non-uniform ( n − 1)-tuples, i.e. k ( n − 1) / 2 − 1 b y Lemma 2.1(i),(v). By Lemma 3.9(ii),(iii), | U out ∩ P in | = | P out ∩ U in | = k − 1 (remo ving the case where the c i are all equal). By Lemma 3.9(i), | P out ∩ P in | equals the num b er of negasymmetric non-uniform ( n − 1)-tuples that are also uniform-alternating, i.e. k − 1 b y Lemma 2.1(iv),(vi). The result follo ws from Lemma 3.10. iv)b) n > 4 ; n o dd and k ev en . By Corollary 3.4(iii),(iv), | U out | = | U in | equals the num b er of non-uniform non-alternating left-sns ( n − 1)- tuples, i.e. 2 k ( n − 1) / 2 − 4 b y Lemma 2.1(xi). Also, b y Lemma 3.9(iv), | U out ∩ U in | = 4 k − 4 since if n ≡ 1 (mo d 4) there are k c hoices for c 0 , and tw o choices eac h for c 1 and c 3 , giving 4 k p ossibilities, t wo of which are uniform and tw o of which are alternating. An analogous argumen t applies if n ≡ 3 (mo d 4). By Corollary 3.8(ii), | P out | = | P in | equals the num b er of uniform or alternating negasymmetric ( n − 1)-tuples, i.e. k b y Lemma 2.1(v),(vii). By Corollary 3.8(ii), P out and P in only contain edges out-going/in- going from/to uniform or alternating ( n − 1)-tuples, and by Corol- lary 3.4(iii),(iv), U out and U in only contain edges that are out-going/in- going from/to non-uniform non-alternating ( n − 1)-tuples, and hence | U out ∩ P in | = | P out ∩ U in | = ∅ . By Corollary 3.8(ii), P out and P in only contain edges out-going/in- going from/to uniform or alternating negasymmetric ( n − 1)-tuples, whic h by Lemma 2.1(vii) are uniform-alternating. Hence an edge in P out ∩ P in m ust b e uniform-alternating. An edge in P out ∩ P in cannot b e negasymmetric, and there are k − 2 non-negasymmetric uniform- alternating n -tuples by Lemma 2.1(iii),(v). Hence | P out ∩ P in | = k − 2. The result follo ws from Lemma 3.10. iv)c) n > 4 ; n ev en . By Corollary 3.4(v),(vi), | U out | = | U in | equals the n umber of non-uniform-alternating left-sns ( n − 1)-tuples, i.e. k n/ 2 − k b y Lemma 2.1(x). Additionally , b y Lemma 3.9(iv), | U out ∩ U in | = k ( k − 1), by c ho osing c 0 and c 2 to b e distinct. If k is o dd then, by Corollary 3.8(i), | P out | = | P in | equals the num- b er of negasymmetric non-uniform ( n − 1)-tuples, i.e. k ( n − 2) / 2 − 1 by Lemma 2.1(i),(v). 16 By Lemma 3.9(ii),(iii), | U out ∩ P in | = | P out ∩ U in | = k − 1, by ensuring that c 0 , c 1 and c 2 are not all equal to 0. By Lemma 3.9(i), | P out ∩ P in | = 0, i.e. the num b er of non-uniform alternating ( n − 1)-tuples in whic h ev ery en try is either 0 or k / 2 when k is o dd. If k is even then, by Corollary 3.8(iii), | P out | = | P in | equals the n umber of uniform-alternating negasymmetric ( n − 1)-tuples, i.e. 2 by Lemma 2.1(vi). By Corollary 3.8(iii), P out and P in resp ectiv ely only contain edges out- going from or in-going to uniform-alternating ( n − 1)-tuples, and b y Corollary 3.4(v),(vi), U out and U in only contain edges that are out- going/in-going from/to non-uniform-alternating ( n − 1)-tuples, and hence | U out ∩ P in | = | P out ∩ U in | = ∅ . Finally , it is immediate that | P out ∩ P in | = 2, i.e. the num b er of non- uniform alternating ( n − 1)-tuples in which ev ery entry is either 0 or k / 2 when k is ev en. The result follo ws from Lemma 3.10. The bounds resulting from Theorem 3.11 are tabulated for small k and n in T able 1. Note that the n um b ers giv en in brac kets are the b ounds deriv ed from [10, Theorem 3.10], and are pro vided for comparison purp oses. T able 1: Bounds — new and (old) — on the p eriod of an N O S k ( n ) n k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 k = 9 2 3 5 10 14 21 27 36 (3) (5) (10) (14) (21) (27) (36) 3 11 25 58 99 165 245 356 (11) (27) (58) (101) (165) (247) (356) 4 32 113 288 614 1152 1987 3200 (35) (119) (298) (629) (1173) (2015) (3236) 5 105 471 1510 3790 8295 16205 29340 (113) (495) (1538) (3851) (8355) (16319) (29444) 6 324 1961 7620 23024 58296 130339 264600 (347) (2015) (7738) (23219) (58629) (130815) (265316) 7 1032 8007 38760 139330 410928 1047053 2389680 (1067) (8127) (38938) (139751) (411429) (1048063) (2390756) 8 3141 32393 194270 837884 2878491 8382499 21512844 (3227) (32639) (194938) (839159) (2881029) (8386559) (21519716) 9 9645 130311 975010 5034970 20170815 67096589 193693860 (9761) (130815) (975938) (5037551) (20174403) (67104767) (193703684) 17 4 Concluding remarks Although for n > 2 the new b ounds are sharp er than those previously kno wn, there remains a gap b et w een the largest kno wn perio d of a N O S k ( n ) and the bound, even for small v alues of n . The curren t state of knowledge for small n and k > 2 is summarised in T able 2, where the upp er bound from Theorem 3.11 is given in brack ets b eneath the largest kno wn p eriod of a N OS k ( n ). The sequences with the largest kno wn perio d are deriv ed from [10, Lemma 3.9]. The v alues in b old represen t maximal v alues. T able 2: Largest kno wn p erio ds for an N O S k ( n ) (and bounds) n k = 3 k = 4 k = 5 k = 6 k = 7 k = 8 2 3 5 10 14 21 27 (3) (5) (10) (14) (21) (27) 3 10 22 56 89 162 225 (11) (25) (58) (99) (165) (245) 4 31 93 278 550 1109 1835 (32) (113) (288) (614) (1152) (1987) 5 96 386 1432 3362 8008 14858 (105) (471) (1510) (3790) (8295) (16205) 6 294 1586 7162 20441 55518 119895 (324) (1961) (7620) (23024) (58296) (130339) 7 897 6476 36220 123895 393991 965569 (1032) (8007) (38760) (139330) (410928) (1047053) 8 2727 26333 181550 749422 2748581 7766075 (3141) (32393) (194270) (837884) (2878491) (8382499) While it w as already kno wn [10, Note 3.12] ho w to construct optimal p erio d negativ e orientable sequences for n = 2, it should b e clear that determining the optimal p erio d for a N O S k ( n ) remains an open question for all n > 2 and all k . Addressing this question remains an area for future research. 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