Bounds on the Number of Numerical Semigroups of a Given Genus

Bounds on the Num b er of Numerical Semigr oups of a Giv en Gen us Maria Bras-Amor´ os ∗ Octob er 31, 2018 Abstract Com bin atorics on m ultisets is used to deduce new upp er and low er b ounds on the num b er of numerical semig roups of each given gen u s, signif- ican t ly impro ving existing ones. In particular, it is prov ed that the num b er n g of numerical semi groups of genus g satisfies 2 F g 6 n g 6 1 + 3 · 2 g − 3 , where F g denotes the g th Fibonacci num b er. 1 In tro duction Let N 0 denote the s e t of all non-negative integers. A numeric al semigr oup is a subset Λ of N 0 containing 0, closed under summation and with finite complement in N 0 . The elements in the c o mplemen t N 0 \ Λ are ca lled the gap s of the numerical semigroup and | N 0 \ Λ | is its genus . The lar gest gap is the F r ob enius numb er o f Λ and it is at mo st t wo times the g e n us minu s one. If it equals this b ound then the n umer ical se mig roup is said to b e symmetric. Some results have b een proved related to the n umber o f numerical semi- groups of a given F robenius num b e r [1] and the n umber of symmetric semigr oups of a given F robenius num b er (and thus, the n um ber of symmetric semigr oups of a given ge n us) [4, 8]. In this work w e address the pro ble m of counting the nu m ber of numerical semigro ups of a given genus. W e denote by n g the num b er of numerical semigro ups of g en us g . It is ea s y to chec k that n 0 = 1 and n 1 = 1. The v alues up to n 16 where computed by Niv aldo Medeir os a nd Shizuo Kak utani, a nd the v alues up to n 50 can b e found in [2 ]. It is proved in [3] tha t any numerical semigr oup can b e repr e s en ted by a unique Dyck path of order g iv en by its genus and thus n g 6 C g where C g denotes the Cata lan num b er, C g = 1 g +1  2 g g  . It is conjectured in [2] that the sequence ( n g ) asy mptotically b ehav es like the Fib onacci sequence. More precisely , n g > n g − 1 + n g − 2 , for g > 2 ; lim g →∞ n g − 1 + n g − 2 n g = 1; lim g →∞ n g n g − 1 = φ , where φ is the golden ratio. See [5] for further r esults in this dir e ction. ∗ This work was partly supp orted by the Spanish M inistry of Education through pro j ec ts TSI2007-6540 6-C03-01 ”E-AEGIS” and CONSOLIDER CSD2007-00004 ”ARES”, and by the Go vernmen t of Catalonia under grant 2005 SGR 00446. 1 Let F i denote the i th Fibo nacci n umber starting by F 0 = 0, F 1 = 1. W e prov e tha t 2 F g 6 n g 6 1 + 3 · 2 g − 3 . 2 Some Results on Com binatorics Lemma 1. The multisets A g define d r e cu rsive ly by A 2 = { 1 , 3 } , A g = { g + 1 } ∪   [ m ∈ A g − 1 { 0 , 1 , . . . , m − 1 }   \ { g − 2 } for g > 2 (se e Figur e 1) satisfy, if g > 2 , A g =    { 2 F g − 2 z }| { 0 , 0 , . . . , 0 } ∪ { 2 F g − 3 z }| { 1 , 1 , . . . , 1 } ∪ { 2 F g − 4 z }| { 2 , 2 , . . . , 2 } ∪ . . . ∪ { 2 F 2 z }| { g − 4 , g − 4 } ∪ { 2 F 1 z }| { g − 3 , g − 3 }    ∪ { g − 1 , g + 1 } and | A g | = 2 F g . Pr o of. Bo th results ca n b e prov ed by induction and are a consequence from the fact that, for i > 2, F i = 1 + P i − 2 j =1 F j . This in turn can be proved by induction. Indeed, it is o b vious for i = 2. If i > 2, by the induction h yp othesis F i − 1 = 1 + P i − 3 j =1 F j and hence F i = F i − 1 + F i − 2 = 1 + P i − 2 j =1 F j . A 2 = { 1 , 3 } A 3 = { 0 , 0 , 2 , 4 } A 4 = { 0 , 0 , 1 , 1 , 3 , 5 } A 5 = { 0 , 0 , 0 , 0 , 1 , 1 , 2 , 2 , 4 , 6 } A 6 = { 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 2 , 2 , 3 , 3 , 5 , 7 } A 7 = { 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 2 , 3 , 3 , 4 , 4 , 6 , 8 } Figure 1: Firs t multisets A g as in Le mma 1. Lemma 2. The multisets B g define d r e cursively by B 2 = { 1 , 3 } , B g = { 0 , g + 1 } ∪   [ m ∈ B i − 1 { 1 , 2 , . . . , m }   \ { g , g − 2 } for g > 2 (se e Figur e 2) satisfy, if g > 2 , 2 B g = { 0 } ∪    { 3 · 2 g − 4 z }| { 1 , 1 , . . . , 1 } ∪ { 3 · 2 g − 5 z }| { 2 , 2 , . . . , 2 } ∪ . . . ∪ { 3 · 2 0 z }| { g − 3 , g − 3 , g − 3 }    ∪ { g − 2 , g − 1 , g + 1 } and | B g | = 1 + 3 · 2 g − 3 . Pr o of. Bo th results ca n b e prov ed by induction and are a consequence from the fact that, for i > 0, 2 i = 1 + P i − 1 j =0 2 j . This in turn ca n b e prov ed by induction. B 2 = { 1 , 3 } B 3 = { 0 , 1 , 2 , 4 } B 4 = { 0 , 1 , 1 , 1 , 2 , 3 , 5 } B 5 = { 0 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 3 , 4 , 6 } B 6 = { 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 2 , 2 , 2 , 3 , 3 , 3 , 4 , 5 , 7 } B 7 = { 0 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 2 , 3 , 3 , 3 , 3 , 3 , 3 , 4 , 4 , 4 , 5 , 6 , 8 } Figure 2: Firs t multisets B g as in Le mma 2. 3 T aking out Generators from a Semigroup Every num erical semigro up can b e gener ated by a finite set o f elements and a minimal set of genera tors is unique (see for instance [4]). Given a numeri- cal semigr oup Λ o f genus g and F rob enius num b er f , Λ ∪ { f } is a numerical semigroup and its genus is g − 1. So, any num erical s emigroup of gen us g can be obta ined from a numerical semigroup of genus g − 1 by removing one ele- men t larger than its F rob enius nu m ber . It is easy to chec k that when removing such an element from a numerical semigroup, the set o btained is a numerical semigroup if and only if the r emo ved element b elongs to the set of minimal generator s. This gives a recur siv e pr ocedure to o btain all numerical s emigroups of genus g from a ll numerical s e mig roups of ge nus g − 1 b y tak ing out, o ne b y one, each genera tor tha t is larger tha n the F rob enius num b er for each numerical semigroup. W e ca n think of a tree whose ro ot cor responds to the num erical semigr oup N 0 , each numerical semig r oup o f genus g is a no de a t dista nce g f rom the r oot, and the children of a numerical semigr oup a re the numerical semigr oups o b- tained when removing o ne by one ea ch of its minimal generator s which are larger than its F rob enius num be r. This constr uction was alr eady considere d in [6, 8, 7]. W e depicted this tree in Fig ure 3 . W e wrote < λ i 1 , λ i 2 , . . . , λ i n > to de - note the numerical semig roup g enerated by λ i 1 , λ i 2 , . . . , λ i n . W e used bo ldfa c e letters fo r the minimal generator s that are larger than the F ro benius n umber . 3 < 1 > < 2 , 3 > < 3 , 4 , 5 > < 2 , 5 > < 4 , 5 , 6 , 7 > < 3 , 5 , 7 > < 3 , 4 > < 2 , 7 > < 5 , 6 , 7 , 8 , 9 > . . . < 4 , 6 , 7 , 9 > . . . < 4 , 5 , 7 > . . . < 4 , 5 , 6 > . . . < 3 , 7 , 8 > . . . < 3 , 5 > . . . < 2 , 9 > . . . Figure 3: Recursive c onstruction of n umeric a l semigro ups o f genus g from n u- merical semigroups of genus g − 1. W e say that a numerical semigroup is or dinary if it is equal to { 0 } ∪ { i ∈ N 0 : i > c } f o r some non-negative integer c . Lemma 3. L et Λ b e a non-or dinary numeric al semigr oup. Su pp ose that { λ i 1 < λ i 2 < . . . < λ i k } ar e the minimal gener ators of Λ which ar e lar ger than the F r ob enius nu mb er. Then the nu mb er of minimal gener ators of t he nu meric al semigr oup Λ \ { λ i j } which ar e lar ger tha n its F r ob enius nu mb er i s • at le ast k − j , • at most k − j + 1 . Pr o of. It is o b vious that the num b er of minimal ge ner ators w hich are larger than the F rob enius n umber is at least k − j b ecause all elements in Λ \ { λ i j } which are minimal g enerators in Λ are also minimal gener a tors in Λ \ { λ i j } a nd the new F ro benius num b er is λ i j . The elemen ts in Λ \ { λ i j } whic h are not minimal gener ators in Λ and become minimal generator s in Λ \ { λ i j } must b e of the form λ i j + λ r for some λ r ∈ Λ. Let λ 1 be the smallest non- zero element of Λ. If λ r > λ 1 then λ i j + λ r − λ 1 > λ i j and hence λ i j + λ r = λ 1 + λ s for some λ s ∈ Λ \ { λ i j } , and λ i j + λ r is not a minimal generato r of Λ \ { λ i j } . So, the only element that is not a minimal generator of Λ a nd that may b e a minimal generato r of Λ \ { λ i j } is λ i j + λ 1 . Lemma 4 . The or dinary semigr oup Λ = { 0 , g + 1 , g + 2 , . . . } has minimal set of gener ators { g + 1 , g + 2 , . . . , 2 g + 1 } and 1. Λ \ { g + 1 } has g + 2 minimal gener ators lar ger than its F r ob enius numb er. 2. Λ \ { g + 2 } has g minimal gener ators lar ger than its F r ob enius nu mb er. 4 3. Λ \ { g + r } , with r > 2 , has g − r + 1 minimal gener ators lar ger than its F r ob enius numb er. Pr o of. The firs t item is obvious. As proved in Lemma 3 the o nly element that is not a minima l generator of Λ and that may b e a minimal generato r of Λ \ { λ i j } is λ i j + λ 1 . It is easy to prov e that if r = 2 then λ i j + λ 1 is a minimal g e ne r ator while if r > 2 , it is not. Theorem 5. The num b er n g of numeric al semigr ou ps of genu s g satisfies 2 F g 6 n g for al l g > 2 and 2 F g 6 n g 6 1 + 3 · 2 g − 3 for al l g > 3 . Pr o of. Set A 0 = B 0 = { 1 } , A 1 = B 1 = { 2 } and consider A g and B g defined as befo re for g > 2. Consider tw o trees A a nd B that r espectively hav e A g and B g as the nodes a t dista nc e g from its roo t, with the elemen t a ∈ A g having a children and the element b ∈ B g having b children. It is then easy to check that, by Lemma 3 and Lemma 4, the tr ee in Figure 3 co ntains A as a subtree and is contained in B . Th us, | A g | 6 n g 6 | B g | . Now b y Lemma 1 and Lemma 2 it follows the res ult. In T able 1 one can co mpare for g up to 3 0 the actua l v alues of n g with the b ounds given in Theorem 5 and also with the b ound given by the Ca ta lan nu m ber s prov e d in [3]. The v alues n g are from [2 ]. References [1] J¨ orgen Backelin. On the n umber of s emigroups of na tur al n umber s. Math. Sc and. , 6 6(2):197–21 5 , 1990. [2] M. Bra s-Amor´ os. Fib onacci-like b eha vio r o f the num b e r of numerical semi- groups o f a given genus. Semigr oup F orum . (ar Xiv:math/061263 4 ) [3] Mar ia Bras-Amor ´ os and Anna de Mier. Repres en tation o f numerical semi- groups by Dyck paths. Semigr oup F orum , 75(3):67 7 –682, 20 07. [4] R. F r¨ ob e r g, C. Gottlieb, and R. H¨ agg kvist. On n umerica l semigro ups. S emi- gr oup F orum , 35 (1):63–83, 1987 . [5] Jo rge L . Ra m ´ ırez Alfons ´ ın. Some rema rks on g aps. [6] J. C. Ros ales. F amilies of n umerical semig roups closed under finite intersec- tions a nd for the frob enius num b er. Houst on J ournal of Mathematics . [7] J. C. Rosales, P . A. Garc ´ ıa-S´ anc hez, J. I. Garc ´ ıa-Garc ´ ıa, and J. A. Jim´ enez Madr id. The oversemigroups of a numerical semigr oup. Semigr oup F orum , 67(1):14 5–158, 20 03. [8] J. C. Rosales, P . A. Garc ´ ıa-S´ anc hez, J. I. Garc ´ ıa-Garc ´ ıa, and J. A. Jim´ enez Madrid. F undamen tal ga ps in numerical semigroups. J. Pur e Appl. Alge br a , 1 89(1-3):301– 313, 2 004. 5 g 2 F g n g 1 + 3 · 2 g − 3 C g 0 1 1 1 1 1 2 2 2 2 3 4 4 4 5 4 6 7 7 14 5 10 12 13 42 6 16 23 25 132 7 26 39 49 429 8 42 67 97 1430 9 68 118 193 4862 10 110 204 385 16796 11 178 343 769 58786 12 288 592 1537 20801 2 13 466 1001 3073 74290 0 14 754 1693 6145 2674 440 15 1220 2857 12289 96948 45 16 1974 4806 24577 353576 70 17 3194 8045 49153 12964 4790 18 5168 13 467 98305 47763 8700 19 8362 22 464 19660 9 17672 63190 20 13530 3739 6 393217 656412 0420 21 21892 6219 4 786433 24466 267020 22 35422 103246 15728 65 91482 563640 23 57314 170963 31457 29 34305961 3650 24 92736 282828 62914 57 12899041 47324 25 15005 0 46722 4 12582 913 48619 46401452 26 24278 6 77083 2 25165 825 18367 35307215 2 27 39283 6 12702 67 50331 649 69533 55091600 4 28 63562 2 20910 30 10066 3297 263 74795175 0360 29 102 8 458 3437839 201326 5 93 1002242 216651368 30 166 4 080 5646773 402653 1 85 3814986 502092304 T able 1: V alues of 2 F g , n g , 1 + 3 · 2 g − 3 , and C g for g up to 30 . 6