CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS

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1 Submitted to the Bulleti of the Australia Mathematical Society doi: /s... CLOSED FORM FORMULA FOR THE NUMBER OF RESTRICTED COMPOSITIONS GAŠPER JAKLIČ, VITO VITRIH ad EMIL ŽAGAR Abstract I this paper, compositios of a atural umber are studied. The umber of restricted compositios is give i a closed form, ad some applicatios are preseted Mathematics subject classificatio. primary 05A17; secodary 68R05. Keywords ad phrases: compositio, partitio, restricted, maximal elemet, miimal elemet. 1. Itroductio Compositios ad partitios of a atural umber frequetly appear i research ad i practical applicatios. Although the umber of compositios or partitios, satisfyig particular requiremets, ca be obtaied from their geeratig fuctios, this is a serious drawback, sice it requires symbolic computatioal facilities, exact computatios, ad because of computatioal complexity ivolved. I this paper, we preset a closed form formula for the umber of restricted compositios, ad give some applicatios of the results. Let us be more precise. The list of atural umbers t i, which sum up to a atural umber, is a iteger compositio of. The set of all such lists, where the orderig of the summads matters, is the set of all iteger compositios of. The set of restricted iteger compositios of is the subset of all compositios that satisfy some additioal restrictios, e.g., o the umber of summads, o the values of summads,... Let a, b, N with a b. Let C(, a, b deote the umber of compositios of, such that summads t i are atural umbers, bouded as a t i b, for all i. Furthermore, let C(, k, a, b deote the umber of those restricted compositios of, where the umber of summads is equal to k, k t i =, a t j b, j=1, 2,...,k. Clearly, i=1 C(, a, b= a k= b C(, k, a, b.

2 2 G. Jaklič, V. Vitrih ad E. Žagar It is trivial to prove that C( := C(, 1, =2 1 ad C(, k := C(, k, 1, = ( 1 k 1. There are also kow formulas for the special cases ( ka+k 1 a C(, k, a, =, C(, a, = C(, k, a,. k 1 Also a obvious recursive relatio for the geeral case C(, k, a, b= a i= b k=1 C(i, k 1, a, b is right at had. Nevertheless, the geeratig fuctios are kow for both C(, k, a, b ad C(, a, b. They are of the form ([2, 6] k (z a b a+1 1 ad, (1 j=1 a b a+1 respectively. We are also iterested i a closed-form formula for the umber of compositios of with more tha oe maximal (or miimal elemet. We will deote them by Max( ad Mi(, respectively. Agai, there are the geeratig fuctios kow for C( Max( ad C( Mi( ad are of the form ( ( 2 z j 2 ( ad ( 2 z j 2 1 2z+z j+1 z j, respectively ([6]. It is quite easy to obtai closed form formulas at least for C(, 1, b ad C(, a, b, a > 1. Namely, by (1 1 Sice 1 b the coefficiet at z becomes where b = =0 C(, 1, b z. j=1 = 1 2 z+z b+1= ( (2 z z b+1 i = ( i=0 i=0 i ( i ( 1 j 2 i j z i j z j (b+1, j j=0 C(, 1, b=g(, b g( 1, b, g(, b := i, j i+ j b= ( i ( 1 j 2 i j. j

3 The umber of restricted compositios 3 Similarly C(, a, b=g(, a, b g( 1, a, b, a>1, where (( i j g(, a, b= ( 1 l. j l i, j,l i+ j (a 1+l (b a+1= But it seems that derivig a explicit formula for C(, k, a, b is a far more difficult problem. The paper is orgaized as follows. I Sectio 2 closed form formulae for the umber of restricted compositios ad restricted partitios are obtaied. They are used as a basis for studyig two related problems i Sectio 3. The paper is cocluded by some examples i Sectio Restricted compositios I this sectio, our aim is to fid a combiatorial closed-form expressio for C(, k, a, b. THEOREM 2.1. Let a b ad b k a. To each compositio of assig a vector i = (i 2, i 3,...,i b, where i j deotes the frequecy of the umber j i the compositio. Moreover, let α j := k ( j 1 j=2, 3,...,b. The (a (b (c C(, k, 1, b= b (l j+1 i l,β j := k l= j+1 i 2 =α 2, i 3,...,i b max{0,α j } i j mi{β j,γ j } b i l,γ j := l= j+1 b ( k l 1 l=2 j=2 i j i l. k b l= j+1 (l 1 i l j 1 C(, k, a, b=c( k(a 1, k, 1, b (a 1. If kb ka+(b a Nad N 0, the C(, k, a, b= ( ka+k 1 k 1 k 1 k 1. PROOF. At first, ote that the frequecy of umber 1 is b l=2 l i l ad so the umber of summads i the compositio is Furthermore, there are exactly k(i := b (l 1i l. (2 l=2 b ( k(i l 1 l=2 i l j=2 i j,

4 4 G. Jaklič, V. Vitrih ad E. Žagar differet compositios with the same vector i. Sice the umber of summads has to be k, the oly admissible compositios are those with k(i = k. Therefore, the relatios b b b k i l ( j 1 l i l k i l 0, j=2, 3,...,b, l= j l= j have to be satisfied. With the help of (2, we obtai the appropriate rages for umbers i j, max{0,α j } i j mi{β j,γ j }, 3 j b, i 2 :=α 2 =γ 2. The first formula is therefore prove. I order to show that a additioal coditio, which requires the summads i the compositio to be at least a, does ot icrease the difficulty of the problem, let us defie a fuctio k f : (t 1,...,t k, t i =, a t i b i=1 k (s 1,..., s k, s i = k(a 1, 1 s i b (a 1, i=1 f : (t 1, t 2,...,t k (t 1 (a 1, t 2 (a 1,...,t k (a 1, which is clearly a bijectio ad thus C(, k, a, b=c( k(a 1, k, 1, b (a 1. To prove the last statemet of the theorem, assume C(, k, a, b=c(m, k, a 2, m. The m=+k(m b ad a 2 = a+m b. Hece m= kb k 1, a 2= ka+(b a. k 1 If m Nad a 2 N 0, the C(m, k, a 2, m is well defied ad it follows ( ka+k 1 C(, k, a, b=c(m, k, a 2, m=c(m ka 2, k, 0, m ka 2 =. k 1 This result ca be used to derive some iterestig properties of restricted compositios. COROLLARY 2.2. The followig formulae hold true: l= j ( k C(, k, 1, 2=, C(, 1, 2= k ( k C(, k, a, a+1=, C(, a, a+1= ka k= 2 a k= a+1 ( k 2 ( k =, k k ( k ka. k=0

5 The umber of restricted compositios 5 Suppose ow that oe is iterested i restricted partitios. The list of atural umbers, which sum up to ad where the orderig of summads is ot importat, is the set of iteger partitios of. The partitios, where the umber of summads is equal to k ad where they are bouded betwee a ad b, will be deoted by P(, k, a, b. The followig corollary follows directly from Theorem 2.1. COROLLARY 2.3. Let 1 a b ad b k a. To each partitio of assig a vector i=(i 2, i 3,...,i b, where i j deotes the frequecy of the umber j i the partitio. Moreover, letα j,β j adγ j, j=2, 3,...,b, be as i Theorem 2.1. The (a P(, k, 1, b= 1, (b i 2 =α 2, i 3,...,i b max{0,α j } i j mi{β j,γ j } P(, k, a, b=p( k(a 1, k, 1, b (a Two related problems It is iterestig to cosider the problem of coutig the compositios, where more tha oe maximal (or miimal summad exists. A applicatio will be give i the last sectio. Usig Theorem 2.1, oe ca prove the followig theorem. THEOREM 3.1. Let Max( deote the umber of all compositios of, such that there are at least two maximal summads, ad let Mi( deote the umber of all compositios of, such that there are at least two miimal summads. The 2 i Max(=1+ 2 i Mi( = i=2ν i =2 iν i k= iν i i 1 iνi i+1 i=1ν i =2 k=sig( iν i ( k+νi ν i ( k+νi ν i C( iν i, k, 1, i 1, C( iν i, k, i+1, iν i. PROOF. Let us deote the value of maximal summads by i ad the frequecy of i i the compositio byν i. If i=1, the there is exactly oe appropriate compositio. Let ow i {2, 3,..., 2 } adνi {2, 3,..., i }. Cosider ow the summads, which are smaller tha i, ad deote the umber of these summads by k := k(i,ν i. Clearly iνi i 1 k iνi. The there are C( iν i, k, 1, i 1 differet possible compositios amog them. But ow the maximal summads could be arraged through the sequece of summads, which implies ( k+ν i ν i possibilities of where to set theseνi maximal summads. To prove the secod formula, let i deote the value of miimal summads. Therefore i {1, 2,..., 2 }, adνi {2, 3,..., i }. Let ow k deote the umber of summads which are greater tha i. If iν i = 0, the k=0 ad there is exactly oe such compositio. Suppose ow iν i > 0. If iν i i+1 = 0, there is o appropriate compositio, cotaiigν i summads i, otherwise k ca be ay umber betwee 1

6 6 G. Jaklič, V. Vitrih ad E. Žagar ad ( iν i i+1. Further, there are exactly k+νi ν C( iνi i, k, i+1, iν i compositios, cotaiigν i summads i ad k summads greater tha i. TABLE 1. Values Max( ad Mi( for Max( Mi( Let Max C ( (Mi C ( deote the umber of compositios of, such that there is exactly oe maximal (miimal summad, respectively. Sice Max(+Max C (=C(=2 1 ad Mi(+Mi C (=C(, Max( ad Mi( ca be computed also via Max C ( ad Mi C (. COROLLARY 3.2. Let Max( ad Mi( be as i Theorem 3.1. The Max C (= Mi C (= i=2 i k= i i 1 i i+1 i=1 k=sig( i (k+1 C( i, k, 1, i 1, (k+1 C( i, k, i+1, i. PROOF. The expressios ca be obtaied similarly as i the proof of Theorem 3.1. Although it seems easier to obtai Max( ad Mi( from Max C ( ad Mi C (, let us ote, that the time complexity icreases this way. The ext importat questio is the asymptotic behavior of Max( ad Mi( for large itegers. Numerical examples ad Table 1 idicate the followig cojecture. CONJECTURE 3.3. Let Max( ad Mi( be as i Theorem 3.1. The Max(+1 Mi(+1 lim = lim = 2. Max( Mi( 4. Examples A iterestig applicatio of Max( arises i umerical aalysis, i particular i asymptotic aalysis of the geometric Lagrage iterpolatio problem by Pythagoreahodograph (PH curves ([1, 3]. Here the umber of cases of the problem cosidered, that eed to be studied, ca be sigificatly reduced by kowig Max( i advace. More precisely, if the geometric iterpolatio (see [4], e.g. by PH curves of degree

7 The umber of restricted compositios 7 is cosidered, the ukow iterpolatig parameters t i, i=1, 2,..., 1, have to lie i D= { } (t i 1 i=1 R 1 t 0 := 0<t 1 < t 2 < <t 1 < 1=: t. It turs out that the iterpolatio problem requires the aalysis of a particular oliear system of equatios ivolvig the ukow t i oly at the boudary ofd. Quite clearly, if the poit ir 1 is to be o the boudary ofd, at least two cosecutive t i have to coicide (but ot all of them, sice t 0 = 0 ad t = 1. Thus the umber of cases cosidered is equal to C(+1 2=2 2 (see Figure 1, e.g.. t 0 0 t 3 1 FIGURE 1. All possible cases for =3. Some further observatios reduce the problem oly to the aalysis of particular parts of the boudary. Let 0, t i 1 t i, ν i := max {i j t l+1= t l, j l i 1}, otherwise, 0 j i 1 where i=1, 2,...,. It turs out that if the sequece (ν i i=1 has a uique maximum, the correspodig choice of parameters (t i 1 i=1 ca be skipped i the aalysis. But the umber of sequeces (ν i i=1 for which the maximum is ot uique is precisely Max(+1. Let us coclude the paper with a aother example. I high order parametric polyomial approximatio of circular arcs ([5], e.g., the coefficiets of the optimal solutio ivolve the umber of restricted partitios of a atural umber. Namely, the coefficiets of the parametric polyomial approximat p(t=(x(t, y(t T, where x(t := α k t k, y(t := β k t k, are of the form α k = j=0 k( k k=0 k=0 ( k P( 2 π j, k, k cos 2 +π j, k is eve, 0, k is odd,

8 8 G. Jaklič, V. Vitrih ad E. Žagar ad β k = k( k j=0 where P(, k, b := k l=1 P(,l, 1, b. 0, k is eve, ( k P( 2 π j, k, k si 2 +π j, k is odd,, 5. Ackowledgemet We would like to thak Prof. Marko Petkovšek ad Prof. Herbert Wilf for their valuable suggestios. Refereces [1] Rida T. Farouki. Pythagorea-hodograph curves: algebra ad geometry iseparable, Geometry ad Computig, Volume 1 (Spriger, Berli, [2] Philippe Flajolet ad Robert Sedgewick. Aalytic combiatorics (Cambridge Uiversity Press, Cambridge, [3] Gašper Jaklič, Jerej Kozak, Marjeta Krajc, Vito Vitrih, ad Emil Žagar. Geometric Lagrage iterpolatio by plaar cubic Pythagorea-hodograph curves. Comput. Aided Geom. Desig 25 (9 (2008, [4] Gašper Jaklič, Jerej Kozak, Marjeta Krajc, ad Emil Žagar. O geometric iterpolatio by plaar parametric polyomial curves. Math. Comp. 76 (260 (2007, [5] Gašper Jaklič, Jerej Kozak, Marjeta Krajc, ad Emil Žagar. O geometric iterpolatio of circle-like curves. Comput. Aided Geom. Desig 24 (5 (2007, [6] N. J. A. Sloa. The Olie Ecyclopedia of Iteger Sequeces, jas/ sequeces (2008. Gašper Jaklič, FMF ad IMFM, Uiversity of Ljubljaa, Jadraska 19, Ljubljaa, Sloveia, PINT, Uiversity of Primorska, Muzejski trg 2, Koper, Sloveia gasper.jaklic@fmf.ui-lj.si Vito Vitrih, PINT, Uiversity of Primorska, Muzejski trg 2, Koper, Sloveia vito.vitrih@upr.si Emil Žagar, FMF ad IMFM, Uiversity of Ljubljaa, Jadraska 19, Ljubljaa, Sloveia emil.zagar@fmf.ui-lj.si

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