closed form formula for the number of restricted compositions

Submitted to the Bulletin of the Australian Mathematical Societydoi:10.1017/S . . .CLOSED FORM FORMULA FOR THE NUMBER OFRESTRICTED COMPOSITIONSGAŠPER JAKLIČ, VITO VITRIH □∨ and EMIL ŽAGARAbstractIn this paper, compositions of a natural number are studied. The number of restricted compositions isgiven in a closed form, and some applications are presented.2000 Mathematics subject classification. primary 05A17; secondary 68R05.Keywords and phrases: composition, partition, restricted, maximal element, minimal element.1. IntroductionCompositions and partitions of a natural number n frequently appear in researchand in practical applications. Although the number of compositions or partitions,satisfying particular requirements, can be obtained from their generating functions,this is a serious drawback, since it requires symbolic computational facilities, exactcomputations, and because of computational complexity involved. In this paper, wepresent a closed form formula for the number of restricted compositions, and givesome applications of the results.Let us be more precise. The list of natural numbers t i , which sum up to a naturalnumber n, is an integer composition of n. The set of all such lists, where the ordering ofthe summands matters, is the set of all integer compositions of n. The set of restrictedinteger compositions of n is the subset of all compositions that satisfy some additionalrestrictions, e.g., on the number of summands, on the values of summands, . . . Leta, b, n∈N with a≤b≤n. Let C(n, a, b) denote the number of compositions of n, suchthat summands t i are natural numbers, bounded as a≤t i ≤ b, for all i. Furthermore, letC(n, k, a, b) denote the number of those restricted compositions of n, where the numberof summands is equal to k,k∑t i = n, a≤t j ≤ b, j=1, 2,...,k.Clearly,i=1C(n, a, b)=⌊ n a⌋ ∑k=⌈ n b⌉C(n, k, a, b).

8 G. Jaklič, V. Vitrih and E. Žagarand⎧⎪⎨β k =∑⎪⎩k(n−k)j=0where˜P(n, k, b) := ∑ kl=1P(n,l, 1, b).0, k is even,( k˜P( 2 )πj, k, n−k) sin2n +π n j , k is odd, ,5. AcknowledgementWe would like to thank Prof. Marko Petkovšek and Prof. Herbert Wilf for theirvaluable suggestions.References[1] Rida T. Farouki. Pythagorean-hodograph curves: algebra and geometry inseparable, Geometryand Computing, Volume 1 (Springer, Berlin, 2008).[2] Philippe Flajolet and Robert Sedgewick. Analytic combinatorics (Cambridge University Press,Cambridge, 2009).[3] Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, Vito Vitrih, and Emil Žagar. ‘Geometric Lagrangeinterpolation by planar cubic Pythagorean-hodograph curves’. Comput. Aided Geom. Design 25 (9)(2008), 720–728.[4] Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, and Emil Žagar. ‘On geometric interpolation byplanar parametric polynomial curves’. Math. Comp. 76 (260) (2007), 1981–1993.[5] Gašper Jaklič, Jernej Kozak, Marjeta Krajnc, and Emil Žagar. ‘On geometric interpolation ofcircle-like curves’. Comput. Aided Geom. Design 24 (5) (2007), 241–251.[6] N. J. A. Sloan. The Online Encyclopedia of Integer Sequences, http://www.research.att.com/∼njas/sequences (2008).Gašper Jaklič, FMF and IMFM, University of Ljubljana, Jadranska 19, Ljubljana,Slovenia,PINT, University of Primorska, Muzejski trg 2, Koper, Sloveniae-mail: gasper.jaklic@fmf.uni-lj.siVito Vitrih, PINT, University of Primorska, Muzejski trg 2, Koper, Sloveniae-mail: vito.vitrih@upr.siEmil Žagar, FMF and IMFM, University of Ljubljana, Jadranska 19, Ljubljana,Sloveniae-mail: emil.zagar@fmf.uni-lj.si