* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Full text
Survey
Document related concepts
Georg Cantor's first set theory article wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Abuse of notation wikipedia , lookup
Large numbers wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Factorization wikipedia , lookup
History of the function concept wikipedia , lookup
Non-standard calculus wikipedia , lookup
Function (mathematics) wikipedia , lookup
Series (mathematics) wikipedia , lookup
Collatz conjecture wikipedia , lookup
Transcript
COMBINATORIAL INTERPRETATION OF AN ANALOG OF GENERALIZED BINOMIAL COEFFICIENTS IV!. J. HO DEL Duke University, Durham, North Carolina 27706 1. INTRODUCTION Defining f/^fn;r,s) as the number of sequences of nonnegative integers (1.1) {a7,a2,-~,an} such that (1.2) -s < aj+1 -a,- < r (1<i<n-1), where r and s are arbitrary positive integers, and (1.3) a7 = /, an = k, the author [2] has shown that the generating function 00 mm 4 n(r+s),j+nr\ <t>j,r,s(x-y) = E £ fj,j+nr-m(n + I'V^y™ n=0 m=0 can be expressed in terms of generalized binomial coefficients cr+s(n,k) defined by / (1.4) For the cases r=1 n - £ cr+s(n,k)xk . E** \ h=0 ) k=0 or s=1 we have explicit formulas for fji<(n;r,s), 1 (1.5) fj,k(n + \ .r*s 1;1,s)=Y,cs+1(-t-1,j-t) t=o namely s-1 cs+1(n + t,n + t-k)-X\{h + 1)cs+1(n + t,n+t - k-h-2) h=o and k (1.6) fj/k(n+7;rJ) = r~7 Y^cr+lf-'t~^k-^ cr+1(n•f-tn-f-t-/)-J2(h + 1)cr+1(n + t,n + h=0 These formulas generalize a result of Carlitz [1] for r=s= 1. We now define an analog of cr+s(n,k), n > 0, by n I (1.7) Letting fk(m,n;r,s) r+s 0<8) satisfying (1.9) \ n(r+s) TI £ q(r-h»xh U £ M\ h=0 J k=0 denote the number of sequences of integers [a1fa2f'',an } -$ < a,+i-a; < r (1 <i<n-1) where r and s are nonnegative integers, (1.10) Cr+s(n,k;q)x a7 = 0, an = k and 360 , t-j-h-2) DEC 1974 COMBINATORIAL INTERPRETATION OF AN ANALOG OF GENERALIZED BINOMIAL COEFFICIENTS 361 n (1 11) - E */ m i=1 we show in this paper that cr+s(n,k;q) = J^ fnr-k(™, n + Vv)qm (1-12) . From (1.12) we obtain a partition identity. 2. COMBINATORIAL INTERPRETATION OF cr+s(n,k;q) From the definition of fk(m,n;r,s) it follows that (2.1) and fk(m,l;r,s) = dk,0$m,0 r+s fk(mtr) + 1;r,s) = ] T fk+s-h(m - k, w l • h=0 Now (2.1) had (2.2) imply respectively (2.2) Y*fk(m<1;r's>clm (2.3) = b k,0 M and r+s (2.4) £ fk(m,n + 1; r,s)qm = ^ m £ fk+s„h(m,n;r,s)qm+k . h=0 m Let 00 n(r+s) Z Z E fnr-k(™,n + 1;r,s)qmxkyn . n=0 k=0 m = tfx>Y>'*> Using (2.3) and (2.4) we get r+s °° (n+1)(r+s) <t>(x,y;q) - / ^ ^ j ; h=0 n=0 k=0 \ I r+s £ 5 ] = 7 +W fnr-k+h(m,n + Wig"*"***/ m £ \ h=0 <TV <S>(xq1,yqr;q). ) By iteration 7 00 #**-'*> = E /?=0 n y ; \ r+s E ^ ' * * K \ h=0 } °° = n(r+s) ^+s(nfk;q)xkyn . E E n=0 k=0 Equating coefficients we have cr+s(n,k;q) = J^ fnr-k(mfn + t;r,s)qm . (2.5) m 1 APPLICATION TO PARTITIONS Assuming the parts of a partition to be written in ascending order, let ur(k,m,n) denote the number of partitions of m into at most n parts with the minimum part at most r, the maximum part k and the difference between consecutive parts at most r. Define vr(kfm,n) to be the number of partitions of m into k parts with each part at most n and each part occurring at most r times. We show that (3.1) Proof. ur(k,m,n) = vr(k,m,n) (n?1), It is easy to see that (3.2) By (2.5) and (1.7) we have ur(k,m,n) = fk(m,n + 1;r,0) . nr £ £ f«(m'n + 1;rf)qmJ k=0 m nr = Y, cr(n,nr-k;qlS k=0 n [ - II j=1\ r \ £ qh'xh h=0 J -, R9 *"£ COMBINATORIAL INTERPRETATION OF AN ANALOG OF GENERALIZED BINOMIAL COEFFICIENTS Thus the generating function for ur(k,m,n) n P r u c u 1crM sa/ ^ is n ( V <^V J . (3.3) " \ ,= 0 / But it is well known (see for example [3, p. 10] for r = 1) that the generating function for vr(k,m,n) is also (3.3). Hence we have (3.1). This identity is also evident from the Ferrers graph. To illustrate (3.1) and (3.2) let m = 7, n = 4, k = 3 and r = 2. The sequences enumerated by fs(7,5; 2,0) are 0,0,1,3,3 , 0,0,2,2.3 and 0,1,1,2,3 . The function U2(3,7,4) counts the corresponding partitions, namely 132, 223 and 1223. The partitions which 1/2(3,7,4) enumerates are 223, 132 and 124. From the graphs we observe that 132 is the conjugate of 223, 223 is the conjugate of 132 and 1223 is the conjugate of 124. REFERENCES 1. L Garlitz, "Enumeration of Certain Types of Sequences/' Mathematische Nachrichten, Vol. 49 (1971), pp. 125-147. 2. M J . Hodel, "Enumeration of Sequences of Nonnegative Integers/'Mathematische Nachrichten, Vol. 59 (1974) pp. 235-252. 3. P.A.M. MacMahon, Combinatory Analysis, Vol. 2, Cambridge, 1916. izkkkkkk [Continued from page 354.] SPECIAL CASES Putting r= 1, s = 0, we obtain the generating function for the Fibonacci sequence (see [3] and Riordan [6]). Putting r = 2, s = -1, we obtain the generating function for the Lucas sequence (see [3] and Carlitz [1]). Other results in Riordan [6] carry over to the //-sequence. The //-sequence (and the Fibonacci and Lucas sequences), and the generalized Fibonacci and Lucas sequences are all special cases of the ^sequence studied by the author in [ 4 ] . More particularly, \HB) = \wn(r,r+s;1,-1)\ and so \fn\ = \wn(1,1;1,-1)), \an\ = | wn(2, 1;1,-1)\ . Interested readers might consult the article by Kolodner [5] which contains material somewhat similar to that in [ 3 ] , though the methods of treatment are very different. REFERENCES 1. 2. 3. 4. 5. 6. L. Carlitz, "Generating Functions for Powers of Certain Sequences of Numbers," Duke Math. J. 29 (4) (1962) pp. 521-538. A. Horadam, " A Generalized Fibonacci Sequence," Amer. Math. Monthly, 68 (5) (1961), pp. 455-459. A. Horadam, "Generating Functions for Powers of a Certain Generalized Sequence of Numbers," Duke Math, J., 32 (3) (1965), pp. 437-446. A. Horadam, "Basic Properties of a Certain Generalized Sequence of Numbers," The Fibonacci Quarterly, Vol. 3, No. 3 (October 1965), pp. 161-176. I. Kolodner, "On a Generating Function Associated with Generalized Fibonacci Sequences," The Fibonacci Quarterly, Vol. 3, Noa 4 (December 1965), pp. 272-278. J. Riordan, "Generating Functions for Powers of Fibonacci Numbers," Duke Math. J., 29 (1) (1962), pp. 5-12.