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On Certain Sums of Stirling Numbers with Binomial Coefficients 1 2 Journal of Integer Sequences, Vol. 18 (2015), 3 Article 15.9.6 47 6 23 11 On Certain Sums of Stirling Numbers with Binomial Coefficients H. W. Gould Department of Mathematics West Virginia University Morgantown, WV 26506 USA [email protected] Harris Kwong Department of Mathematical Sciences SUNY Fredonia Fredonia, NY 14063 USA [email protected] Jocelyn Quaintance Department of Computer Science University of Pennsylvania Philadelphia, PA 19104 USA [email protected] Abstract We study two sums involving the Stirling numbers and binomial coefficients. We find their closed forms, and discuss the connection between these sums. Dedicated to the memory of our mentors, Professors Leonard Carlitz and Albert Nijenhuis 1 1 Introduction Stirling numbers of the first and second kind, denoted by s(n, k) and S(n, k) respectively, in Riordan’s [8] popular notation, have long fascinated mathematicians. They were named for James Stirling [13] who used them in 1730. In 1852 Schl¨afli [10] studied relations between s(n, k) and S(n, k). Then in 1960 Gould [3] extended Schl¨afli’s work by discovering the pair of dual relations n n + m n − m (−1)nS(m,m − n) = s(n + k, k), n − k n + k k=0 Xn n + m n − m (−1)ns(m,m − n) = S(n + k, k). n − k n + k k X=0 Prompted by a recent problem [7] in the Amer. Math. Monthly that asked the readers to n k 2n find a closed form expression for k=0(−1) n+k s(n + k, k), Gould tried to relate this sum to the first of his dual sums. By choosing m = n + 1 and noting that −1 =(−1)n+k, the P n−k first of Gould’s relations yields n 2n +1 (−1)k s(n + k, k)= S(n +1, 1)=1, n − k k X=0 which is not quite the proposed result but suggested that we study a wider range of sums. Motivated by this and other experiments using Maple, we study the following sums: n m + 2n + m f (n) = (−1)k s(n + k, k), m n + k k=0 nXm + 2n + m F (n) = (−1)k S(n + k, k), m n + k k=0 Xn 2n + m g (n) = (−1)k s(n + k, k), m n − k k=0 Xn 2n + m G (n) = (−1)k S(n + k, k). m n − k k X=0 For m ≥ 0, the closed forms for fm(n) and Fm(n) are easy to obtain, but the sums gm(n) and Gm(n) are more complicated. We also study the case of m< 0. We shall derive formulas for these sums, and discuss their connections. To simplify the notation, define fm(n)= f−m(n), Fm(n)= F−m(n), gm(n)= g−m(n), and Gm(n)= G−m(n). Consequently, throughout this paper, unless otherwise stated, m will denote a nonnegative b b b b integer, and n a positive integer. 2 2 Closed forms for fm(n) and Fm(n) We start with f0(n) and determine its value using a combinatorial argument. Theorem 1. For any positive integer, n 2n f (n)= (−1)k s(n, k)=(2n − 1)!!, 0 n + k k X=0 where (2n − 1)!! denotes the double factorial (2n − 1)(2n − 3) ··· 3 · 1. Proof. Recall that the unsigned Stirling number of the first kind c(n, k)=(−1)n−ks(n, k) counts the number of n-permutations with k disjoint cycles. Then n 2n f (n) = (−1)k s(n + k, k) 0 n + k k=0 Xn 2n = (−1)n−k c(n + k, k) n − k k=0 Xn 2n = (−1)j c(2n − j,n − j). j j=0 X Let S be the set of 2n-permutations with n cycles, and Ai be the subset of permutations of S with i as a fixed point. Then Ai1 ∩ Ai2 ∩···∩ Aij | = c(2n − j,n − j). It follows from the principle of inclusion-exclusion that f (n) is precisely the number of 2n- 0 permutations without fixed points. Notice that if a permutation in S has no fixed point, it must be a permutation with exactly n transpositions (that is, 2-cycles). After lining up 2n objects, we can take the elements two at a time to form n transpositions. Since the order within each transposition does not matter, it is just a matter of calculating the order among the transpositions; hence, there are (2n)! = (2n − 1)(2n − 3) ··· 3 · 1=(2n − 1)!! n!2n permutations of 2n with exactly n transpositions. Thus, f0(n)=(2n − 1)!!. The proof suggests we should examine the combinatorial interpretation of fm(n). Let c∗(n, k) denote the number of n-permutations with k cycles and no fixed points. It is called the unsigned associated Stirling number of the first kind ([2, p. 256] and [8, p. 73]). In a similar fashion, we can define S∗(n, k) as the number of partitions of an n-set into k subsets with no singleton subset as any part. The number S∗(n, k) is the associated Stirling number of the second kind ([2, p. 221] and [8, p. 77]). See [6] for a more thorough discussion of the associated Stirling numbers. 3 m ∗ Lemma 2. The identity fm(n)=(−1) c (2n + m,n + m) holds for any integers m and n such that n + m ≥ 1. Proof. Let S be the set of (2n + m)-permutations with n + m cycles, and Ai be the subset of permutations of S with i as a fixed point. Then Ai1 ∩ Ai2 ∩···∩ Aij = c(2n + m − j,n + m − j). Therefore, according to the principle of inclusion-exclus ion, n m + 2n + m c∗(2n + m,n + m) = (−1)j c(2n + m − j,n + m − j) j j=0 X 0 2n + m = (−1)n+m−k c(n + k, k) n + m − k k=n+m X n m + 2n + m = (−1)m (−1)k s(n + k, k). n + k k X=0 ∗ m Therefore, c (2n + m,n + m)=(−1) fm(n). By using an almost identical argument, we obtain a similar result for the associated Stirling numbers of the second kind. n+m ∗ Lemma 3. The identity Fm(n)=(−1) S (2n + m,n + m) holds for any integers m and n such that n + m ≥ 1. These combinatorial interpretations allow us to determine the exact values of fm(n) and Fm(n) for m ≥ 0. Again, due to their similarity, we shall only prove the first result. Theorem 4. For any integer n ≥ 1, f0(n)=(2n − 1)!!, and fm(n)=0 if m> 0. m ∗ Proof. Lemma 2 states that fm(n)=(−1) c (2n + m,n + m). If m > 0, it is clear that ∗ ∗ 2n + m< 2(n + m), hence c (2n + m,n + m) = 0. When m = 0, we have f0(n)= c (2n,n), which counts the number of permutations with exactly n transpositions. Hence, f0(n) = (2n − 1)!!. n Theorem 5. For any integer n ≥ 1, F0(n)=(−1) (2n − 1)!!, and Fm(n)=0 if m> 0. The same conclusions can be drawn from generating functions. Lemma 6. Let T be a nonempty set of positive integers. Define cT (n, k) as the number of n-permutations with k disjoint cycles whose lengths belong to T . Then xn xj c (x,y) := c (n, k) yk = exp y . T T n! j n,k≥ j∈T ! X0 X 4 Proof. The result follows easily from the exponential formula [15]. Alternatively, we can prove it directly as follows. For any n-permutation, let nj denotes the number of j-cycles. It is a routine exercise to show that n! cT (n, k)= , T nj j∈T nj! j X where the summation T is taken over all nj Q≥ 0, where j ∈ T , such that j∈T nj = k, and jnj = n. Then j∈T P P n jnj nj P x k x y cT (n, k) y = n n! T nj! j j n,k≥ n,k≥ j∈T X0 X0 X Y 1 xjy nj = . T nj! j n,k≥ j∈T X0 X Y Noting that this is in the form of a convolution, we determine that 1 xjy nj xjy xj cT (x,y)= = exp = exp y . nj! j j j j∈T n ≥0 j∈T j∈T ! Y Xj Y X Lemma 7. Let T be a nonempty set of positive integers. Define ST (n, k) as the number of ways to partition an n-set into k subsets with cardinalities belonging to T . Then xn xj S (x,y) := S (n, k) yk = exp y . T T n! j! n,k≥ j∈T ! X0 X Proof. The proof is identical to that of Lemma 6, except that n! ST (n, k)= . T nj j∈T nj!(j!) X Q For our purpose, we need T = N − {1}. We find xn xj c∗(x,y) := c∗(n, k) yk = exp y = (1 − x)−ye−xy, (1) n! j n,k≥ j≥2 ! X0 X and n j x x x S∗(x,y) := S∗(n, k) yk = exp y = ey(e −1−x). (2) n! j! n,k≥ j≥2 ! X0 X From the generating function c∗(x,y), it is clear that the coefficient of xryt is zero if r< 2t. m ∗ x2n n Hence, fm(n)=(−1) c (2n + m,n + m)=0if m> 0.
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