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COMBINATORIAL SEQUENCES ARISING FROM A RATIONAL INTEGRAL
"... Abstract. We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2adic valuation of these integers that has a combinatorial interpretation. k=l 1. ..."
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Abstract. We present analytical properties of a sequence of integers related to the evaluation of a rational integral. We also discuss an algorithm for the evaluation of the 2adic valuation of these integers that has a combinatorial interpretation. k=l 1.
THE RATIO MONOTONICITY OF THE BOROSMOLL POLYNOMIALS
"... Abstract. In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the BorosMoll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are logconcave. In this paper, we show that the BorosMoll polynomials poss ..."
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Abstract. In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the BorosMoll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are logconcave. In this paper, we show that the BorosMoll polynomials possess the ratio monotone property which implies the logconcavity and the spiral property. We conclude with a conjecture which is stronger than Moll’s conjecture on the ∞logconcavity. 1.
A Remarkable Sequence of Integers
, 2008
"... A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and numbertheoretical nature. ..."
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A survey of properties of a sequence of coefficients appearing in the evaluation of a quartic definite integral is presented. These properties are of analytical, combinatorial and numbertheoretical nature.
The independence polynomial of a graph  a survey
, 2005
"... A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. There are three different kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, t ..."
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A stable (or independent) set in a graph is a set of pairwise nonadjacent vertices. The stability number α(G) is the size of a maximum stable set in the graph G. There are three different kinds of structures that one can see observing behavior of stable sets of a graph: the enumerative structure, the intersection structure, and the exchange structure. The independence polynomial of G I(G; x) = α(G) � k=0 skx k = s0 + s1x + s2x 2 +... + sα(G)x α(G), defined by Gutman and Harary (1983), is a good representative of the enumerative structure (sk is the number of stable sets of cardinality k in a graph G). One of the most general approaches to graph polynomials was proposed by Farrell (1979) in his theory of Fpolynomials of a graph. According to Farrell, any such polynomial corresponds to a strictly prescribed family of connected subgraphs of the respective graph. For the matching polynomial of a graph G, this family consists of all the edges of G, for the independence polynomial of G, this family includes all the stable sets of G. In fact, various aspects of combinatorial information concerning a graph is stored in the coefficients of a specific graph polynomial. In this paper, we survey the most important results referring the independence polynomial of a graph.
Ratio Monotonicity of Polynomials Derived from Nondecreasing Sequences
"... The ratio monotonicity of a polynomial is a stronger property than logconcavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x + 1), which leads to the logconcavity of P(x + c) for any c ≥ 1 due to Llamas and MartínezBernal. ..."
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The ratio monotonicity of a polynomial is a stronger property than logconcavity. Let P(x) be a polynomial with nonnegative and nondecreasing coefficients. We prove the ratio monotone property of P(x + 1), which leads to the logconcavity of P(x + c) for any c ≥ 1 due to Llamas and MartínezBernal. As a consequence, we obtain the ratio monotonicity of the BorosMoll polynomials obtained by Chen and Xia without resorting to the recurrence relations of the coefficients. Keywords: logconcavity, ratio monotonicity, BorosMoll polynomials. 1
THE REVERSE ULTRA LOGCONCAVITY OF THE BOROSMOLL POLYNOMIALS
"... Abstract. We prove the reverse ultra logconcavity of the BorosMoll polynomials. We further establish an inequality which implies the logconcavity of the sequence {i!di(m)} for any m ≥ 2, where di(m) are the coefficients of the BorosMoll polynomials Pm(a). This inequality also leads to the fact t ..."
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Abstract. We prove the reverse ultra logconcavity of the BorosMoll polynomials. We further establish an inequality which implies the logconcavity of the sequence {i!di(m)} for any m ≥ 2, where di(m) are the coefficients of the BorosMoll polynomials Pm(a). This inequality also leads to the fact that in the asymptotic sense, the BorosMoll sequences are just on the borderline between ultra logconcavity and reverse ultra logconcavity. We propose two conjectures on the logconcavity and reverse ultra logconcavity of the sequence {di−1(m)di+1(m)/di(m) 2} for m ≥ 2. 1.
Jour. Math. Comp. Anal. 134 (2001) 113126. AN INTEGRAL WITH THREE PARAMETERS. PART 2.
"... Abstract. In this paper we use the exact expression for the integral x I(a, b; r):= ..."
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Abstract. In this paper we use the exact expression for the integral x I(a, b; r):=
Logconcavity and qLogconvexity Conjectures on the Longest Increasing Subsequences of Permutations
, 806
"... Abstract. Let Pn,k be the number of permutations π on [n] = {1, 2,..., n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx k. We propose some conje ..."
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Abstract. Let Pn,k be the number of permutations π on [n] = {1, 2,..., n} such that the length of the longest increasing subsequences of π equals k, and let M2n,k be the number of matchings on [2n] with crossing number k. Define Pn(x) = ∑ k Pn,kx k and M2n(x) = ∑ k M2n,kx k. We propose some conjectures on the logconcavity and qlogconvexity of the polynomials Pn(x) and M2n(x).
A Proof of Moll’s Minimum Conjecture
, 904
"... Abstract. Let di(m) denote the coefficients of the BorosMoll polynomials. Moll’s minimum conjecture states that the sequence {i(i+1)(d 2 i(m) −di−1(m)di+1(m))}1≤i≤m attains its minimum with i = m. This conjecture is a stronger than the logconcavity conjecture proved by Kausers and Paule. We give a ..."
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Abstract. Let di(m) denote the coefficients of the BorosMoll polynomials. Moll’s minimum conjecture states that the sequence {i(i+1)(d 2 i(m) −di−1(m)di+1(m))}1≤i≤m attains its minimum with i = m. This conjecture is a stronger than the logconcavity conjecture proved by Kausers and Paule. We give a proof of Moll’s conjecture by utilizing the spiral property of the sequence {di(m)}0≤i≤m, and the logconcavity of the sequence {i!di(m)}0≤i≤m.