Partially 2-colored permutations and the Boros–Moll polynomials

Chen, William Y. C.; Pang, Sabrina X. M.; Qu, Ellen X. Y.

doi:10.1007/s11139-011-9359-1

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…Kauers and Paule [17] proved this conjecture based on recurrence relations which were found by using a computer algebra approach. Chen, Pang and Qu [11] gave a combinatorial proof of the log-concavity of P m (a) by introducing the structure of partially 2-colored permutations. Chen and Gu [9] proved the reverse ultra log-concavity of the sequence {d i (m)} 0≤i≤m .…”

Section: Introductionmentioning

confidence: 99%

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2-Log-Concavity of the Boros–Moll Polynomials

Chen¹,

Xia²

2013

Proceedings of the Edinburgh Mathematical Society

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The Boros-Moll polynomials P m (a) arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that P m (a) is 2-log-concave for any m ≥ 2. Let d i (m) be the coefficient of a i in P m (a). We also show that the sequence {i (i+1)This leads another proof of Moll's minimum conjecture.As will be seen, the 2-log-concavity of P m (a) implies the log-concavity of a sequence considered by Moll [18,21].Since log-concavity implies unimodality, the above property leads to another proof of Moll's minimum conjecture [21] for the sequence {i(i+1)(d 2 i (m)−d i−1 (m)d i+1 (m)} 1≤i≤m . By comparing the first entry with the last entry, we deduce that this sequence attains its minimum at i = m which equals 2 −2m m(m+1) 2m m 2 . This conjecture was confirmed by Chen and Xia [14] by using a result of Chen and Gu [9] and the spiral property of the Boros-Moll sequences [13].

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Section: Introductionmentioning

confidence: 99%

“…Chen et al . [15] found a combinatorial proof of the log-concavity of P m (a) by introducing the structure of partially 2-coloured permutations. Chen and Gu [10] proved the reverse ultra log-concavity of the sequence {d i (m)} 0 i m .…”

Section: Introductionmentioning

confidence: 99%

2-Log-Concavity of the Boros–Moll Polynomials

Chen¹,

Xia²

2013

Proceedings of the Edinburgh Mathematical Society

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“…Hence P n (x) are log-concave and ratio monotone. A combinatorial interpretation of the log-concavity of P n (x) has been found by Chen, Pang and Qu [5].…”

Section: Introductionmentioning

confidence: 77%

Brändén's Conjectures on the Boros–Moll Polynomials

Chen

Dou

Yang

2012

International Mathematics Research Notices

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“…Log-convex sequences have also received attention (e.g., [9,10]). Some combinatorial proofs for them have emerged in turn (e.g., [8,58]). Log-concavity of the convolution of sequences has been studied in [3,Section 6] implicitly.…”

Section: Introductionmentioning

confidence: 99%

Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

Groß¹,

Mansour²,

Tucker³

et al. 2015

SIAM J. Discrete Math.

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Abstract. We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

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Partially 2-colored permutations and the Boros–Moll polynomials

Cited by 4 publications

References 14 publications

2-Log-Concavity of the Boros–Moll Polynomials

2-Log-Concavity of the Boros–Moll Polynomials

Brändén's Conjectures on the Boros–Moll Polynomials

Log-Concavity of Combinations of Sequences and Applications to Genus Distributions

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