Factorials and Stirling numbers in the algebra of formal Laurent series II: za−zb=t

AbstractIn Part I, Stirling numbers of both kinds were used to define a binomial (Laurent) series of integer degree in the formal variable x. The binomial series in turn served as coefficient of tn in a formal series that reasonably well reflects the properties of (1+t)x. Analogously, generalized Stirling numbers (like central factorial numbers) are now used to define a kind of generalized Catalan series. By a different method, the Catalan series can be shown to generate a formal series that has the properties of z(t)x, where z(t)a−z(t)b=t. As in the case of ordinary Stirling numbers, not all the necessary coefficients can be described by generalized Stirling numbers alone. But they can all be explicitly expressed as an ordinary double sum of powers and factorials

Inverses of Motzkin and Schr\"oder Paths

We suggest three applications for the inverses: For the inverse Motzkin matrix we look at Hankel determinants, and counting the paths inside a horizontal band, and for the inverse Schr\"oder matrix we look at the paths inside the same band, but ending on the top side of the band

Catalan Traffic and Integrals on the Grassmannians of Lines

We prove that certain numbers occurring in a problem of paths enumeration, studied by Niederhausen (Catlan Traffic at the Beach, The Electronic Journal of Combinatorics, 9, (2002), 1--17), are top intersection numbers in the cohomology ring of the grassmannians of the lines in the complex projective (n+1)-space.Comment: Excerpt from my Ph.D. Thesis; to appear on "Discrete Mathematics

New Constraints on Supersymmetry Using Neutrino Telescopes

We demonstrate that megaton-mass neutrino telescopes are able to observe the signal from long-lived particles beyond the Standard Model, in particular the stau, the supersymmetric partner of the tau lepton. Its signature is an excess of charged particle tracks with horizontal arrival directions and energy deposits between 0.1 and 1 TeV inside the detector. We exploit this previously-overlooked signature to search for stau particles in the publicly available IceCube data. The data shows no evidence of physics beyond the Standard Model. We derive a new lower limit on the stau mass of $320$ GeV (95\% C.L.) and estimate that this new approach, when applied to the full data set available to the IceCube collaboration, will reach world-leading sensitivity to the stau mass ($m_{\tilde{\tau}}=450\,\mathrm{GeV}$)