Matrices of Formal Power Series Associated to Binomial Posets
Gábor Hetyei
Mathematics Department UNC Charlotte Charlotte NC 28223
DOI: 10.1007/s10801-005-2506-2
Abstract
We introduce an operation that assigns to each binomial poset a partially ordered set for which the number of saturated chains in any interval is a function of two parameters. We develop a corresponding theory of generating functions involving noncommutative formal power series modulo the closure of a principal ideal, which may be faithfully represented by the limit of an infinite sequence of lower triangular matrix representations. The framework allows us to construct matrices of formal power series whose inverse may be easily calculated using the relation between the Möbius and zeta functions, and to find a unified model for the Tchebyshev polynomials of the first kind and for the derivative polynomials used to express the derivatives of the secant function as a polynomial of the tangent function.
Pages: 65–104
Keywords: keywords partially ordered set; binomial; noncommutative formal power series; tchebyshev polynomial; derivative polynomial
Full Text: PDF
References
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1979. HETYEI
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1979. HETYEI
2. James W. Brewer, “Power series over commutative rings,” Lecture Notes in Pure and Applied Mathematics, Vol. 64, Marcel Dekker, New York, 1981.
3. R. Ehrenborg and M. Readdy, “Sheffer posets and r-signed permutations,” Ann. Sci. Math. Québec 19 (1995) 173-196.
4. L. Gerritzen and R. Holtkamp, “On Gr\ddot obner bases of noncommutative power series,” Indag. Math. (N.S.) 4 (1998) 503-519.
5. G. Haigh, “A natural approach to Pick's theorem,” Math. Gaz. 64 (1980) 173-180.
6. G. Hetyei, “Tchebyshev posets,” Discrete & Comput. Geom. 32 (2004) 493-520.
7. G. Hetyei, “Orthogonal polynomials represented by CW-spheres,” Electron. J. Combin. 11(2) (2004) #R4, 28.
8. M.E. Hoffman, “Derivative polynomials for tangent and secant,” Amer. Math. Monthly 102 (1995) 23-30.
9. M.E. Hoffman, “Derivative polynomials, euler polynomials, and associated integer sequences,” Electron. J. Comb. 6 (1999) #R21.
10. R. Holtkamp, personal communication.
11. D.E. Knuth and T.J. Buckholtz, “Computation of tangent, Euler and Bernoulli numbers,” Math. Comp. 21 (1967) 663-688.
12. C. Krichnamachary and Rao M. Bhimasena, “On a table for calculating Eulerian numbers based on a new method,” Proc. London Math. Soc. 22(2) (1923) 73-80.
13. V. Reiner, “Upper binomial posets and signed permutation statistics,” Europ. J. Combin. 14 (1993) 581-588.
14. R.P. Stanley, Combinatorics and Commutative Algebra, 2nd ed., Birkh\ddot auser Boston, 1996.
15. R.P. Stanley, Enumerative Combinatorics, Vol. I, Cambridge University Press, Cambridge, 1997.
16. R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999.
17. R.P. Stanley, “A survey of Eulerian posets,” in: Polytopes: Abstract, Convex, and Computational, T. Bisztriczky, P. McMullen, R. Schneider and A.I. Weiss (eds.), NATO ASI Series C, Vol. 440, Kluwer Academic Publishers, 1994, pp. 301-333.