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In combinatorics, the Eulerian number (,) is the number of permutations of the numbers 1 to in which exactly elements are greater than the previous element (permutations with "ascents"). Leonhard Euler investigated them and associated polynomials in his 1755 book Institutiones calculi differentialis .
The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω) – the third cyclotomic field.To see that the Eisenstein integers are algebraic integers note that each z = a + bω is a root of the monic polynomial
The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. The latter is the function in the definition. The latter is the function in the definition. They also occur in combinatorics , specifically when counting the number of alternating permutations of a set with an even number of elements.
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Euler numbers, integers occurring in the coefficients of the Taylor series of 1/cosh t; Eulerian numbers count certain types of permutations. Euler number (physics), the cavitation number in fluid dynamics. Euler number (algebraic topology) – now, Euler characteristic, classically the number of vertices minus edges plus faces of a polyhedron.
When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. [1]
Euler numbers; Euler number (physics) Euler–Bernoulli beam theory; Euler–Lagrange equation; Euler–Lotka equation; Euler–Maclaurin formula; Euler–Maruyama method; Euler–Poisson–Darboux equation; Euler–Rodrigues formula; Euler–Tricomi equation; Euler's constant; Euler's continued fraction formula; Euler's critical load; Euler's ...
Software tools for working with finite sums involving Stirling numbers and Eulerian numbers are provided by the RISC Stirling.m package utilities in Mathematica. Other software packages for guessing formulas for sequences (and polynomial sequence sums) involving Stirling numbers and other special triangles is available for both Mathematica and ...
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