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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 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.
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
Euler diagram numbers with many divisors Image title Euler diagram of abundant, primitive abundant, highly abundant, superabundant, colossally abundant, highly composite, superior highly composite, weird and perfect numbers under 100 in relation to composite and deficient numbers by CMG Lee.
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When P is an Eulerian poset of rank d + 1 such that P − 1 is simplicial, the toric h-vector coincides with the ordinary h-vector constructed using the numbers f i of elements of P − 1 of given rank i + 1. In this case the toric h-vector of P satisfies the Dehn–Sommerville equations =.
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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.
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