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  2. Eulerian number - Wikipedia

    en.wikipedia.org/wiki/Eulerian_number

    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 .

  3. Euler numbers - Wikipedia

    en.wikipedia.org/wiki/Euler_numbers

    The Euler numbers appear in the Taylor series expansions of the secant and hyperbolic secant functions. 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.

  4. Eisenstein integer - Wikipedia

    en.wikipedia.org/wiki/Eisenstein_integer

    3 and each rational prime congruent to 1 mod 3 are equal to the norm x 2 − xy + y 2 of an Eisenstein integer x + ωy. Thus, such a prime may be factored as ( x + ωy )( x + ω 2 y ) , and these factors are Eisenstein primes: they are precisely the Eisenstein integers whose norm is a rational prime.

  5. Euler function - Wikipedia

    en.wikipedia.org/wiki/Euler_function

    The coefficient in the formal power series expansion for / gives the number of partitions of k.That is, = = ()where is the partition function.. The Euler identity, also known as the Pentagonal number theorem, is

  6. List of topics named after Leonhard Euler - Wikipedia

    en.wikipedia.org/wiki/List_of_topics_named_after...

    Euler's number, e = 2.71828 . . . , the base of the natural logarithm; Euler's idoneal numbers, a set of 65 or possibly 66 or 67 integers with special properties; Euler numbers, integers occurring in the coefficients of the Taylor series of 1/cosh t; Eulerian numbers count certain types of permutations.

  7. BEST theorem - Wikipedia

    en.wikipedia.org/wiki/BEST_theorem

    Here t w (G) is the number of arborescences, which are trees directed towards the root at a fixed vertex w in G. The number t w (G) can be computed as a determinant, by the version of the matrix tree theorem for directed graphs. It is a property of Eulerian graphs that t v (G) = t w (G) for every two vertices v and w in a connected Eulerian ...

  8. Dying To Be Free - The Huffington Post

    projects.huffingtonpost.com/dying-to-be-free...

    The last image we have of Patrick Cagey is of his first moments as a free man. He has just walked out of a 30-day drug treatment center in Georgetown, Kentucky, dressed in gym clothes and carrying a Nike duffel bag.

  9. h-vector - Wikipedia

    en.wikipedia.org/wiki/H-vector

    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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