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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. 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. 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

  5. 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.

  6. 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.

  7. Euler product - Wikipedia

    en.wikipedia.org/wiki/Euler_product

    In practice all the important cases are such that the infinite series and infinite product expansions are absolutely convergent in some region ⁡ >, that is, in some right half-plane in the complex numbers. This already gives some information, since the infinite product, to converge, must give a non-zero value; hence the function given by the ...

  8. de Bruijn sequence - Wikipedia

    en.wikipedia.org/wiki/De_Bruijn_sequence

    Goal: to construct a B(2, 4) de Bruijn sequence of length 2 4 = 16 using Eulerian (n − 1 = 4 − 1 = 3) 3-D de Bruijn graph cycle. Each edge in this 3-dimensional de Bruijn graph corresponds to a sequence of four digits: the three digits that label the vertex that the edge is leaving followed by the one that labels the edge.

  9. Triangular array - Wikipedia

    en.wikipedia.org/wiki/Triangular_array

    Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial. [10] Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered. [11]

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