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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. 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. Lucky numbers of Euler - Wikipedia

    en.wikipedia.org/wiki/Lucky_numbers_of_Euler

    Leonhard Euler published the polynomial k 2 − k + 41 which produces prime numbers for all integer values of k from 1 to 40. Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS). [1] Note that these numbers are all prime numbers. The primes of the form k 2 − k + 41 are

  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. Euler characteristic - Wikipedia

    en.wikipedia.org/wiki/Euler_characteristic

    hence has Betti number 1 in dimensions 0 and n, and all other Betti numbers are 0. Its Euler characteristic is then χ = 1 + (−1) n ; that is, either 0 if n is odd , or 2 if n is even . The n dimensional real projective space is the quotient of the n sphere by the antipodal map .

  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. Euler's totient function - Wikipedia

    en.wikipedia.org/wiki/Euler's_totient_function

    A perfect totient number is an integer that is equal to the sum of its iterated totients. That is, we apply the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and add together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.

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