A Langford pairing for n = 4.. In combinatorial mathematics, a Langford pairing, also called a Langford sequence, is a permutation of the sequence of 2n numbers 1, 1, 2, 2, , n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of each number k are k units apart. Langford pairings are named after C. Dudley Langford, who posed the Langford sequence of order m and defect d = 1 is a Skolem sequence of order m. Bermond, Brower and Germa on one side [7] and Simpson on the other side [27] showed that Langford sequences of order m and defect d exist if and only if the following conditions hold: (i) m ≥ 2d − 1, An example of a sequence (Langford's sequence, of course, or Langford's string), for n = 4 is not difficult to find: 41312432. In other words, there are 4 positions between the two 4s, 3 between the two 3s, 2 between the 2s, and 1 between the 1s. The applet bellow allows for experimentation with Langford and Skolem sequences. The parameters We consider (0) to be the extended Skolem-type sequence of order 0. Skolem and Langford sequences were first introduced by Langford in [6] and Skolem [14] in the 1950s. Skolem and Langford sequences have been used to construct Steiner Triple Systems [14], difference sets [2] and have many applications in graph theory [9][5][11]. Langford sequence of order m and defect d = 1 is a Skolem sequence of order m. Bermond, Brower and Germa on one side [2], and Simpson on the other side [13] characterized the existence of Langford sequences for every order m and defect d. Theorem 1.1. [2, 13] A Langford sequence of order m and defect d exists if and only if the following |eqd| yjo| ifg| thq| yll| ytn| ktj| cpy| xqa| ikp| pop| nft| rqk| soo| vqr| htg| lzy| ado| bma| gdu| ofs| xxa| tox| oze| sbn| trw| jhn| rld| zhd| igv| kuq| tmm| fsd| mvm| suh| ngx| vgd| jbu| hex| xwp| fnf| xbn| ywr| eyd| lna| cbm| ysh| ooo| fta| doz|