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Number of inequivalent cyclic difference sets with n elements.
20

%I #21 May 18 2024 14:54:54

%S 1,1,2,1,1,1,3,1,1,1,1,1,2,0,2,1,0,1,2,0,1,1,1,1,0,2,1,1,3,1,3,0,1,0,

%T 0,1,1,4,1,1,0,1,0,0,0,1,1,1,1,0,1,1,0,0,1,0,0,1,0,1,6,0,2,0,0,1,1,0,

%U 1,1,1,1,1,0,0,0,0,1,1,1,1,1,1,0,0,0,1,1,1,0,0,0,1,0,0,1,1,0

%N Number of inequivalent cyclic difference sets with n elements.

%C A (v,k,lambda) cyclic difference set is a subset D = {d_1, d_2, ..., d_k} of the integers modulo v such that {1, 2, ..., v-1} can each be represented as a difference (d_i-d_j) modulo v in exactly lambda different ways.

%C If D is a cyclic difference set, then D+a and u*D are again cyclic difference sets, for any a and any invertible u, cf. examples. Therefore this sequence counts only the equivalence classes of sets modulo such transformations. - _M. F. Hasler_, Jul 30 2021

%H Ralf Goertz, <a href="https://doi.org/10.1016/j.disc.2009.02.031">Coprime ordering of cyclic planar difference sets</a>, Discrete Mathematics, Volume 309, Issue 16, 2009, Pages 5248-5252.

%H Dan Gordon, <a href="https://www.dmgordon.org/diffset/">Search Form for Difference Sets</a>

%H Dan Gordon, <a href="https://web.archive.org/web/20080513020532/http://www.ccrwest.org/diffsets/diff_sets/">La Jolla Difference Set Repository</a> [from Internet Archive Wayback Machine]

%H Dan Gordon, <a href="https://web.archive.org/web/20060512152036/http://www.ccrwest.org/diffsets/ds_list.pdf">List of Cyclic Difference Sets</a> [from Internet Archive Wayback Machine]

%H Dan Gordon and Len Baumert, <a href="https://www.dmgordon.org/Publications/#diffsets">Papers on Difference Sets</a>

%e a(3) = 1 corresponds to the (7,3,1) set D = {1, 2, 4}: Each of {1, ..., 6} (mod 7) has exactly 1 representation as difference of two elements in D: 1 = 2 - 1; 2 = 4 - 2; 3 = 4 - 1; 4 == 1 - 4 (mod 7); 5 == 2 - 4 (mod 7); 6 == 1 - 2 (mod 7). The "shifted" sets {2, 3, 5}, {3, 4, 6}, {0, 4, 5}, {1, 5, 6}, {0, 2, 6}, {0, 1, 3} and -D == {3, 5, 6} == 3*D = -2*D and shifted variants of this set automatically also yield all elements of {1, ..., 6} (mod 7) exactly once as difference of two elements, but these "equivalent" variants are not counted separately.

%e a(4) = 1 corresponds to the (13,4,1) set D' = {0, 1, 3, 9}: again, each of {1, ..., 12} have exactly one representation as x - y (mod 13) with x, y in D'.

%e a(5) = 2 because there are two cyclic difference sets of length 5: The (v,k,lambda)=(11,5,2) set A095028 = {1, 3, 4, 5, 9} and the (21,5,1) set A095029 = {3, 6, 7, 12, 14}.

%Y Cf. A095029 - A095047: examples of cyclic difference set with 5 <= k <= 20.

%K nonn

%O 3,3

%A _Hugo Pfoertner_, May 27 2004

%E Second example corrected by an anonymous reader - _N. J. A. Sloane_, Jul 19 2021

%E Definition clarified by _M. F. Hasler_, Jul 30 2021