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%I A263626 #99 Dec 22 2024 16:50:10
%S A263626 1,1,0,1,1,4,0,19,10,360,0,25112,720,6975976,0,7030565576,703760,
%T A263626 25473667704064,0,333377175390634880,9168331776,
%U A263626 15881995583375102901824,0,2775171714758797872451279104,1601371799340544,1790676527377592615808645709379328,0,4291738728096066980970104665682237420544,3837878966366932639744,38401679870039848205115101680007123726281532416,0
%N A263626 Number of inequivalent self-complementary Seidel matrices of order n.
%C A263626 From _Petros Hadjicostas_, Mar 01 2021: (Start)
%C A263626 On p. 9 of their arxiv paper, Greaves et al. (2014-2015) state that "The number of self-complementary Seidel matrices is known explicitly" and refer to Sozański (1980). The authors apparently refer to formula (20) in Theorem 2 (p. 139) in Sozański (1980). (See also p. 218 in Greaves et al. (2016).)
%C A263626 The so-called "Seidel matrix" of a graph G was apparently introduced by the Dutch mathematician Johan Jacob Seidel (1967, 1968, and with van Lint in 1966). They are not related to the so-called Euler-Seidel (or just Seidel) matrices named after Leonhard Euler and Philipp Ludwig von Seidel.
%C A263626 A Seidel matrix S = S(G) of a simple labeled graph G with vertex set V = {v_1, ..., v_n} is an n x n symmetric matrix with rows and columns corresponding to the n vertices such that S(i,j) is defined in the following way: S(i,j) = 0 iff i = j; S(i,j) = -1 iff vertex v_i is connected to vertex v_j (i <> j), and S(i,j) = 1 iff vertex v_i is not connected to vertex v_j (i <> j). Thus, S(G) = J - I - 2*A(G), where J is an n x n matrix of 1's, I is the n x n identity matrix, and A(G) is the adjancecy matrix of the labeled graph G.
%C A263626 Two Seidel matrices S1 and S2 are said to be equivalent if S1 can be obtained from S2 by a sequence of simultaneous row i to j and column i to j permutations or by a simultaneous multiplication of a row i and a column i by -1.
%C A263626 Thus, Seidel matrices S1 and S2 are equivalent iff there is a symmetric signed permutation matrix P such that S1 = P*S2*P^T.
%C A263626 A Seidel matrix S is called "self-complementary" if S is equivalent to -S. Thus, S is self-complementary iff there is a symmetric signed permutation matrix P such that S = -P*S*P^T. The sequence a(n) counts equivalence classes of self-complementary n x n Seidel matrices.
%C A263626 In terms of a labeled graph G with Seidel matrix S and vertex set V, a simultaneous row i to j and column i to j permutation of S corresponds to exchanging labels between vertices v_i and v_j. On the other hand, a simultaneous multiplication of row i and column i by -1 corresponds to what Mallows and Sloane (1975) call "switching at node" v_i: if A(i) is the set of vertices in V to which v_i is connected and B(i) is the set of vertices in V to which v_i is not connected, "switching at node v_i" means switching the sets A(i) and B(i) (the vertices connected with the vertices non-connected to v_i). Of course, either A(i) or B(i) may be empty.
%C A263626 Mallows and Sloane (1975) proved that the number of equivalence classes of Seidel matrices of order n is equal to the number of unlabeled even graphs (known colloquially as unlabeled "Euler graphs"). See sequence A002854.
%C A263626 The equivalence class of graphs corresponding to a self-complementary Seidel matrix is closed under complementation.
%C A263626 For simplicity, in the examples below that illustrate equivalence classes Seidel matrices, we only include unlabeled graphs on n = 3 or n = 4 nodes rather than labeled ones. (End)
%H A263626 G. Greaves, J. H. Koolen, A. Munemasa, and F. Szöllősi, <a href="http://arxiv.org/abs/1403.2155">Equiangular lines in Euclidean spaces</a>, arXiv:1403.2155 [math.CO], 2014-2015.
%H A263626 G. Greaves, J. H. Koolen, A. Munemasa, and F. Szöllősi, <a href="https://doi.org/10.1016/j.jcta.2015.09.008">Equiangular lines in Euclidean spaces</a>, Journal of Combinatorial Theory, Series A, 138 (2016), 208-235.
%H A263626 C. L. Mallows and N. J. A. Sloane, <a href="http://www.jstor.org/stable/2100368">Two-graphs, switching classes and Euler graphs are equal in number</a>, SIAM J. Appl. Math., 28 (1975), 876-880. (<a href="http://neilsloane.com/doc/MallowsSloane.pdf">copy</a> at N. J. A. Sloane's home page)
%H A263626 Johan Jacob Seidel, <a href="https://doi.org/10.1016/B978-0-12-189420-7.50009-8">Strongly regular graphs on L2-type and of triangular type</a>, Nederl. Akad. Wet., Proc., Ser. A, 70 (1967), 188-196. (<a href="https://www.zbmath.org/?q=an%3A0161.20802">Zentralblatt 161 #20802</a>)
%H A263626 Johan Jacob Seidel, <a href="https://doi.org/10.1016/0024-3795(68)90008-6">Strongly regular graphs with (-1,1,0) adjacency matrix having eigenvalue 3</a>, Linear Algebra and its Applications, 1(2) (1968), 281-298. (<a href="https://www.zbmath.org/?q=an%3A0159.25403">Zentralblatt 159 #25403</a>)
%H A263626 Tadeusz Sozański, <a href="https://doi.org/10.1002/jgt.3190040202">Enumeration of weak isomorphism classes of signed graphs</a>, J. Graph Theory 4 (1980), no. 2, 127-144. (<a href="https://www.zbmath.org/?q=ai%3Asozanski.tadeusz+se%3A00000494">Zentralblatt 434 #05059</a>)
%H A263626 Jacobus Hendricus van Lint and Johan Jacob Seidel, <a href="https://pure.tue.nl/ws/portalfiles/portal/1655945/593474.pdf">Equilateral point sets in elliptic geometry</a>Nederl. Akad. Wetensch. Proc. Ser. A, 69 (= Indag. Math., 28) (1966), 335-348. (<a href="https://www.zbmath.org/?q=an%3A0138.41702">Zentralblatt 138 #41702</a>)
%H A263626 Wikipedia, <a href="https://en.wikipedia.org/wiki/Seidel_adjacency_matrix">Seidel adjacency matrix</a>.
%F A263626 From _Petros Hadjicostas_, Mar 01 2021: (Start)
%F A263626 a(n) = 0 for 3 == n mod 4 (see the references).
%F A263626 Conjecture (due to _Peter Luschny_): a(n) = Sum_{k=0}^{n*(n-1)/2} (-1)^k*A341941(n,k) = [o.g.f. of n-th row of A341941](x=-1).
%F A263626 a(n) = Sum_{p in M2(n)} 2^(I2(p) - I1(p))/f(p), where
%F A263626 p = (p[1], ..., p[n]) is a partition of n written in frequency or multiplicity notation with p[i] >= 0 for i = 1..n and Sum_{i=1..n} i*p[i] = n;
%F A263626 M2(n) is the set of all partitions p of n with either (a) p[i] = 0 for all odd i or (b) p[1] = 1 or 2, and p[i] = 0 for i >= 2 with 0 <> (i mod 4);
%F A263626 f(p) = Product_{i=1..n} p[i]!*i^p[i];
%F A263626 J(p) = Sum_{j=1..floor((n-1)/2)} p[2*j+1];
%F A263626 I1(p) = Sum_{i=1..n} p[i] - if(J(p) == 0, 0, 1); and
%F A263626 I2(p) = (Sum_{1 <= i, j <= n} p[i]*p[j]*gcd(i,j) - J(p))/2.
%F A263626 (This is modification of formula (20) in Theorem 2 (p. 139) in Sozański (1980).)
%F A263626 Note that, if we take sum over all partitions of n, rather than just over M2(n), then we get A002854(n).
%F A263626 If, on the other hand, we consider Sum_{p in M1(n)} 2^I2(p)/f(p), where M1(n) is the set of all partitions p of n with p[1] in {0,1} and p[i] = 0 for i >= 2 with 0 <> (i mod 4), then we get A000171(n).
%F A263626 Finally, if we consider Sum_{p} 2^I2(p)/f(p) over all partitions p of n, then we get A000088(n). (End)
%e A263626 From _Petros Hadjicostas_, Mar 01 2021: (Start)
%e A263626 For n = 3, we have the following A000088(3) = 4 unlabeled graphs on 3 nodes:
%e A263626 (1) o (2) o (3) o (4) o
%e A263626 / \ / / \
%e A263626 o o o o o o o-- o
%e A263626 Graphs (1) and (2) form one equivalence class and graphs (3) and (4) form another equivalence class (under the Seidel-Mallows-Sloane "switching" equivalence relation). The complement of (1) is (4) and the complement of (2) is (3), so there are no self-complementary classes; i.e., a(3) = 0. (Here, the number of equivalence classes is A002854(3) = 2.)
%e A263626 For n = 4, the A000088(4) = 11 unlabeled graphs are listed in Mallows and Sloane (1975). There are A002854(4) = 3 equivalence classes, only one of which is self-complementary:
%e A263626 (1) o o (2) o -- o (3) o--o (4) o o (5) o--o
%e A263626 | | | | /| | /|
%e A263626 o o o o o--o o--o o--o
%e A263626 Thus, a(4) = 1. (End)
%o A263626 (PARI) a(n)={local(p=vector(n));
%o A263626 my(S=0, J() = sum(j=0, floor((n-1)/2), p[2*j+1]),
%o A263626 I1() = sum(i=1, n, p[i]) - if(J() == 0, 0, 1),
%o A263626 I2() = (sum(i=1, n, sum(j=1, n, p[i]*p[j]*gcd(i,j))) - J())/2,
%o A263626 M2() = sum(i=1, n, if(1 == (i%2), p[i], 0))*(abs((p[1]-1)*(p[1]-2)) + sum(j=2, n, if(0!=(j%4), p[j], 0))),
%o A263626 M() = I2() - I1(),
%o A263626 inc()=!forstep(i=n, 1, -1, p[i]<n\i && p[i]++ && return; p[i]=0), t); until(inc(), t=0; for( i=1, n, if( n < t+=i*p[i], until(i++>n, p[i]=n); next(2))); t==n && S+=if(M2() == 0, 2^M()/prod(i=1, n, i^p[i]*p[i]!), 0)); S} \\ This is a modification of _M. F. Hasler_'s PARI program from A002854. - _Petros Hadjicostas_, Mar 02 2021
%Y A263626 Cf. A000088, A000171, A002854, A085618 (same ?), A341941.
%K A263626 nonn
%O A263626 1,6
%A A263626 _N. J. A. Sloane_, Oct 24 2015
%E A263626 Terms a(13) to a(21) copied from Table I (p. 140) in Sozański (1980). - _Petros Hadjicostas_, Feb 27 2021
%E A263626 Name edited by and terms a(22) to a(31) from _Petros Hadjicostas_, Mar 01 2021