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A problem of Zarankiewicz: maximal number of 1's in a symmetric n X n matrix of 0's and 1's with 0's on the main diagonal and no "rectangle" with 1's at the four corners.
+10
8
0, 2, 6, 8, 12, 14, 18, 22, 26, 32, 36, 42, 48, 54, 60, 66, 72, 78, 84, 92, 100, 104, 112, 118, 126, 134, 142, 152, 160, 170, 180, 184, 192, 204, 212, 220, 226, 234, 244, 254
COMMENTS
In other words, the pattern
1...1
.....
1...1
is forbidden.
Such matrices are adjacency matrices of squarefree graphs (cf. A006786). The number of matrices with a(n) ones is given by A191966 and A335820 (up to permutations of rows/columns). - Max Alekseyev, Jan 29 2022
REFERENCES
B. Bollobas, Extremal Graph Theory, pp. 309ff.
Number of n X n symmetric (0,1) matrices that achieve the record mentioned in A191965.
+10
6
1, 1, 1, 12, 15, 900, 6615, 90720, 1995840, 1360800, 197920800, 359251200, 1297296000, 7264857600, 119870150400, 2615348736000, 29640619008000, 533531142144000, 101370917007360000, 101370917007360000, 425757851430912000, 3325168819675422720000
PROG
(Sage) a191966 = lambda n: sum( factorial(n) // g.automorphism_group(return_group=False, order=True) for g in graphs.nauty_geng(options=f'-c -f {n} {oeis(6855)(n)}:0') ) # Max Alekseyev, Jan 29 2022
Triangle T(n,k) read by rows: the number of symmetric binary n X n matrices with k ones and no all-1 2 X 2 submatrix.
+10
5
1, 1, 1, 1, 2, 2, 2, 1, 3, 6, 10, 9, 9, 4, 1, 4, 12, 28, 46, 72, 80, 80, 60, 16, 1, 5, 20, 60, 140, 296, 500, 780, 1005, 1085, 992, 560, 170, 1, 6, 30, 110, 330, 876, 1956, 4020, 7140, 11480, 16248, 19608, 20560, 16500, 9720, 3276, 360, 1, 7, 42, 182, 665, 2121, 5852, 14792, 33117, 68355, 126994, 214158
COMMENTS
There are 2^(n^2) binary n X n matrices (entries of {0,1}). There are 2^(n*(n+1)/2) symmetric binary matrices. There are A184948(n,k) symmetric binary n X n matrices with k ones. This sequence is the triangle T(n,k) of symmetric binary n x n matrices with k ones but no 2 X 2 submatrix with all entries = 1. [So in the display of these matrices there is no rectangle with four 1's at the corners.]
The row lengths minus 1 are 0, 1, 3, 6, 9, 12, 17, 21, 24, 29, ... and indicate the maximum number of 1's than can be packed into a symmetric binary n X n matrix without creating an all-1 quadrangle/submatrix of order 2.
FORMULA
T(n,0) = 1.
T(n,1) = n.
T(n,5) = n*(n-1)*(n-2)*(13*n^2-n-24)/60. T(n,6) = n*(n-1)*(n-2)*(19*n^3-18*n^2-97*n+60)/180. T(n,7) = n*(n-1)*(n-2)*(n-3)*(58*n^3+75*n^2-223*n+180)/1260. - Conjectured by R. J. Mathar, Mar 11 2022; proved by Max Alekseyev, Apr 02 2022 G.f.: F(x,y) = Sum_{n,k} T(n,k)*x^n/n!*y^k = exp( Sum_G x^n(G) * y^u(G) / |Aut(G)| ), where G runs over the connected squarefree graphs with loops, n(G) is the number of nodes in G, u(G) the number of ones in the adjacency matrix of G, and Aut(G) is the automorphism group of G. It follows that F(x,y) = exp(x) * (1 + x*y + x^2*y^2 + (2/3*x^3 + x^2)*y^3 + (5/12*x^4 + 3/2*x^3)*y^4 + (13/60*x^5 + 3/2*x^4 + 3/2*x^3)*y^5 + (19/180*x^6 + 7/6*x^5 + 8/3*x^4 + 2/3*x^3)*y^6 + (29/630*x^7 + 3/4*x^6 + 19/6*x^5 + 10/3*x^4)*y^7 + O(y^8)), implying the above formulas for T(n,k). - Max Alekseyev, Apr 02 2022
EXAMPLE
The triangle starts
1;
1 1;
1 2 2 2;
1 3 6 10 9 9 4;
1 4 12 28 46 72 80 80 60 16;
1 5 20 60 140 296 500 780 1005 1085 992 560 170;
...
To place 4 ones, one can place 2 of them in C(n,2) ways on the diagonal and the other 2 in n*(n-1)/2 ways outside the diagonal, avoiding one matrix that builds an all-1 submatrix, which are C(n,2)*(n*(n-1)/2-1) matrices. One can place all 4 on the diagonal in C(n,4) ways. One can place 2 outside the diagonal (the other 2 mirror symmetrically) in C(n*(n-1)/2,2) ways. Sum of the 3 terms is T(n,4) = C(n,3)*(5*n+3)/2. - R. J. Mathar, Mar 10 2022
Number of labeled simple graphs on n vertices without cycles of length 4.
+10
4
1, 2, 8, 54, 548, 7984, 163440, 4599908, 174204728, 8721120744, 568964631296, 47787888342520, 5112015062311008, 689824902243337856, 116423739687724785152, 24387469030487505651984, 6296486009090647137387200, 1991072810881504185092485408
CROSSREFS
A006786 counts isomorphism classes.
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