|
%I #68 Oct 11 2019 02:45:34
%S 1,3,299,28349043,21262618727925419,426789461753903103302333992563,
%T 576797123806621878513443912437627670334052360619,
%U 110627172261659730424051586605958905845740712964061737226074854597705843
%N Number of chains of binary matrices of order n.
%C For n >= 1, a(n) is the number of chains of n X n (0, 1) matrices.
%C a(n) is also the number of chains in the power set of n^2 elements.
%C a(n) is the n^2-th term of A007047.
%C A chain of binary (crisp or Boolean or logical) matrices of order n can be thought of as a fuzzy matrix of order n.
%C a(n) is the number of distinct n X n fuzzy matrices.
%C a(n) is the sum of the n^2-th row of triangle A038719.
%H Rajesh Kumar Mohapatra, <a href="/A328044/b328044.txt">Table of n, a(n) for n = 0..10</a>
%H S. R. Kannan and Rajesh Kumar Mohapatra, <a href="https://arxiv.org/abs/1909.13678">Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques</a>, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
%H V. Murali, <a href="https://doi.org/10.1016/j.fss.2006.03.005">Combinatorics of counting finite fuzzy subsets</a>, Fuzzy Sets and Systems, 157(17)(2006), 2403-2411.
%H V. Murali and B. Makamba, <a href="https://doi.org/10.1080/03081070512331318356">Finite Fuzzy Sets</a>, International Journal of General Systems, Vol. 34 (1) (2005), pp. 61-75.
%H R. B. Nelsen and H. Schmidt, Jr., <a href="http://www.jstor.org/stable/2690450">Chains in power sets</a>, Math. Mag., 64 (1991), 23-31.
%F Let T(n, k) denote the number of chains of binary matrices of order n of length k, T(0, 0) = 1, T(0, k) = 0 for k > 0, thus T(n, k) = A038719(n, k).
%F a(n) = Sum_{k=0..n^2} T(n, k); a(0) = 1.
%F a(n) = A007047(n^2) = A007047(A000290(n)).
%p # P are the polynomials defined in A007047.
%p A328044 := n -> 2^(n^2)*subs(x=1/2, P(n^2, x)):
%p seq(A328044(n), n=0..7); # _Peter Luschny_, Oct 10 2019
%t Array[2 PolyLog[-#^2, 1/2] - 1 &, 8, 0] (* _Michael De Vlieger_, Oct 05 2019, after _Jean-François Alcover_ at A007047 *)
%t Table[2*PolyLog[-n^2, 1/2] - 1 , {n, 0, 29}]
%Y Cf. A000079 (subsets of an n-set), A007047 (chains in power set of an n-set).
%Y Cf. A000290 (squares), A002416 (binary relations on an n-set), A038719 (chains of length k in poset).
%K nonn
%O 0,2
%A S. R. Kannan, _Rajesh Kumar Mohapatra_, Oct 03 2019
| 