A328044 Number of chains of binary matrices of order n.

1, 3, 299, 28349043, 21262618727925419, 426789461753903103302333992563, 576797123806621878513443912437627670334052360619, 110627172261659730424051586605958905845740712964061737226074854597705843

Offset: 0

Author

S. R. Kannan, Rajesh Kumar Mohapatra, Oct 03 2019

Keywords

• nonn

Comments

For n >= 1, a(n) is the number of chains of n X n (0, 1) matrices.

a(n) is also the number of chains in the power set of n^2 elements.

a(n) is the n^2-th term of A007047.

A chain of binary (crisp or Boolean or logical) matrices of order n can be thought of as a fuzzy matrix of order n.

a(n) is the number of distinct n X n fuzzy matrices.

a(n) is the sum of the n^2-th row of triangle A038719.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20 (first 11 terms from Rajesh Kumar Mohapatra)
• S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of Non-Equivalent Classes of Fuzzy Matrices Using Combinatorial Techniques, arXiv preprint arXiv:1909.13678 [math.GM], 2019.
• V. Murali, Combinatorics of counting finite fuzzy subsets, Fuzzy Sets and Systems, 157(17)(2006), 2403-2411.
• V. Murali and B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34 (1) (2005), pp. 61-75.
• R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 23-31.

Crossrefs

Cf. A000079 (subsets of an n-set), A007047 (chains in power set of an n-set).

Cf. A000290 (squares), A002416 (binary relations on an n-set), A038719 (chains of length k in poset).

Programs

Maple

# P are the polynomials defined in A007047.
A328044 := n -> 2^(n^2)*subs(x=1/2, P(n^2, x)):
seq(A328044(n), n=0..7); # Peter Luschny, Oct 10 2019

Mathematica

Array[2 PolyLog[-#^2, 1/2] - 1 &, 8, 0] (* Michael De Vlieger, Oct 05 2019, after Jean-François Alcover at A007047 *)
Table[2*PolyLog[-n^2, 1/2] - 1 , {n, 0, 29}]

Formula

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).

a(n) = Sum_{k=0..n^2} T(n, k); a(0) = 1.

a(n) = A007047(n^2) = A007047(A000290(n)).