

A328044


Number of chains of binary matrices of order n.


10



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

0,2


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^2th 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^2th row of triangle A038719.


LINKS

Rajesh Kumar Mohapatra, Table of n, a(n) for n = 0..10
S. R. Kannan and Rajesh Kumar Mohapatra, Counting the Number of NonEquivalent 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), 24032411.
V. Murali and B. Makamba, Finite Fuzzy Sets, International Journal of General Systems, Vol. 34 (1) (2005), pp. 6175.
R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1991), 2331.


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)).


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 JeanFrançois Alcover at A007047 *)
Table[2*PolyLog[n^2, 1/2]  1 , {n, 0, 29}]


CROSSREFS

Cf. A000079 (subsets of an nset), A007047 (chains in power set of an nset).
Cf. A000290 (squares), A002416 (binary relations on an nset), A038719 (chains of length k in poset).
Sequence in context: A282195 A334177 A303388 * A119065 A119069 A119059
Adjacent sequences: A328041 A328042 A328043 * A328045 A328046 A328047


KEYWORD

nonn


AUTHOR

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


STATUS

approved



