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A328044 Number of chains of binary matrices of order n. 10
1, 3, 299, 28349043, 21262618727925419, 426789461753903103302333992563, 576797123806621878513443912437627670334052360619, 110627172261659730424051586605958905845740712964061737226074854597705843 (list; graph; refs; listen; history; text; internal format)
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^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

Rajesh Kumar Mohapatra, Table of n, a(n) for n = 0..10

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.

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 Jean-Fran├žois Alcover at A007047 *)

Table[2*PolyLog[-n^2, 1/2] - 1 , {n, 0, 29}]

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).

Sequence in context: A104821 A282195 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

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Last modified April 5 19:56 EDT 2020. Contains 333260 sequences. (Running on oeis4.)