A330302 Number of chains of 2-element subsets of {0,1, 2, ..., n} that contain no consecutive integers.

1, 1, 3, 51, 18731, 408990251, 921132763911411, 324499299994016295527283, 25190248259800264134073495741338539, 576797123806621878513443912437627670334052360619

Offset: 0

Author

S. R. Kannan, Rajesh Kumar Mohapatra, Jan 01 2020

Keywords

• nonn

Comments

For n >= 1, a(n) is the number of chains of binary reflexive symmetric matrices of order n.

The number of chains of strictly upper triangular or strictly lower triangular matrices.

Also, number of chains in power set of (n^2-n)/2 elements.

a(n) is the number of distinct reflexive symmetric fuzzy matrices of order n.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..29
• 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 Syst., 157(17)(2006), 2403-2411.
• V. Murali and B. Makamba, Finite Fuzzy Sets, Int. J. Gen. Syst., Vol. 34 (1) (2005), pp. 61-75.
• R. B. Nelsen and H. Schmidt, Jr., Chains in power sets, Math. Mag., 64 (1) (1991), 23-31.

Crossrefs

Cf. A000079, A161680, A038719, A007047, A328044.

Programs

Maple

# P are the polynomials defined in A007047.
a:= n -> (m-> 2^m*subs(x=1/2, P(m, x)))(n*(n-1)/2):
seq(a(n), n=0..9);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 4,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n<2, 1, b(n*(n-1)/2)-1):
seq(a(n), n=0..10);  # Alois P. Heinz, Feb 11 2020

Mathematica

Array[2 PolyLog[-(#^2-#)/2, 1/2] - 1 &, 10, 0]
Table[2*PolyLog[-(n^2-n)/2, 1/2] - 1, {n, 0, 29}]
Table[LerchPhi[1/2, -(n^2-n)/2, 2]/2, {n, 0, 19}]

Formula

a(n) = A007047((n^2-n)/2) = A007047(A161680(n)).