

A360743


Number of idempotent binary relation matrices E on [n] such that E contains an identity matrix of order n1 and (E  I_n)^2 = 0.


3



1, 2, 9, 52, 435, 5046, 81501, 1823144, 56572263, 2435930410, 145888123953, 12173595399516, 1418664206897691, 231298954644947294, 52860840028599821445, 16957903154151836822608, 7647128139328190245443279, 4852236755345544324027858258
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

A Boolean relation matrix R is said to be convergent in its powers if in the sequence {R,R^2,R^3, ...} there is an m such that R^m = R^(m+1).
An idempotent Boolean relation matrix E is said to have a proper power primitive iff there is a convergent relation R with limit matrix E where R is not equal to E.
If an idempotent Boolean relation matrix E contains an identity matrix of order n1 and (EI_n)^2 = 0 then E has no proper power primitive. The converse is not true for n>=4. Consider {{1,0,1,0}, {0,1,0,1}, {0,0,0,0}, {0,0,0,0}}. The converse is erroneously stated and proved in Rosenblatt, Theorem 4.


LINKS



FORMULA

E.g.f.: x*A'(x) + A(x) where A(x) = Sum_{n>=0} x^n/n! exp((2^n1)*x) is the e.g.f. for A001831.


MAPLE

a:= n> (n+1)*add(binomial(n, k)*(2^k1)^(nk), k=0..n):


MATHEMATICA

nn = 16; A[x_] := Sum[x^n/n! Exp[(2^n  1) x], {n, 0, nn}]; Range[0, nn]! CoefficientList[Series[A[x] + x D[A[x], x], {x, 0, nn}], x]


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



