A class of Artinian local rings of homogeneous type

‎Let $I$ be an ideal in a regular local ring $(R,n)$‎, ‎we will find‎ ‎bounds on the first and the last Betti numbers of‎ ‎$(A,m)=(R/I,n/I)$‎. ‎if $A$ is an Artinian ring of the embedding‎ ‎codimension $h$‎, ‎$I$ has the initial degree $t$ and $mu(m^t)=1$‎, ‎we call $A$ a {it $t-$extended stretched local ring}‎. ‎This class of‎ ‎local rings is a natural generalization of the class of stretched local rings studied by Sally‎, ‎Elias and Valla‎. ‎For a $t-$extended stretched local ring‎, ‎we show that ${h+t-2choose t-1}-h+1leq tau(A)leq {h+t-2choose‎ ‎t-1}$ and $ {h+t-1choose t}-1 leq mu(I) leq {h+t-1choose t}$‎. ‎Moreover $tau(A)$ reaches the upper bound if and only if $mu(I)$‎ ‎is the maximum value‎. ‎Using these results‎, ‎we show when‎ ‎$beta_i(A)=beta_i(gr_m(A))$ for each $igeq 0$‎. ‎Beside‎, ‎we will‎ ‎investigate the rigid behavior of the Betti numbers of $A$ in the‎ ‎case that $I$ has initial degree $t$ and $mu(m^t)=2$‎. ‎This class‎ ‎is a natural generalization of {it almost stretched local rings}‎ ‎again studied by Elias and Valla‎. ‎Our research extends several‎ ‎results of two papers by Rossi‎, ‎Elias and Valla‎.