In a Noetherian local ring $R$, an ideal $I$ is called an *almost complete intersection ideal* if $\mu(I)=\text{ht}(I)+1$. 


**Q) Is it true that $I$ is generated by a $d$-sequence?** 



Almost complete intersection ideal and $d$-sequence
Due to the comments to the previous answer (see the next answer) I give the following counterexample for the case where $d$-sequences are defined without considering permutations:


Let $R=K[x,y]$ and consider the ideal $(x^2,y^2,xy)$. It can be seen that no permutation of $x^2,y^2,xy$ is a d-sequence, e.g. $(xy):x^2y^2\neq (xy):x^2$. But in order to prove that the ideal is not generated by any $d$-sequence it is better to use theorems. In view of the theorems of the cited papers below, the ideal generated by a $d$-sequence is an ideal of linear type i.e. its Rees Algebra and Symmetric Algebra are equal. Now, by, e.g., using Macaulay2, one can see that the ideal $(x^2,y^2,xy)$ is not of linear type.


[Theorem, page 472, *Craig Huneke*, MR 608547 [**Powers of ideals generated by weak $d$-sequences**](http://dx.doi.org/10.1016/0021-8693(81)90276-3), *J. Algebra* **68** (1981), no. 2, 471--509.] 


[Page 5, *J. Herzog, A. Simis, and W. V. Vasconcelos*, MR 686942 [**Koszul homology and blowing-up rings**](http://www.ams.org/mathscinet-getitem?mr=686942), *Commutative algebra (Trento, 1981)* 79--169.]


