TITLE: Least permutations needed to permute from decreasing order to increasing order
QUESTION [10 upvotes]: You are asked to permute the neighboring sub-sequence of the sequence $n,n-1,n-2,\cdots,1$ until the sequence is brought to the increasing order.  
By permute the neighboring sub-sequence I mean for example:
$5,4,3,2,1 \to 5,3,4,2,1 $ or $5,4,3,2,1\to 5,2,4,3,1$ or $5,4,3,2,1\to5,2,1,4,3$. 
What is the least number of permutations needed?
Edit
A first nontrivial case I can come up with is:
$$
54321\to52143\to14523\to12345
$$
Edit 2
A second nontrivial case I come up with to show $T(n)<f(n)$ is possible regarding the answer by @Brian M. Scott:
$$
7654321\to7632154\to7215634\to1567234\to1234567
$$
maybe $\frac{n+1}{2}$?

REPLY [2 votes]: A detailed canonical answer is required

As far as I understood, you are asking about a minimal number $d(\iota)$ of transpositions (swaps of adjacent blocks) required to sort (that is, transform to identity) the completely reverted permutation $\iota$. It equals $\lceil (n+1)/2\rceil$ for $n\ge 3$, see Theorem 4.2 in [EEKSW]. This is a known special case of a general well-studied problem called Sorting by Transpositions in computational biology for DNA sequences introduced by Bafna and Pevzner [BP] in 1995 and shown to be NP-hard by Bulteau et al. [BFR] in 2010. See [FLRTV] for a detailed review on this problem and its variants and refer to [S, ES, MS , DMW] for more background on genome rearrangements. A number of studies designs approximation algorithms and bounds for $d(\pi)$, see the beginning of [BFT] for related references. For instance, Eriksson et al. [EEKSW] provided an algorithm that sorts any given permutation on $n$ elements by at most $2n/3$ transpositions. On the other hand, as far as I know, no tight upper bound for the necessary number of steps in Sorting by Transpositions is known. The best-known ﬁxed-ratio algorithm being a 1.375-approximation [EH]. 
References
[BFR] Laurent Bulteau, Guillaume Fertin, Irena Rusu, Sorting by Transpositions Is Difficult, SIDMA, 26:3 (2012), 1148-1180, arXiv version.
[BP] V. Bafna and P. A. Pevzner, Sorting permutations by transpositions, in Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, 1995, 614–623.
[DMW] Z. Dias, J. Meidanis, M. E. Walter Reversal and transposition distance of linear chromosomes. In: String Processing and Information Retrieval: A South American Symposium (SPIRE 98), 1998.
[EEKSW] H. Eriksson, K. Eriksson, J. Karlander, L. Svensson, J. Wästlund, Sorting a bridge hand, Discrete Math. 241:1 (2001), 289-300.
[EH] I. Elias, T. Hartman, A 1.375-approximation algorithm for sorting by transpositions, IEEE/ACM Trans Comput. Biol. Bioinform., 3:4, 369-379, 2006. doi:10.1109/TCBB.2006.44.
[ES] N. El-Mabrouk,  D. Sankoff, Genome rearrangement. In: T.Jiang, T. Smith, Y. Xu, and M. Q. Zhang, editors, Current Topics in Computational Molecular Biology. MIT Press, 2002.
[FLRTV] G. Fertin, A. Labarre, I. Rusu, É. Tannier, S. Vialette, Combinatorics of Genome Rearrangements, The MIT Press, Cambridge, MA, 2009.
[MS] J. Meidanis, J. Setubal. Introduction to Computational Biology. PWS Publishing Co., 1997.
[S] R. Shamir. Algorithms in molecular biology: Lecture notes, 2002.