%I A093055
%S A093055 1,1,3,2,2,6,2,2,2,10,1,1,2,2,15,1,5,4,2,1,21,4,2,6,10,2,4,28,2,8,8,8,
               2,
%T A093055 4,2,36,1,1,6,2,1,3,6,2,45,5,7,6,6,5,19,4,8,1,55,2,4,2,2,2,2,10,2,4,2,
%U A093055 66,2,2,12,8,10,14,6,8,6,2,4,78,3,5,8,4,1,1,10,6,3,7,2,4,91,1,7,2,2,1
%N A093055 Triangle T(j,k) read by rows, where T(j,k) = number of non-singleton 
               cycles in the in-situ transposition of a rectangular j X k matrix.
%C A093055 The first row and the first column are excluded, i.e. j>=k, k>1. a(1)=T(2,
               2), a(2)=T(3,2),a(3)=T(3,3), a(4)=T(4,2),a(5)=T(4,3),a(6)=T(4,4), 
               a(7)=T(5,2),.......
%D A093055 Esco G. Cate, David W. Twigg, Algorithm 513 Analysis of In-Situ Transposition. 
               ACM Transactions on Mathematical Software, Vol. 3, No. 1, March 1977, 
               pp. 104-110
%D A093055 D. E. Knuth, The Art of Computer Programming, Vol. 1 (3rd ed.). Fundamental 
               Algorithms. Addison-Wesley 1997. Ch. 1.3.3 Exercise 12: Transposing 
               a rectangular matrix. p. 182, answer p. 523.
%D A093055 S. Laflin, M. A. Brebner, Algorithm 380; In-situ tranposition of a rectangular 
               matrix. [F1], Communications of the ACM, Vol. 13, No. 5, May 1970, 
               pp. 324-326
%H A093055 E. G. Cate and D. W. Twigg, <a href="http://www.netlib.org/toms/513">
               ACM algorithm 513.</a> Revision of algorithm 380.
%H A093055 P. F. Windley, <a href="http://www3.oup.co.uk/computer_journal/hdb/Volume_02/
               Issue_01/">Transposing Matrices in a digital computer.</a> The Computer 
               Journal, Volume 2, Issue 1, April 1959, pp. 47-48
%H A093055 Dave Rusin, <a href="http://mathforum.org/kb/plaintext.jspa?messageID=96226">
               Problem with permutation cycles. </a> Posting in newsgroup sci.math 
               Oct 11, 1997
%e A093055 Transposition of a 3 X 7 matrix, written as one-dimensional vector: first 
               line: before transposition, 2nd line: after transposition
%e A093055 (1.2..3..4.5..6..7)(8..9.10.11.12.13.14)(15.16.17.18.19.20.21)
%e A093055 (1.8.15)(2.9.16)(3.10.17)(4.11.18)(5.12..19)(6.13.20)(7.14.21)
%e A093055 The following exchange cycles have to be performed: 2->4->10->8, 3->7->
               19->15, 5->13->17->9, 6->16, 12->14->20->18;
%e A093055 11 remains fixed.
%e A093055 4 cycles of length 4 + 1 cycle of length 2 -> A093055(17)=T(7,3)=5, length 
               of longest cycle: A093056(17)=4, number of fixed elements besides 
               first and last: A093057(17)=1.
%Y A093055 Cf. A093056 length of longest cycle, A093057 number of singleton cycles, 
               T(n, n)=A000217(n-1) exchanges in transposition of square matrix.
%Y A093055 Sequence in context: A112196 A021035 A007567 this_sequence A106335 A065474 
               A111702
%Y A093055 Adjacent sequences: A093052 A093053 A093054 this_sequence A093056 A093057 
               A093058
%K A093055 nonn,tabl
%O A093055 1,3
%A A093055 Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 19 2004