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%I #33 Jun 28 2022 12:52:00
%S 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,
%T 2,4,2,36,1,1,6,2,1,3,6,2,45,5,7,6,6,5,19,4,8,1,55,2,4,2,2,2,2,10,2,4,
%U 2,66,2,2,12,8,10,14,6,8,6,2,4,78,3,5,8,4,1,1,10,6,3,7,2,4,91,1,7,2,2,1
%N 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 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 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.
%H Esco G. Cate, David W. Twigg, <a href="https://doi.org/10.1145/355719.355729">Algorithm 513: Analysis of In-Situ Transposition</a>, ACM Transactions on Mathematical Software, Vol. 3, No. 1, March 1977, pp. 104-110.
%H E. G. Cate and D. W. Twigg, <a href="http://www.netlib.org/toms/513">ACM algorithm 513</a>, Revision of algorithm 380.
%H S. Laflin, M. A. Brebner, <a href="https://doi.org/10.1145/362349.362368">Algorithm 380; In-situ transposition of a rectangular matrix [F1]</a>, Communications of the ACM, Vol. 13, No. 5, May 1970, pp. 324-326.
%H Dave Rusin, <a href="https://groups.google.com/d/msg/sci.math/w4eeI0JqrFQ/oLTlECNTHcQJ">Problem with permutation cycles</a>, Posting in newsgroup sci.math Oct 11, 1997.
%H P. F. Windley, <a href="https://doi.org/10.1093/comjnl/2.1.47">Transposing Matrices in a digital computer</a>, The Computer Journal, Volume 2, Issue 1, April 1959, pp. 47-48.
%e Transposition of a 3 X 7 matrix, written as one-dimensional vector: first line: before transposition, 2nd line: after transposition
%e (1.2..3..4.5..6..7)(8..9.10.11.12.13.14)(15.16.17.18.19.20.21)
%e (1.8.15)(2.9.16)(3.10.17)(4.11.18)(5.12..19)(6.13.20)(7.14.21)
%e 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 11 remains fixed.
%e 4 cycles of length 4 + 1 cycle of length 2 -> a(17) = T(7,3) = 5, length of longest cycle: A093056(17) = 4, number of fixed elements besides first and last: A093057(17) = 1.
%Y Cf. A093056 length of longest cycle, A093057 number of singleton cycles, T(n,n) = A000217(n-1) exchanges in transposition of square matrix.
%K nonn,tabl
%O 1,3
%A _Hugo Pfoertner_, Mar 19 2004
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