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Search: id:A093055
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| A093055 |
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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. |
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3
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| 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, 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, 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
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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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),.......
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REFERENCES
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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. 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.
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
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LINKS
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E. G. Cate and D. W. Twigg, ACM algorithm 513. Revision of algorithm 380.
P. F. Windley, Transposing Matrices in a digital computer. The Computer Journal, Volume 2, Issue 1, April 1959, pp. 47-48
Dave Rusin, Problem with permutation cycles. Posting in newsgroup sci.math Oct 11, 1997
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EXAMPLE
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Transposition of a 3 X 7 matrix, written as one-dimensional vector: first line: before transposition, 2nd line: after transposition
(1.2..3..4.5..6..7)(8..9.10.11.12.13.14)(15.16.17.18.19.20.21)
(1.8.15)(2.9.16)(3.10.17)(4.11.18)(5.12..19)(6.13.20)(7.14.21)
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;
11 remains fixed.
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.
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CROSSREFS
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Cf. A093056 length of longest cycle, A093057 number of singleton cycles, T(n, n)=A000217(n-1) exchanges in transposition of square matrix.
Sequence in context: A112196 A021035 A007567 this_sequence A106335 A065474 A111702
Adjacent sequences: A093052 A093053 A093054 this_sequence A093056 A093057 A093058
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KEYWORD
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nonn,tabl
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AUTHOR
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Hugo Pfoertner (hugo(AT)pfoertner.org), Mar 19 2004
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