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A134832
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Triangle of succession numbers for circular permutations.
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13
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1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 4, 0, 0, 1, 8, 5, 10, 0, 0, 1, 36, 48, 15, 20, 0, 0, 1, 229, 252, 168, 35, 35, 0, 0, 1, 1625, 1832, 1008, 448, 70, 56, 0, 0, 1, 13208, 14625, 8244, 3024, 1008, 126, 84, 0, 0, 1, 120288, 132080, 73125, 27480, 7560, 2016, 210, 120, 0, 0, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,12
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COMMENTS
| Imagine seating n people numbered 1,2,...n around a circular table. There are only n!/n=(n-1)! inequivalent permutations due to the action of the cyclic group Z_n. a(n,k) enumerates such circular permutations which have precisely k successor pairs (i,i+1). Due to cyclicity (n,1) is also counted as successor pair. See the Charalambides reference.
This is an example of a Sheffer triangle of the Appell type denoted by (((1-ln(1-x))/e^x,x). This explaines the e.g.f. for column nr. k given below. For Sheffer a- and z-sequences see the W. Lang link under A006232.
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REFERENCES
| Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15).
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LINKS
| W. Lang, First 10 rows and more.
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FORMULA
| a(n,k)= binomial(n,k)*a(n-k,0), k>=1 with a(n-k,0):=A000757(n), n>=0.
E.g.f. column k: ((1-ln(1-x))/e^x)*(x^k)/k!, k>=0. From the Sheffer property.
Recurrence a(n,k) = (n/k)*a(n-1,k-1), n >= k >= 1, (from the Sheffer a-sequences [1,0,0,...] due to the Appell type).
Recurrence a(n,0)= n*sum(z(j)*a(n-1,j),j=0..n-1), n>=1; a(0,0):=1, with the Sheffer z-sequence z(j):= A135808(j).
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EXAMPLE
| [1];[0,1];[0,0,1];[1,0,0,1];[1,4,0,0,1];...
Recurrence: 15=a(6,2) = (6/2)*a(5,1)=3*5 (from Sheffer a-sequence)
Recurrence: 36=a(6,0)=6*(0+0+(1/3)*10+0+0+(8/3)*1) =6*6 (from Sheffer z-sequence).
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CROSSREFS
| Cf. A000142 (row sums are factorials). A134833 (alternating row sums).
Sequence in context: A151905 A078669 A046783 * A123163 A194794 A177893
Adjacent sequences: A134829 A134830 A134831 * A134833 A134834 A134835
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008
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