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 A134832 Triangle of succession numbers for circular permutations. 13
 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; text; internal format)
 OFFSET 0,12 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-log(1-x))/e^x,x). This explains the e.g.f. for column nr. k given below. For Sheffer a- and z-sequences see the W. Lang link under A006232. REFERENCES Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15). LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. Wolfdieter Lang, First 10 rows and more. 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-log(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). EXAMPLE Triangle begins: [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). MATHEMATICA A000757[n_] := (-1)^n + Sum[(-1)^k*n!/((n-k)*k!), {k, 0, n-1}]; a[n_, n_] = 1; a[n_, 0] := A000757[n]; a[n_, k_] := a[n, k] = n/k*a[n-1, k-1]; Table[a[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2013 *) CROSSREFS Cf. A000142 (row sums are factorials), A134833 (alternating row sums). Sequence in context: A254156 A344386 A046783 * A123163 A194794 A337967 Adjacent sequences: A134829 A134830 A134831 * A134833 A134834 A134835 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Jan 21 2008 STATUS approved

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