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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-Fran├žois Alcover, Oct 02 2013 *)

CROSSREFS

Cf. A000142 (row sums are factorials), A134833 (alternating row sums).

Sequence in context: A335951 A254156 A046783 * A123163 A194794 A317448

Adjacent sequences:  A134829 A134830 A134831 * A134833 A134834 A134835

KEYWORD

nonn,easy,tabl

AUTHOR

Wolfdieter Lang, Jan 21 2008

STATUS

approved

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Last modified August 11 03:07 EDT 2020. Contains 336421 sequences. (Running on oeis4.)