0,12

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.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15).

W. Lang, First 10 rows and more.

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).

[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).

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

Sequence in context: A096623 A078669 A046783 this_sequence A123163 A058305 A020808

Adjacent sequences: A134829 A134830 A134831 this_sequence A134833 A134834 A134835

nonn,easy,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Jan 21 2008