A000239 One-half of number of permutations of [n] with exactly one run of adjacent symbols differing by 1. (Formerly M2758 N1109)

1, 1, 3, 8, 28, 143, 933, 7150, 62310, 607445, 6545935, 77232740, 989893248, 13692587323, 203271723033, 3223180454138, 54362625941818, 971708196867905, 18347779304380995, 364911199401630640, 7624625589633857940, 166977535317365068775, 3824547112283439914893, 91440772473772839055238

Offset

1,3

Comments

First differences seem to be in A000130. - Ralf Stephan, Aug 28 2003

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 264, Table 7.6.2.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Jean-François Alcover, Table of n, a(n) for n = 1..40

Example

The permutation 3 2 1 4 5 7 6 has three such runs: 3-2-1, 4-5 and 7-6.
For n<=3 all permutations have one such run. For n=4, 16 have one run, two have no such runs (2413 and 3142), and 6 have two runs (1243, 2134, 2143, 3412, 3421), so a(4) = 16/2 = 8.

Mathematica

S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t + 2*t^2}[[n+1]], (n+1-t)* S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4]]; A000239 = Join[{1}, Table[Coefficient[S[n], t, 1]/2, {n, 1, 20}] // Accumulate // Rest] (* Jean-François Alcover, Feb 06 2016, from successive accumulation of A000130 *)

Crossrefs

This is a diagonal of the irregular triangle in A010030.

Sequence in context: A000776 A327030 A355986 * A268302 A345177 A342139

Adjacent sequences: A000236 A000237 A000238 * A000240 A000241 A000242

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Entry revised by N. J. A. Sloane, Apr 14 2014

More terms from Jean-François Alcover, Feb 06 2016

Status

approved