A000239 One-half of number of permutations of [n] with exactly one run of adjacent symbols differing by 1.
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
Keywords
Examples
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.
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
Crossrefs
This is a diagonal of the irregular triangle in A010030.
Programs
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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 *)
Extensions
Entry revised by N. J. A. Sloane, Apr 14 2014
More terms from Jean-François Alcover, Feb 06 2016
Comments