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A111282
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Number of permutations avoiding the patterns {1432,2431,3412,3421,4132,4231,4312,4321}; number of strong sorting class based on 1432.
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3
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1, 2, 6, 16, 42, 110, 288, 754, 1974, 5168, 13530, 35422, 92736, 242786, 635622, 1664080, 4356618, 11405774, 29860704, 78176338, 204668310, 535828592, 1402817466, 3672623806, 9615053952, 25172538050, 65902560198, 172535142544
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OFFSET
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1,2
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COMMENTS
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a(n-1) is the sum, over all Boolean n-strings, of the product of the lengths of the runs. For example, the Boolean 7-string (0,1,1,0,1,1,1) has four runs, whose lengths are 1,2,1 and 3, contributing a product of 6 to a(6). The 4 Boolean 2-strings contribute to a(3) as follows: 00 and 11 both contribute 2 and 01 and 10 both contribute 1. - David Callan, Jul 22 2008
The sequence 0, 2, 0, 0, 1, 2, 6, 16, 42, 110, 288, 754, 1974, ... with g.f. H(x) = 2*x+(x^4-x^5+x^6)/(1-3*x+x^2) is the number of "splitted indecomposable weakly threshold graphs" on n nodes [Barrus, 2016]. - N. J. A. Sloane, Jul 25 2017
Number of permutations of length n>0 avoiding the partially ordered pattern (POP) {2>1, 2>4} of length 4. That is, number of length n permutations having no subsequences of length 4 in which the second element is larger than the first and fourth elements. - Sergey Kitaev, Dec 09 2020
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LINKS
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M. Albert, R. Aldred, M. Atkinson, C. Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) #R31.
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FORMULA
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a(n) = 3a(n-1) - a(n-2), n > 3.
G.f.: (1 - x + x^2)/(1 - 3x + x^2).
a(n) = F(2n+1) + F(2n-2) + 0^n. (End)
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EXAMPLE
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x + 2*x^2 + 6*x^3 + 16*x^4 + 42*x^5 + 110*x^6 + 288*x^7 + ...
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MATHEMATICA
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a[1] = 1; a[2] = 2; a[3] = 6; a[n_] := a[n] = 3a[n - 1] - a[n - 2]; Table[a[n], {n, 28}] (* Robert G. Wilson v *)
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PROG
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(Haskell)
a111282 n = a111282_list !! (n-1)
a111282_list = 1 : a025169_list
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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