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 A117158 Number of permutations avoiding the consecutive pattern 1234. 23
 1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463, 25874432578888440471, 548178875969847203202 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321. REFERENCES F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pages 156-157. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..450  (terms n = 0..60 from Ray Chandler) A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes. Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155. Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125. Steven Finch, Pattern-Avoiding Permutations. [Archived version] Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission] Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285. Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3) and set m=4. - N. J. A. Sloane, Aug 11 2014 Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14. Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018. Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020). FORMULA From Sergei N. Gladkovskii, Nov 30 2011: (Start) E.g.f.: 2/(exp(-x) + cos(x) - sin(x)) = 1/W(0) with continued fraction W(k) = 1 + (x^(2*k))/(f + f*x/(4*k + 1 - x - (4*k + 1)*b/R)), where R := x^(2*k) + b -(x^(4*k+1))/(c + (x^(2*k+1)) + x*c/T); T := 4*k + 3 - x - (4*k + 3)*d/(d +(x^(2*k+1))/W(k+1)), and f := (4*k)!/(2*k)!; b := (4*k + 1)!/(2*k + 1)!; c := (4*k + 2)!/(2*k + 1)!; and d :=(4*k + 3)!/(2*k + 2)!. (End) a(n) ~ n! / (sin(r)*r^(n+1)), where r = 1.0384156372665563... is the root of the equation exp(-r) + cos(r) = sin(r). - Vaclav Kotesovec, Dec 11 2013 MAPLE b:= proc(u, o, t) option remember; `if`(u+o=0, 1,       `if`(t<2, add(b(u+j-1, o-j, t+1), j=1..o), 0)+       add(b(u-j, o+j-1, 0), j=1..u))     end: a:= n-> b(n, 0, 0): seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013 MATHEMATICA a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]), {x, 0, 30}], x^n]*n! (* second program: *) b[u_, o_, t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *) CROSSREFS Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693, A230051, A324132. Column k=0 of A220183. Column k=7 of A242784. Sequence in context: A113228 A201693 A063255 * A317128 A185334 A290280 Adjacent sequences:  A117155 A117156 A117157 * A117159 A117160 A117161 KEYWORD nonn AUTHOR Steven Finch, Apr 26 2006 STATUS approved

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