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A231211 Number of permutations of [n] avoiding simultaneously consecutive patterns 123, 1432, 2431, and 3421. 2
1, 1, 2, 5, 14, 46, 177, 790, 4024, 23056, 146777, 1027850, 7852184, 64985116, 579191277, 5530869310, 56336971744, 609708912976, 6986749484177, 84510154473170, 1076016705993704, 14385283719409636, 201475033030143477, 2950048762311387430, 45073424916825354064 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of permutations of [n] avoiding simultaneously consecutive step patterns up, up and up, down, down.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250

A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns

FORMULA

a(n) ~ (1+exp(Pi/2)) * (2/Pi)^(n+1) * n!. - Vaclav Kotesovec, Aug 28 2014

EXAMPLE

a(3) = 5: 132, 213, 231, 312, 321.

a(4) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.

a(5) = 46: 13254, 14253, 14352, ..., 54231, 54312, 54321.

a(6) = 177: 132546, 132645, 142536, ..., 654231, 654312, 654321.

MAPLE

b:= proc(u, o, t) option remember; `if`(t=4, 0, `if`(u+o=0, 1,

      add(b(u+j-1, o-j, [2, 4, 2][t]), j=1..o)+

      add(b(u-j, o+j-1, [1, 3, 4][t]), j=1..u)))

    end:

a:= n-> b(n, 0, 1):

seq(a(n), n=0..30);

# second Maple program

n:=40: c[0, 0]:=1: for i to n-1 do c[i, 0]:=0 end do: for i to n-1 do for j to i do c[i, j] := c[i, j-1] + c[i-1, i-j] + 1 end do end do: 1, seq(c[k, k]/2, k=1..n-1); # Sergei N. Gladkovskii, Jul 27 2015

MATHEMATICA

b[u_, o_, t_] := b[u, o, t] = If[t == 4, 0, If[u + o == 0, 1,

    Sum[b[u + j - 1, o - j, {2, 4, 2}[[t]]], {j, 1, o}] +

    Sum[b[u - j, o + j - 1, {1, 3, 4}[[t]]], {j, 1, u}]]];

a[n_] := b[n, 0, 1];

a /@ Range[0, 30] (* Jean-Fran├žois Alcover, Dec 22 2020, after Alois P. Heinz *)

CROSSREFS

Column k=0 of A231210.

Cf. A049774, A177479.

Sequence in context: A275424 A328429 A107268 * A006216 A148337 A149899

Adjacent sequences:  A231208 A231209 A231210 * A231212 A231213 A231214

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Nov 05 2013

STATUS

approved

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Last modified August 13 18:03 EDT 2022. Contains 356107 sequences. (Running on oeis4.)