A242785 Number of permutations of [n] avoiding the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down.

1, 1, 2, 5, 21, 70, 450, 4326, 34944, 209863, 1573632, 21824925, 302273664, 2854894485, 60269056512, 1207441809209, 19346879737625, 252773481889854, 2918333808555034, 69792946997645295, 982945842995115000, 16085109561896603059, 402131210857811703926

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..150

Example

a(4) = 21 because there are 4! = 24 permutations of {1,2,3,4} and only 3 of them do not avoid the consecutive step pattern up, down, down given by the binary expansion of 4 = 100_2: (1,4,3,2), (2,4,3,1), (3,4,2,1).

Maple

a:= proc(n) option remember; local b, m, r, h;
      if n<2 then return 1 fi;
      m:= iquo(n, 2, 'r'); h:= 2^ilog2(n);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t=m and r=0, 0, add(b(u-j, o+j-1, irem(2*t, h)), j=1..u))+
      `if`(t=m and r=1, 0, add(b(u+j-1, o-j, irem(2*t+1, h)), j=1..o)))
      end; forget(b);
      b(n, 0, 0)
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := a[n] = Module[{b, m, r, h},
     If[n < 2, Return[1]]; {m, r} = QuotientRemainder[n, 2];
     h = 2^(Length@IntegerDigits[n, 2] - 1);
     b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     If[t == m && r == 0, 0,
          Sum[b[u - j, o + j - 1, Mod[2t, h]], {j, 1, u}]] +
     If[t == m && r == 1, 0,
          Sum[b[u + j - 1, o - j, Mod[2t+1, h]], {j, 1, o}]]];
     b[n, 0, 0]];
a /@ Range[0, 30] (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)

Crossrefs

Column k=0 of A242783.

Main diagonal of A242784.

Cf. A335308.

Sequence in context: A000941 A000131 A328041 * A359672 A228385 A152801

Adjacent sequences: A242782 A242783 A242784 * A242786 A242787 A242788

Keyword

nonn

Author

Alois P. Heinz, May 22 2014

Status

approved