A231410 Number of permutations of [n] with exactly n-3 (possibly overlapping) occurrences of some of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).

6, 11, 29, 99, 367, 1543, 7901, 41759, 241361, 1647843, 11321131, 83279563, 710717285, 6009605795, 53680350389, 549737059971, 5519982252151, 58008028652479, 693065960525741, 8057982367331159, 97381078055591177, 1329697914765988419, 17567989325451095443

Offset

3,1

Links

Alois P. Heinz, Table of n, a(n) for n = 3..200

Example

a(3) = 6: 123, 132, 213, 231, 312, 321.
a(4) = 11: 1243, 1342, 2341 (UUD), 1324, 1423, 2314, 2413, 3412 (UDU), 2134, 3124, 4123 (DUU).
a(5) = 29: 12435, 12534, 13245, ..., 51243, 51342, 52341.
a(6) = 99: 124356, 125346, 126345, ..., 623514, 624513, 634512.
a(7) = 367: 1243576, 1243675, 1253476, ..., 7346125, 7356124, 7456123.

Maple

b:= proc(u, o, t, c) option remember;  `if`(u+o<c, 0,
      `if`(u+o=0, `if`(c=0, 1, 0),
      add(b(u+j-1, o-j, [2, 3, 3, 6, 6, 3][t],
              `if`(t in [5, 6], c-1, c)), j=1..o)+
      add(b(u-j, o+j-1, [4, 5, 5, 4, 4, 5][t],
              `if`(t=3, c-1, c)), j=1..u)))
    end:
a:= n-> add(b(j-1, n-j, 1, n-3), j=1..n):
seq(a(n), n=3..25);

Mathematica

b[u_, o_, t_, c_] := b[u, o, t, c] = If[u + o < c, 0,
     If[u + o == 0, If[c == 0, 1, 0],
     Sum[b[u + j - 1, o - j, {2, 3, 3, 6, 6, 3}[[t]],
          If[5 <= t <= 6, c - 1, c]], {j, 1, o}] +
     Sum[b[u - j, o + j - 1, {4, 5, 5, 4, 4, 5}[[t]],
          If[t == 3, c - 1, c]], {j, 1, u}]]];
a[n_] := Sum[b[j - 1, n - j, 1, n - 3], {j, 1, n}];
a /@ Range[3, 25] (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)

Crossrefs

Diagonal of A231384.

Sequence in context: A273794 A273857 A034489 * A285917 A105508 A114960

Adjacent sequences: A231407 A231408 A231409 * A231411 A231412 A231413

Keyword

nonn

Author

Alois P. Heinz, Nov 08 2013

Status

approved