A097899 -id:A097899 - cp's OEIS frontend

A228614 Number of permutations of [n] having a shortest ascending run of length one.

0, 1, 1, 5, 18, 101, 611, 4452, 36287, 333395, 3382758, 37688597, 456839351, 5989023768, 84421235807, 1273482972215, 20470309460322, 349326503482301, 6307682420743595, 120157254334350828, 2408293016265606623, 50663563124372167787, 1116225038923857181614

Offset: 0

Alois P. Heinz, Aug 27 2013

nonn

			a(1) = 1: 1.
a(2) = 1: 21.
a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 18: 1243, 1342, 1432, 2134, 2143, 2341, 2431, 3124, 3142, 3214, 3241, 3421, 4123, 4132, 4213, 4231, 4312, 4321.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=1 of A064315.

g:= proc(u, o) option remember; `if`(u+o<2, u,
      add(b(u-i, o+i-1), i=1..u) +add(g(u+i-1, o-i), i=1..o))
    end:
b:= proc(u, o) option remember; `if`(u+o<2, 1-o,
      u*(u+o-1)! +add(g(u+i-1, o-i), i=1..o))
    end:
a:= n-> add(b(j-1, n-j), j=1..n):
seq(a(n), n=0..25);

g[u_, o_] := g[u, o] = If[u + o < 2, u,
     Sum[b[u - i, o + i - 1], {i, u}] +
     Sum[g[u + i - 1, o - i], {i, o}]];
b[u_, o_] := b[u, o] = If[u + o < 2, 1 - o, u*(u + o - 1)! +
     Sum[g[u + i - 1, o - i], {i, o}]];
a[n_] := Sum[b[j - 1, n - j], {j, n}];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

a(n) = A000142(n) - A097899(n).

E.g.f.: 1/(1-x) - sqrt(3)*exp(-x/2) / (2*cos(sqrt(3)*x/2+Pi/6)).

A186735 Number of permutations of [n] with no ascending runs of length 1 or 2.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 20, 69, 180, 1930, 12611, 61051, 566129, 5179750, 38348469, 376547340, 4169246332, 41559058969, 465750294781, 5905176350849, 72848728572828, 946103621115633, 13501160406995728, 195518567272213262, 2918439778172724571, 46559546190633191495

Offset: 0

Alois P. Heinz, Aug 29 2013

nonn

			a(0) = 1: the empty permutation.
a(3) = 1: 123.
a(4) = 1: 1234.
a(5) = 1: 12345.
a(6) = 20: 123456, 124356, 125346, 126345, 134256, 135246, 136245, 145236, 146235, 156234, 234156, 235146, 236145, 245136, 246135, 256134, 345126, 346125, 356124, 456123.

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

Cf. A064315, A097899.

Cf. A000142, A185652, A228614.

A[n_, k_] := A[n, k] = Module[{b}, b[u_, o_, t_] := b[u, o, t] =
     If[t + o <= k, (u + o)!,
     Sum[b[u + i - 1, o - i, Min[k, t] + 1], {i, 1, o}] +
     If[t <= k, u*(u + o - 1)!,
     Sum[b[u - i, o + i - 1, 1], {i, 1, u}]]];
  Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a[n_] := n! - A[n, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Sep 03 2021, after Alois P. Heinz in A064315 *)

a(n) = A000142(n) - A228614(n) - A185652(n).

cp's OEIS Frontend

A228614 Number of permutations of [n] having a shortest ascending run of length one.

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