A317166 - OEIS

%I #15 Sep 01 2021 16:07:56

%S 1,1,1,5,7,27,241,505,1975,10241,188743,460545,2323679,10836141,

%T 85023209,2734858573,8010483015,45714797671,243112435345,

%U 1632532938001,15831051353773,892173483721887,2978105991739613,19855526019022967,113487352591708591

%N Number of permutations of [n] with distinct lengths of increasing runs.

%H Alois P. Heinz, <a href="/A317166/b317166.txt">Table of n, a(n) for n = 0..66</a>

%F a(A000217(n)) = A317165(n).

%p g:= (n, s)-> `if`(n in s, 0, 1):

%p b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),

%p       `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})

%p        , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))

%p     end:

%p a:= n-> b(n, 0$2, {}):

%p seq(a(n), n=0..24);

%t g[n_, s_] := If[MemberQ[s, n], 0, 1];

%t b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],

%t      If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],

%t      {j, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, o}]];

%t a[n_] := b[n, 0, 0, {}];

%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A000217, A007838, A317165.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jul 23 2018