cp's OEIS Frontend

A317447 Number of permutations of [n] whose lengths of increasing runs are distinct prime numbers.

%I A317447 #11 Mar 29 2021 08:00:56
%S A317447 1,0,1,1,0,19,0,41,110,70,13696,1,44796,155,411064,2122802,251746,
%T A317447 1057634441,4404368,25043183,44848672,19725545894,106293316,
%U A317447 307873058001,50194102,8305023165502,65808841818130,33715371370134,115625740201672616,78940089764191
%N A317447 Number of permutations of [n] whose lengths of increasing runs are distinct prime numbers.
%H A317447 Alois P. Heinz, <a href="/A317447/b317447.txt">Table of n, a(n) for n = 0..100</a>
%p A317447 g:= (n, s)-> `if`(n in s or not (n=0 or isprime(n)), 0, 1):
%p A317447 b:= proc(u, o, t, s) option remember; `if`(u+o=0, g(t, s),
%p A317447       `if`(g(t, s)=1, add(b(u-j, o+j-1, 1, s union {t})
%p A317447        , j=1..u), 0)+ add(b(u+j-1, o-j, t+1, s), j=1..o))
%p A317447     end:
%p A317447 a:= n-> b(n, 0$2, {}):
%p A317447 seq(a(n), n=0..40);
%t A317447 g[n_, s_] := If[MemberQ[s, n] || Not [n == 0 || PrimeQ[n]], 0, 1];
%t A317447 b[u_, o_, t_, s_] := b[u, o, t, s] = If[u + o == 0, g[t, s],
%t A317447      If[g[t, s] == 1, Sum[b[u - j, o + j - 1, 1, s ~Union~ {t}],
%t A317447      {j, 1, u}], 0] + Sum[b[u + j - 1, o - j, t + 1, s], {j, 1, o}]];
%t A317447 a[n_] := b[n, 0, 0, {}];
%t A317447 a /@ Range[0, 40] (* _Jean-François Alcover_, Mar 29 2021, after _Alois P. Heinz_ *)
%o A317447 (Python)
%o A317447 from functools import lru_cache
%o A317447 from sympy import isprime
%o A317447 def g(n, s): return int((n == 0 or isprime(n)) and not n in s)
%o A317447 @lru_cache(maxsize=None)
%o A317447 def b(u, o, t, s):
%o A317447   if u + o == 0: return g(t, s)
%o A317447   c1 = sum(b(u-j, o+j-1, 1, tuple(sorted(s+(t,)))) for j in range(1, u+1)) if g(t, s) else 0
%o A317447   return c1 + sum(b(u+j-1, o-j, t+1, s) for j in range(1, o+1))
%o A317447 def a(n): return b(n, 0, 0, tuple())
%o A317447 print([a(n) for n in range(41)]) # _Michael S. Branicky_, Mar 29 2021 after _Alois P. Heinz_
%Y A317447 Cf. A000040, A317131, A317444, A317445, A317446, A317448.
%K A317447 nonn
%O A317447 0,6
%A A317447 _Alois P. Heinz_, Jul 28 2018