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