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A317165 Number of permutations of [n*(n+1)/2] with distinct lengths of increasing runs. 3

%I #18 Sep 01 2021 22:23:14

%S 1,1,5,241,188743,2734858573,892173483721887,7469920269852025033699,

%T 1841449549508718383891930251607,

%U 14973026148724796464136435753195418043885,4467880642339303169146446437381463615730321314015457,53810913396105573079543194840166969124601447333276658546225661505

%N Number of permutations of [n*(n+1)/2] with distinct lengths of increasing runs.

%F a(n) = A317166(A000217(n)).

%F a(n) >= A317273(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*(n+1)/2, 0$2, {}):

%p seq(a(n), n=0..8);

%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(n+1)/2, 0, 0, {}];

%t Table[a[n], {n, 0, 8}] (* _Jean-François Alcover_, Sep 01 2021, after _Alois P. Heinz_ *)

%Y Cf. A000217, A246292, A317130, A317166, A317273.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 23 2018

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)