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A207821 Number of permutations of [n] that either have a fixed point or a succession, but not both.

%I A207821 #36 Apr 04 2025 01:26:18
%S A207821 0,1,0,5,12,69,370,2609,20552,183249,1817794,19867793,237126320,
%T A207821 3068483277,42788761294,639619513669,10202914060472,172984071549421,
%U A207821 3106257794721534,58892020126278457,1175554242034515780,24643158882899363129,541279064964716455230,12431122899361840993737,297944099946417376956220,7439329384072966947792437
%N A207821 Number of permutations of [n] that either have a fixed point or a succession, but not both.
%C A207821 A succession of a permutation p is the appearance of [k,k+1], e.g. in 23541, 23 is a succession.
%H A207821 Max Alekseyev, <a href="/A207821/b207821.txt">Table of n, a(n) for n = 0..30</a>
%F A207821 a(n) = A209325(n) + A209326(n) = A000166(n) + A000255(n-1) - 2*A209322(n) = 2*A207819(n) - A180191(n) - A002467(n). - _Max Alekseyev_, Apr 03 2025
%e A207821 a(4) = 12 because we have 1324, 1432, 2341, 2431, 3214, 3241, 3412, 3421, 4123, 4132, 4213 and 4312.
%o A207821 (PARI) A207821(n)=my(p,c);sum(k=1,n!,p=numtoperm(n,k);c=(p[1]==1);for(j=2,n,p[j]==j & c<=0 & !c++ & break; p[j]-1==p[j-1] & c>=0 & !c-- & break); c!=0) \\ _M. F. Hasler_, Jan 13 2013
%Y A207821 Cf. A000166, A002467, A180191, A201452, A207819.
%K A207821 nonn
%O A207821 0,4
%A A207821 _Jon Perry_, Jan 10 2013
%E A207821 Values a(1) to a(10) double-checked by _M. F. Hasler_, Jan 13 2013
%E A207821 Inserted a(0) and a(11)-a(13) from _Alois P. Heinz_, Jan 18 2013
%E A207821 a(14)-a(20) from _Alois P. Heinz_, Jul 05 2021
%E A207821 Terms a(21) onward from _Max Alekseyev_, Apr 03 2025