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A319536 Number of signed permutations of length n where numbers occur in consecutive order.

%I A319536 #18 Feb 26 2020 08:15:12
%S A319536 0,2,14,122,1278,15802,225886,3670074,66843902,1349399162,29912161758,
%T A319536 722399486074,18881553923326,531063524702778,15993786127174238,
%U A319536 513533806880120762,17512128958240460286,632099987274779910394,24076353238897830158302
%N A319536 Number of signed permutations of length n where numbers occur in consecutive order.
%C A319536 a(n) also represents the number of reducible signed permutations of length n. A permutation is reducible when an adjacency occurs in the permutation.
%C A319536 The first 8 terms of this sequence were found by exhaustive search of all signed permutations.
%D A319536 Manaswinee Bezbaruah, Henry Fessler, Leigh Foster, Marion Scheepers, George Spahn, Context Directed Sorting: Robustness and Complexity, draft.
%H A319536 Leigh Foster, <a href="/A319536/b319536.txt">Table of n, a(n) for n = 1..50</a>
%F A319536 a(n) = A000165(n) - A271212(n).
%e A319536 Of the 8 signed permutations of length 2: {[1,2], [-1,2], [1,-2], [-1,-2], [2,1], [-2,1], [2,-1], [-2,-1]} only two are reducible: [1,2] and [-2,-1]. Thus a(2) = 2.
%t A319536 Table[(2 n)!!, {n, 1, 20}] - RecurrenceTable[{a[n]==(2n-1)*a[n-1]+2(n-2)*a[n-2], a[0]==1, a[1]==2}, a[n], {n, 1, 20}]
%o A319536 (SageMath)
%o A319536 from ast import literal_eval
%o A319536 def checkFunc(n):
%o A319536     p = SignedPermutations(n)
%o A319536     permlist = p.list()
%o A319536     permset = set(permlist)
%o A319536     for perm in permlist:
%o A319536         perm_literal = literal_eval(str(perm))
%o A319536         for i in range(n-1):
%o A319536             a = perm_literal[i]
%o A319536             if perm_literal[i + 1] == a + 1:
%o A319536                 permset.remove(perm)
%o A319536                 break
%o A319536     print(factorial(n)*(2^n)-len(permset))
%o A319536 # usage: checkFunc({desired permutation length})
%Y A319536 Cf. A000165, A271212.
%K A319536 nonn
%O A319536 1,2
%A A319536 _Leigh Foster_, Sep 22 2018