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A375222 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that exactly one pair k,k stays at its initial locations 2k-1, 2k.

%I A375222 #21 Aug 05 2024 23:34:32
%S A375222 1,0,15,296,10965,609864,47880595,5047886640,688359502089,
%T A375222 117929734950320,24798753695076471,6280419381186155160,
%U A375222 1885582606127524251805,662239984799385248609976,268999138538324585872798395,125133475474486312764311243744,66091677106419135401506827779985
%N A375222 a(n) is the number of permutations of the multiset 1,1, 2,2, ..., n,n such that exactly one pair k,k stays at its initial locations 2k-1, 2k.
%C A375222 1
%H A375222 Alois P. Heinz, <a href="/A375222/b375222.txt">Table of n, a(n) for n = 1..238</a>
%F A375222 a(n) = n * A374980(n-1). - _Alois P. Heinz_, Aug 05 2024
%e A375222 a(3) = 15: The permutations with one stable pair are
%e A375222   [1, 1, 2, 3, 2, 3], [1, 1, 2, 3, 3, 2], [1, 1, 3, 2, 2, 3], [1, 1, 3, 2, 3, 2],
%e A375222   [1, 1, 3, 3, 2, 2], [1, 2, 1, 2, 3, 3], [1, 2, 2, 1, 3, 3], [1, 3, 2, 2, 1, 3],
%e A375222   [1, 3, 2, 2, 3, 1], [2, 1, 1, 2, 3, 3], [2, 1, 2, 1, 3, 3], [2, 2, 1, 1, 3, 3],
%e A375222   [3, 1, 2, 2, 1, 3], [3, 1, 2, 2, 3, 1], [3, 3, 2, 2, 1, 1].
%o A375222 (PARI) a375222(n) = {my(p=vector(2*n,i,1+(i-1)\2), m1=0); forperm (p, q, my(m=0); for (k=1, n, if (q[2*k-1]==k && q[2*k]==k, m++)); m1+=(m==1)); m1}
%Y A375222 Cf. A000680 (all permutations of this multiset), A375223 (at least one stable pair), A374980.
%K A375222 nonn
%O A375222 1,3
%A A375222 _Hugo Pfoertner_, Aug 05 2024
%E A375222 a(8) onwards from _Alois P. Heinz_, Aug 05 2024