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Number of multisets of permutations with a combined total of n moved points spanning an initial interval of positive integers.
2

%I #13 May 06 2023 11:12:53

%S 1,0,1,2,16,94,836,8062,92434,1187780,17103983,271660992,4722454300,

%T 89127765656,1814841101699,39650794527652,925141689393748,

%U 22957155125457704,603681013763369997,16767920412944383544,490550926763623941996,15076432260424342403648,485630804356929583800760

%N Number of multisets of permutations with a combined total of n moved points spanning an initial interval of positive integers.

%H Andrew Howroyd, <a href="/A362767/b362767.txt">Table of n, a(n) for n = 0..200</a>

%e In the following examples permutations are shown using cycle notation.

%e The a(2) = 1 multiset of permutations is {(12)}.

%e The a(3) = 2 multisets of permutations are {(123)}, {(132)}.

%e The a(4) = 16 multisets of permutations are A000166(4) = 9 derangements plus 7 pairs of transpositions which are: {(12), (34)}, {(13), (24)}, {(14), (23)}, {(12), (13)}, {(12), (23)}, {(13), (23)}, {(12), (12)}.

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o R(k,n,b)={EulerT(vector(n, j, binomial(k,j)*polcoef(b,j)))}

%o seq(n)={my(b=serlaplace(exp(-x + O(x*x^n))/(1-x))); concat([1], sum(k=1, n, R(k,n,b) * sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))}

%Y Cf. A000166, A362768 (strict case).

%K nonn

%O 0,4

%A _Andrew Howroyd_, May 04 2023