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A351642 Number of length n word structures with all distinct runs using an infinite alphabet. 4

%I #10 Jan 28 2023 22:07:58

%S 1,1,2,4,10,26,74,218,668,2116,6928,23254,79998,281694,1011956,

%T 3704900,13815692,52386978,201787950,789178950,3130824160,12589367840,

%U 51287685476,211557376938,883067740514,3728494418330,15916998678040,68672820917088,299331260431104

%N Number of length n word structures with all distinct runs using an infinite alphabet.

%C Permuting the symbols will not change the structure.

%C Equivalently, a(n) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1 and all runs are distinct.

%H Andrew Howroyd, <a href="/A351642/b351642.txt">Table of n, a(n) for n = 0..200</a>

%e The a(4) = 10 words are 1111, 1112, 1121, 1122, 1211, 1222, 1123, 1223, 1233, 1234.

%o (PARI) \\ See A351641 for R, S.

%o seq(n)={my(q=S(n)); concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)/r!) )); }

%Y Row sums of A351641.

%Y The initial terms are similar to A206464.

%Y Cf. A351200, A351638.

%K nonn

%O 0,3

%A _Andrew Howroyd_, Feb 15 2022

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Last modified April 20 09:30 EDT 2024. Contains 371799 sequences. (Running on oeis4.)