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A351641 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct runs using exactly k different symbols.

%I A351641 #9 Jan 28 2023 22:08:02
%S A351641 1,0,1,0,1,1,0,1,2,1,0,1,5,3,1,0,1,8,12,4,1,0,1,17,28,22,5,1,0,1,26,
%T A351641 81,68,35,6,1,0,1,45,177,251,135,51,7,1,0,1,76,410,704,610,236,70,8,1,
%U A351641 0,1,121,906,2068,2086,1266,378,92,9,1
%N A351641 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct runs using exactly k different symbols.
%C A351641 Permuting the symbols will not change the structure.
%C A351641 Equivalently, T(n,k) 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, the maximum value is k and all runs are distinct.
%H A351641 Andrew Howroyd, <a href="/A351641/b351641.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A351641 T(n,k) = A351640(n,k)/k!.
%e A351641 Triangle begins:
%e A351641   1;
%e A351641   0, 1;
%e A351641   0, 1,  1;
%e A351641   0, 1,  2,   1;
%e A351641   0, 1,  5,   3,   1;
%e A351641   0, 1,  8,  12,   4,   1;
%e A351641   0, 1, 17,  28,  22,   5,  1;
%e A351641   0, 1, 26,  81,  68,  35,  6, 1;
%e A351641   0, 1, 45, 177, 251, 135, 51, 7, 1;
%e A351641   ...
%e A351641 The T(4,1) = 1 word is 1111.
%e A351641 The T(4,2) = 5 words are 1112, 1121, 1122, 1211, 1222.
%e A351641 The T(4,3) = 3 words are 1123, 1223, 1233.
%e A351641 The T(4,4) = 1 word is 1234.
%o A351641 (PARI) \\ here LahI is A111596 as row polynomials.
%o A351641 LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
%o A351641 S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p, i, y)*LahI(i, y))}
%o A351641 R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
%o A351641 T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
%o A351641 { my(A=T(10)); for(n=1, #A, print(A[n])) }
%Y A351641 Row sums are A351642.
%Y A351641 Partial row sums include A000007, A000012, A351018, A351644.
%Y A351641 Column k=3 is A351643.
%Y A351641 Cf. A111596, A351637, A351640.
%K A351641 nonn,tabl
%O A351641 0,9
%A A351641 _Andrew Howroyd_, Feb 15 2022