

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.


6



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, 81, 68, 35, 6, 1, 0, 1, 45, 177, 251, 135, 51, 7, 1, 0, 1, 76, 410, 704, 610, 236, 70, 8, 1, 0, 1, 121, 906, 2068, 2086, 1266, 378, 92, 9, 1
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OFFSET

0,9


COMMENTS

Permuting the symbols will not change the structure.
Equivalently, T(n,k) is the number of restricted growth strings [s(0), s(1), ..., s(n1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and all runs are distinct.


LINKS



FORMULA



EXAMPLE

Triangle begins:
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, 81, 68, 35, 6, 1;
0, 1, 45, 177, 251, 135, 51, 7, 1;
...
The T(4,1) = 1 word is 1111.
The T(4,2) = 5 words are 1112, 1121, 1122, 1211, 1222.
The T(4,3) = 3 words are 1123, 1223, 1233.
The T(4,4) = 1 word is 1234.


PROG

(PARI) \\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(1)^(nk)*(n!/k!)*binomial(n1, k1))}
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))}
R(q)={[subst(serlaplace(p), y, 1)  p<Vec(q)]}
T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k1)*sum(r=k, n, y^r*binomial(r, k)*(1)^(rk)/r!) ))); [Vecrev(p)  p<v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



