A351637 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct run-lengths using exactly k different symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

1, 0, 1, 0, 1, 0, 1, 2, 0, 1, 2, 0, 1, 4, 0, 1, 10, 6, 0, 1, 12, 6, 0, 1, 18, 12, 0, 1, 26, 18, 0, 1, 56, 96, 24, 0, 1, 64, 102, 24, 0, 1, 100, 186, 48, 0, 1, 132, 264, 72, 0, 1, 192, 420, 120, 0, 1, 350, 1344, 864, 120, 0, 1, 434, 1572, 936, 120

Offset: 0

Author

Andrew Howroyd, Feb 15 2022

Keywords

• nonn
• tabf

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(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and every run has a different length.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1;
  0, 1,   2;
  0, 1,   2;
  0, 1,   4;
  0, 1,  10,   6;
  0, 1,  12,   6;
  0, 1,  18,  12;
  0, 1,  26,  18;
  0, 1,  56,  96, 24;
  0, 1,  64, 102, 24;
  0, 1, 100, 186, 48;
  0, 1, 132, 264, 72;
  ...
The T(6,1) = 1 word is 111111.
The T(6,2) = 10 words are 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222.
The T(6,3) = 6 words are 111223, 111233, 112333, 112223, 122333, 122233.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..958 (rows 0..100)

Crossrefs

Row sums are A351638.

Partial row sums include A000007, A000012, A032020, A351639.

Column k=2 is A216695.

Cf. A000142, A000217, A003056, A008289, A350824, A351641.

Programs

PARI

P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
T(n)={my(u=P(n), v=concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = Sum_{j=1..k} R(n,j)*binomial(k, j)*(-1)^(k-j)/k! for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).

T(n,k) = A350824(n,k)/k!.

T(A000217(n),n) = A000142(n). - Alois P. Heinz, Feb 15 2022