A351645 Triangle read by rows: T(n,k) is the number of length n word structures using exactly k different symbols with all distinct run-lengths and the first run length of a symbol less than that of previous symbols, n >= 0, k = 0..floor(sqrt(8*n+1)-1/2).

1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 0, 1, 5, 1, 0, 1, 6, 1, 0, 1, 9, 2, 0, 1, 13, 3, 0, 1, 28, 16, 1, 0, 1, 32, 17, 1, 0, 1, 50, 31, 2, 0, 1, 66, 44, 3, 0, 1, 96, 70, 5, 0, 1, 175, 224, 36, 1, 0, 1, 217, 262, 39, 1, 0, 1, 308, 428, 71, 2, 0, 1, 425, 619, 105, 3

Offset: 0

Author

Andrew Howroyd, Feb 16 2022

Keywords

• nonn
• tabf

Comments

Permuting the symbols will not change the structure.

The k-th column of A351637 is divisible by k!.

Examples

			Triangle begins:
  1;
  0, 1;
  0, 1;
  0, 1,  1;
  0, 1,  1;
  0, 1,  2;
  0, 1,  5,  1;
  0, 1,  6,  1;
  0, 1,  9,  2;
  0, 1, 13,  3;
  0, 1, 28, 16, 1;
  0, 1, 32, 17, 1;
  0, 1, 50, 31, 2;
  0, 1, 66, 44, 3;
  0, 1, 96, 70, 5;
  ...
The T(8,1) = 1 word is 11111111.
The T(8,2) = 9 words are 11111112, 11111122, 11111211, 11111221, 11111222, 11112111, 11112221, 11121111, 11211111.
The T(8,3) = 2 words are 11111223, 11112223.
In the last example, the word 11111223 corresponds with 6 words in A351637 which are 11111223, 11111233, 11222223, 11233333, 12222233, 12233333.
		

Links

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

Crossrefs

Row sums are A351732.

Cf. A000217, A003056, A008289, A350824, A351637.

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!)^2) ))); [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!)^2 for n > 0, where R(n,k) = Sum_{j=1..A003056(n)} k*(k-1)^(j-1) * j! * A008289(n,j).

T(n,k) = A351637(n,k)/k! = A350824(n,k)/(k!)^2.

T(A000217(n),n) = 1.