A351640 Triangle read by rows: T(n,k) is the number of patterns of length n with all distinct runs and maximum value k.

1, 0, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 10, 18, 24, 0, 1, 16, 72, 96, 120, 0, 1, 34, 168, 528, 600, 720, 0, 1, 52, 486, 1632, 4200, 4320, 5040, 0, 1, 90, 1062, 6024, 16200, 36720, 35280, 40320, 0, 1, 152, 2460, 16896, 73200, 169920, 352800, 322560, 362880

Offset: 0

Author

Andrew Howroyd, Feb 15 2022

Keywords

• nonn
• tabl

Comments

A pattern is a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670.

Examples

			Triangle begins:
  1,
  0, 1;
  0, 1,  2;
  0, 1,  4,   6;
  0, 1, 10,  18,  24;
  0, 1, 16,  72,  96, 120;
  0, 1, 34, 168, 528, 600, 720;
  ...
The T(3,1) = 1 pattern is 111.
The T(3,2) = 4 patterns are 112, 122, 211, 221.
The T(3,3) = 6 patterns are 123, 132, 213, 231, 312, 321.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Crossrefs

Row sums are A351200.

Main diagonal is A000142.

Cf. A000670, A111596, A350824, A351641.

Programs

PARI

\\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
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^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)) ))); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

Formula

T(n,k) = k! * A351641(n,k).