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

0,6

Comments

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

Links

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

Formula

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

Example

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.

Prog

(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])) }

Crossrefs

Row sums are A351200.

Main diagonal is A000142.

Cf. A000670, A111596, A350824, A351641.

Sequence in context: A106579 A287318 A329020 * A173003 A335461 A294411

Adjacent sequences: A351637 A351638 A351639 * A351641 A351642 A351643

Keyword

nonn,tabl

Author

Andrew Howroyd, Feb 15 2022

Status

approved