A050176 T(n,k) = M0(n+1,k,f(n,k)), where M0(p,q,r) is the number of upright paths from (0,0) to (1,0) to (p,p-q) that meet the line y = x-r and do not rise above it and f(n,k) is the least t such that M0(n+1,k,f) is not 0.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 9, 5, 9, 5, 1, 1, 6, 14, 14, 14, 14, 6, 1, 1, 7, 20, 28, 14, 28, 20, 7, 1, 1, 8, 27, 48, 42, 42, 48, 27, 8, 1, 1, 9, 35, 75, 90, 42, 90, 75, 35, 9, 1, 1, 10, 44, 110, 165, 132, 132, 165, 110, 44, 10, 1

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Comments

Let V = (e(1),...,e(n)) consist of q 1's, including e(1) = 1 and p-q 0's; let V(h) = (e(1),...,e(h)) and m(h) = (#1's in V(h)) - (#0's in V(h)) for h = 1,...,n. Then M0(p,q,r) = number of V having r = max{m(h)}.

f(n,k) = -1 if 0 <= k <= [(n-1)/2], else f(n,k) = 2*k-n.

Examples

			Rows:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,  1;
  1,  3,  2,  3,  1;
  1,  4,  5,  5,  4,  1;
  1,  5,  9,  5,  9,  5,  1;
  1,  6, 14, 14, 14, 14,  6,  1;
  1,  7, 20, 28, 14, 28, 20,  7,  1;
  1,  8, 27, 48, 42, 42, 48, 27,  8,  1;
  ...
(all palindromes)
		

Links

• Bruce E. Sagan and Joshua P. Swanson, q-Stirling numbers in type B, arXiv:2205.14078 [math.CO], 2022.

Crossrefs

Cf. A008313.