A193344 Triangle read by rows: T(n,m) (n>=0, 1 <= m <= n+1) = number of unlabeled rigid interval posets with n non-maximal and m maximal elements.

1, 1, 1, 1, 3, 2, 2, 9, 13, 6, 5, 32, 72, 69, 24, 16, 132, 409, 605, 432, 120, 61, 623, 2480, 5016, 5498, 3120, 720, 271, 3314, 16222, 41955, 62626, 54370, 25560, 5040, 1372, 19628, 114594, 363123, 690935, 814690, 584580, 234360, 40320

Offset: 0

Author

N. J. A. Sloane, Aug 26 2011

Keywords

• nonn
• tabl

Examples

			Triangle begins
1
1 1
1 3 2
2 9 13 6
5 32 72 69 24
16 132 409 605 432 120
61 623 2480 5016 5498 3120 720
271 3314 16222 41955 62626 54370 25560 5040
1372 19628 114594 363123 690935 814690 584580 ...
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order 29, pp. 443-461 (2012).

Crossrefs

First column is A138265, second column is A194530.

Programs

Maple

w:= proc(t) option remember;
      `if`(t=0, 1, expand(convert(series(series(z+z*(subs(
       z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))
    end:
T:= (n,m)-> coeff(coeff(w(m+n), z, m), y, n):
seq(seq(T(n, m), m=1..n+1), n=0..10);  # Alois P. Heinz, Aug 27 2011

Mathematica

w[t_] := w[t] = If[t == 0, 1, Expand[Normal[Series[Series[z+z*((w[t-1] /. z -> z+y+y*z)-w[t-1]), {z, 0, t+1}], {y, 0, t+1}]]]]; T[n_, m_] := Coefficient[Coefficient[w[m+n], z, m], y, n]; Table[Table[T[n, m], {m, 1, n+1}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Formula

T(n,m) = [ y^n z^m ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).