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

0,5

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).

Formula

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

Example

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 ...

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 *)

Crossrefs

First column is A138265, second column is A194530.

Sequence in context: A091015 A256097 A058147 * A119954 A100804 A143175

Adjacent sequences: A193341 A193342 A193343 * A193345 A193346 A193347

Keyword

nonn,tabl

Author

N. J. A. Sloane, Aug 26 2011

Status

approved