A193344 -id:A193344 - cp's OEIS frontend

A194530 Number of unlabeled rigid interval posets with n non-maximal and 2 maximal elements.

0, 1, 3, 9, 32, 132, 623, 3314, 19628, 128126, 914005, 7074517, 59050739, 528741491, 5055414317, 51406084221, 553946196892, 6305737560455, 75610546284387, 952559077043183, 12579235034203780, 173759983171005721, 2505751777457313815, 37657189917162605826

Offset: 0

Alois P. Heinz, Aug 28 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200
Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal) (2011).

2nd column of A193344, Column k=2 of A218757.

Cf. A138265.

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:
a:= n-> coeff(coeff(w(2+n), z, 2), y, n):
seq(a(n), n=0..50);

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}]]]]; a[n_] := a[n] = Coefficient[Coefficient[w[2+n], z, 2], y, n]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

a(n) = [ y^n z^2 ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).
From Peter Bala, Aug 21 2023: (Start)
Conjectural g.f.:
1) A(x) = Sum_{n >= 0} n*(Product_{i = 1..n} 1 - 1/(1+x)^i).
2) A(x) = (1/2)*Sum_{n >= 0} n*(n+1)/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A138265. (End)

A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 3, 0, 0, 0, 5, 9, 2, 0, 0, 0, 16, 32, 13, 0, 0, 0, 0, 61, 132, 72, 6, 0, 0, 0, 0, 271, 623, 409, 69, 0, 0, 0, 0, 0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0, 0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0, 0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0

Offset: 0

Joerg Arndt, Nov 05 2012

An ascent sequence is a sequence [d(1), d(2), ..., d(n)] where d(1)=0, d(k)>=0, and d(k) <= asc([d(1), d(2), ..., d(k-1)]) and asc(.) gives the number of ascents of its argument.  Here we consider only those where adjacent digits are unequal.
The rows are the upward diagonals of A193344.
Row sums are A138265.
The column for k=1 is A138265 (i.e. the sum of row n equals the element for k=1 of the row n+1): the length-(n+1) sequences with one zero (which must be at the initial position) are formed by incrementing each digit of the length-n sequences and prepending zero.
The second column is A194530.

			Triangle starts:
1,
0, 1,
0, 1, 0,
0, 1, 1, 0,
0, 2, 3, 0, 0,
0, 5, 9, 2, 0, 0,
0, 16, 32, 13, 0, 0, 0,
0, 61, 132, 72, 6, 0, 0, 0,
0, 271, 623, 409, 69, 0, 0, 0, 0,
0, 1372, 3314, 2480, 605, 24, 0, 0, 0, 0,
0, 7795, 19628, 16222, 5016, 432, 0, 0, 0, 0, 0,
0, 49093, 128126, 114594, 41955, 5498, 120, 0, 0, 0, 0, 0,
0, 339386, 914005, 872336, 363123, 62626, 3120, 0, 0, 0, 0, 0, 0,
...
The A138265(5) = 16 length-5 ascent sequences without flat steps are (dots for zeros):
[ #]     ascent-seq.  #zeros
[ 1]    [ . 1 . 1 . ]  3
[ 2]    [ . 1 . 1 2 ]  2
[ 3]    [ . 1 . 1 3 ]  2
[ 4]    [ . 1 . 2 . ]  3
[ 5]    [ . 1 . 2 1 ]  2
[ 6]    [ . 1 . 2 3 ]  2
[ 7]    [ . 1 2 . 1 ]  2
[ 8]    [ . 1 2 . 2 ]  2
[ 9]    [ . 1 2 . 3 ]  2
[10]    [ . 1 2 1 . ]  2
[11]    [ . 1 2 1 2 ]  1
[12]    [ . 1 2 1 3 ]  1
[13]    [ . 1 2 3 . ]  2
[14]    [ . 1 2 3 1 ]  1
[15]    [ . 1 2 3 2 ]  1
[16]    [ . 1 2 3 4 ]  1
There are 5 sequences with 1 zero, 9 with two zeros and 2 with three zeros, so the row for n==5 is 0, 5, 9, 2, 0, 0.

Joerg Arndt and Alois P. Heinz, Rows n = 0..65, flattened (rows 0..15 from Joerg Arndt)

cp's OEIS Frontend

A194530 Number of unlabeled rigid interval posets with n non-maximal and 2 maximal elements.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A218757 Triangle read by rows: T(n,k) is the number of length-n ascent sequences without flat steps, containing k zeros.

Views

Author

Keywords

Comments

Examples

Links