Konrad Handrich - cp's OEIS frontend

User: Konrad Handrich

Konrad Handrich has authored 2 sequences.

A376894 Stationary differences in A342447: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) which does not depend on k if k>= 2n-2 (for n>0).

1, 3, 14, 61, 273, 1228, 5631, 26141, 123261, 589251, 2855815, 14021038, 69707192

Offset: 1

Rico Zöllner and Konrad Handrich, Oct 22 2024

Number of unlabeled posets A342447(j,k) with j points, without isolated points, with k arcs in the Hasse diagramm missing n points to achieve saturation of the poset i.e. j=2k-n+1.
A342447 is the number of unlabeled posets of j points with k arcs in the Hasse diagram.
A342447(j,k)-A342447(j-1,k) = 0 if j > 2k.
For k >= 2n-2, A342447(2k-n+1,k)-A342447(2k-n,k) does not depend on k.
Therefore we define: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k).
A342447(2k-n,k) = A022016(k) - a(1)-...-a(n) for k >= 2n-2, n>0
Proof will soon be submitted to JOIS.

			See the table of A342447
 1 ;
 1 ;
 1 1 ;
 1 1 3 ;
 1 1 4  8  2 ;
 1 1 4 11 29  12   5 ;
 1 1 4 12 43 105  92   45   12    3 ;
 1 1 4 12 46 156 460  582  487  204   71   14   7 ;
 1 1 4 12 47 170 670 2097 3822 4514 3271 1579 561 186 44 16 4 ;
 ...
The differences between row j and j-1 of column k (convergence indicated by | |):
 0 ;
 0 ;
 0 |1| ;
 0  0 |3| ;
 0  0 |1| 8    2 ;
 0  0  0 |3|  27    12     5 ;
 0  0  0 |1| |14|   93    87      45    12   ... ;
 0  0  0  0   |3|   51   368     537   475   ... ;
 0  0  0  0   |1|  |14|  210    1515  3335   ... ;
 0  0  0  0    0    |3|  |61|    857  6691   ... ;
 0  0  0  0    0    |1|  |14|    258  3683   ... ;
 0  0  0  0    0     0    |3|    |61| 1127   ... ;
 0  0  0  0    0     0    |1|    |14| |273|  ... ;
a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) for n>=1
e.g. for n = 2 -> k = 2n-2 = 2
a(2) = A342447(3,2) - A342447(2,2) = 3 - 0 = 3
for n = 3 -> k >= 2n-2 = 6
a(3) = A342447(10,6) - A342447(9,6) = 745 - 731 = 14

R. P. Stanley, Enumerative Combinatorics I, 2nd. ed.

Cf. A000112, A022016.

Differences of A342447.

a(8)-a(13) from Konrad Handrich, Jan 07 2025

A376633 T(n,k) is the number of nonisomorphic n-element self-dual posets (or partially ordered sets) with k arcs in the Hasse diagram, irregular triangle read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 2, 3, 5, 2, 1, 1, 1, 2, 4, 9, 11, 12, 5, 4, 1, 1, 1, 2, 4, 10, 16, 26, 22, 21, 10, 5, 0, 1, 1, 1, 2, 4, 11, 20, 44, 65, 98, 86, 79, 41, 25, 8, 4, 2, 2, 1, 1, 2, 4, 11, 21, 51, 92, 175, 220, 276, 237, 208, 103, 67, 25, 18, 5, 3, 0, 1, 1, 1, 2, 4, 11, 22, 55, 114, 264, 462, 798, 1015, 1294, 1180, 1035, 676, 477, 243, 149, 57, 36, 13, 8, 2, 4, 1, 1, 1, 2, 4, 11, 22, 56, 121, 303, 614, 1264, 2042, 2348, 3995, 4755, 4272, 3910, 2680, 1977, 1078, 697, 300, 189, 60, 50, 15, 12, 0, 3, 0, 1

Offset: 1

Rico Zöllner and Konrad Handrich, Sep 30 2024

Posets whose Hasse diagram looks the same if it is turned upside down.

The dual poset P* of the poset P is defined by: s ≤ t in P* if and only if t ≤ s in P. If P and P* are isomorphic, then P is called self-dual.

			The table starts:
1 ;
1 1 ;
1 1 1 ;
1 1 2 2 2 ;
1 1 2 3 5 2 1 ;
1 1 2 4 9 11 12 5 4 1 ;
1 1 2 4 10 16 26 22 21 10 5 0 1 ;
1 1 2 4 11 20 44 65 98 86 79 41 25 8 4 2 2 ;
1 1 2 4 11 21 51 92 175 220 276 237 208 103 67 25 18 5 3 0 1 ;
1 1 2 4 11 22 55 114 264 462 798 1015 1294 1180 1035 676 477 243 149 57 36 13 8 2 4 1;
...

R. P. Stanley, Enumerative Combinatorics I, 2nd. ed., pp. 277.

Cf. A000112, A022016, A022017, A342447.

Row sums: A133983.

cp's OEIS Frontend

User: Konrad Handrich

A376894 Stationary differences in A342447: a(n) = A342447(2k-n+1,k)-A342447(2k-n,k) which does not depend on k if k>= 2n-2 (for n>0).

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