A022016 -id:A022016 - cp's OEIS frontend

A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.

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, 1, 1, 4, 12, 47, 173, 731, 2954, 10513, 24584, 40182

Offset: 0

R. J. Mathar, Mar 12 2021

Maximal e for a given n (i.e., the length of the n-th row minus 1) is A002620(n), see Mathematics StackExchange. - Andrey Zabolotskiy, Mar 12 2021

			The table starts
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 ;
...
T(4,0) = 1: the 4-point poset with no relations, 4 isolated points in the Hasse diagram.
T(4,1) = 1: the 4-point poset with one relation, the Hasse diagram has one vertical line and 2 isolated points.
T(4,2) = 4: the 4 posets contributing to A022016(4) = 4, extended by additional isolated point when the number of points is less than 4.
T(4,3) = 8: the 8 posets contributing to A022017(3).
T(4,4) = 2: the "dagaz rune" poset {1<3, 2<3, 1<4, 2<4}
  o o
  |X|
  o o
and the "diamond" poset {1<2, 1<3, 2<4, 3<4}
    o
   / \
  o   o
   \ /
    o

R. J. Mathar, Table of T(n,e) for n<=10
Brendan McKay, Digraphs, posets tables at the bottom of the page.
R. J. Mathar, Poset Illustrations
Mathematics StackExchange, Maximum number of comparisons required to define a partial ordering, 2021.

Cf. A000112 (row sums), A263864, A022016 (convergents down rows), A002620, A342472 (lower bound row length), A342590 (connected), A342589 (labeled), A376633 (self-dual).

T(n,0) = T(n,1) = 1.

T(n,e) = A022016(e) for n >= 2e.

T(0,0) = 1 prepended and "conjecture" removed from A022016 formula. Andrey Zabolotskiy, Mar 12 2021

A022017 Number of connected partially ordered sets with n "lines": pairs (a,b) where a < b and there is no c with a < c < b. The lines form the minimal basis for the partial ordering.

Original entry on oeis.org

1, 3, 8, 29, 103, 442, 1953, 9502, 48533, 262634, 1485764, 8777397, 53869119

Offset: 1

Franklin T. Adams-Watters, Mar 30 2002

The points are unlabeled.

			See Sloane's link.

See A000112 for references and links about partially ordered sets.

N. J. A. Sloane, Illustration of initial terms of A022016 and A022017.
Index entries for sequences related to posets

Cf. A000112, A000608, A022016. Column sums of A342590.

a(6)-a(9) from A342590. - R. J. Mathar, Mar 21 2021

a(10)-a(13) from Rico Zöllner and Konrad Handrich, Nov 19 2024

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.

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).

Original entry on oeis.org

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

cp's OEIS Frontend

A342447 T(n,e) is the number of unlabeled posets of n>=0 points with e>=0 arcs in the Hasse diagram, irregular triangle read by rows.

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A022017 Number of connected partially ordered sets with n "lines": pairs (a,b) where a < b and there is no c with a < c < b. The lines form the minimal basis for the partial ordering.

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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.

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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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