A202182 -id:A202182 - cp's OEIS frontend

A202181 Triangle read by rows: T(n,k) = number of n-element unlabeled N-free posets of height k (1 <= k <= n).

1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 13, 24, 10, 1, 1, 25, 77, 61, 15, 1, 1, 43, 228, 291, 130, 21, 1, 1, 76, 644, 1229, 856, 246, 28, 1, 1, 128, 1776, 4872, 4840, 2136, 427, 36, 1, 1, 216, 4854, 18711, 25107, 15543, 4733, 694, 45, 1, 1, 354, 13184, 70858, 124167, 101538, 43120, 9577, 1071, 55, 1

Offset: 1

N. J. A. Sloane, Dec 13 2011

			Triangle begins:
1
1 1
1 3 1
1 7 6 1
1 13 24 10 1
1 25 77 61 15 1
1 43 228 291 130 21 1
1 76 644 1229 856 246 28 1
1 128 1776 4872 4840 2136 427 36 1
1 216 4854 18711 25107 15543 4733 694 45 1
1 354 13184 70858 124167 101538 43120 9577 1071 55 1
...

Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.

Row sums give A202182. Cf. A202178, A003430, A007453, A053554.

A349367 Number of n-element unlabeled disconnected N-free posets.

Original entry on oeis.org

0, 1, 2, 6, 18, 65, 241, 984, 4250, 19590, 95484, 491459, 2660030, 15100494, 89648378

Offset: 1

Salah Uddin Mohammad, Nov 15 2021

Soheir M. Khamis, Height counting of unlabeled interval and N-free posets, Discrete Math. 275 (2004), no. 1-3, 165-175.
Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, An Exact Enumeration of the Unlabeled Disconnected Posets, J. Int. Seq., Vol. 25 (2022), Article 22.5.4.

Cf. A202182, A202180.

a(n) = A202182(n) - A202180(n).

A350783 Triangle read by rows: T(n,k) is the number of n-element unlabeled N-free posets with k connected components.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 31, 12, 4, 1, 1, 115, 46, 13, 4, 1, 1, 474, 173, 49, 13, 4, 1, 1, 2097, 727, 188, 50, 13, 4, 1, 1, 9967, 3195, 795, 191, 50, 13, 4, 1, 1, 50315, 15017, 3502, 810, 192, 50, 13, 4, 1, 1

Offset: 1

Salah Uddin Mohammad, Jan 16 2022

			Triangle begins:
      1;
      1,     1;
      3,     1,    1;
      9,     4,    1,   1;
     31,    12,    4,   1,   1;
    115,    46,   13,   4,   1,  1;
    474,   173,   49,  13,   4,  1,  1;
   2097,   727,  188,  50,  13,  4,  1, 1;
   9967,  3195,  795, 191,  50, 13,  4, 1, 1;
  50315, 15017, 3502, 810, 192, 50, 13, 4, 1, 1;
  ...

Salah Uddin Mohammad, Md. Shah Noor, and Md. Rashed Talukder, An Exact Enumeration of the Unlabeled Disconnected Posets, J. Int. Seq., Vol. 25 (2022), Article 22.5.4.

Row sums give A202182.
Column 1 is A202180.
Cf. A263864 (all posets), A349488 (disconnected).

A354693 Number of unlabeled prime posets with n elements.

Original entry on oeis.org

1, 0, 0, 1, 4, 28, 234, 2585, 36326, 646405, 14528011, 412212506

Offset: 1

Salah Uddin Mohammad, Jun 03 2022

A poset P is called prime if it is not decomposable. A poset Q is called decomposable if Q can be obtained as the composition (lexicographic product) of the outer poset Q' and the inner posets Qi, 1 <= i <= r, where |Q'| = r > 1 and at least one of the posets Qi is nonsingleton.

S. M. Khamis, On numerical counting of prime, UPO, and the general type of posets according to heights, Congressus Numerantium, 146 (2000), 157-171.
S. M. Khamis, Recognition of prime posets and one of its applications, J. Egypt. Math. Soc., 14 (1) (2006), 5-13.

Cf. A000112, A003430, A202182.

cp's OEIS Frontend

A202181 Triangle read by rows: T(n,k) = number of n-element unlabeled N-free posets of height k (1 <= k <= n).

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A349367 Number of n-element unlabeled disconnected N-free posets.

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A350783 Triangle read by rows: T(n,k) is the number of n-element unlabeled N-free posets with k connected components.

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A354693 Number of unlabeled prime posets with n elements.

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