A006455 Number of partial orders on {1,2,...,n} that are contained in the usual linear order (i.e., xRy => x<y). (Formerly M1805)

1, 1, 2, 7, 40, 357, 4824, 96428, 2800472, 116473461, 6855780268, 565505147444, 64824245807684

Offset

0,3

Comments

Also known as naturally labeled posets. - David Bevan, Nov 16 2023

Also the number of n X n upper triangular Boolean matrices B with all diagonal entries 1 such that B = B^2.

The asymptotic values derived by Brightwell et al. are relevant only for extremely large values of n. The average number of linear extensions (topological sorts) of a random partial order on {1,...,n} is n! S_n / N_n, where S_n is this sequence and N_n is sequence A001035

References

N. B. Hindman, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..12.

S. P. Avann, The lattice of natural partial orders, Aequationes Mathematicae 8 (1972), 95-102.

David Bevan, Gi-Sang Cheon and Sergey Kitaev, On naturally labelled posets and permutations avoiding 12-34, arXiv:2311.08023 [math.CO], 2023.

Graham Brightwell, Hans Jürgen Prömel and Angelika Steger, The average number of linear extensions of a partial order, Journal of Combinatorial Theory A73 (1996), 193-206.

S. R. Finch, Transitive relations, topologies and partial orders

S. R. Finch, Transitive relations, topologies and partial orders, June 5, 2003. [Cached copy, with permission of the author]

Joël Gay and Vincent Pilaud, The weak order on Weyl posets, arXiv:1804.06572 [math.CO], 2018.

L. H. Harper, The Range of a Steiner Operation, arXiv preprint arXiv:1608.07747 [math.CO], 2016.

N. Hindman and N. J. A. Sloane, Correspondence, 1981-1991

Florent Hivert and Nicolas M. Thiéry, Controlling the C3 Super Class Linearization Algorithm for Large Hierarchies of Classes, Order (2022).

Adam King, A. Laubmeier, K. Orans, and A. Godbole, Universal and Overlap Cycles for Posets, Words, and Juggling Patterns, arXiv preprint arXiv:1405.5938 [math.CO], 2014.

D. E. Knuth, POSETS, program for n = 10, 11, 12.

J.-G. Luque, L. Mignot and F. Nicart, Some Combinatorial Operators in Language Theory, arXiv preprint arXiv:1205.3371 [cs.FL], 2012. - N. J. A. Sloane, Oct 22 2012

Index entries for sequences related to posets

Example

a(3) = 7: {}, {1R2}, {1R3}, {2R3}, {1R2, 1R3}, {1R3, 2R3}, {1R2, 1R3, 2R3}.

Crossrefs

Cf. A000112, A001035.

Sequence in context: A363004 A008608 A028441 * A130715 A317723 A340005

Adjacent sequences: A006452 A006453 A006454 * A006456 A006457 A006458

Keyword

hard,more,nice,nonn

Author

N. J. A. Sloane

Extensions

Additional comments and more terms from Don Knuth, Dec 03 2001

Status

approved