[seqfan] Re: more poset enumerations

[Brendan McKay]

Hi Richard,

I take it from these tables that by "number of arcs" you mean the number of
arcs in the Hasse diagram of the poset.  The poset itself is the transitive
closure of the Hasse diagram so it has up to binomial(n,2) arcs.

I have the code to check these and will do so soon.

Brendan.

On 15/3/21 3:21 am, Richard J. Mathar wrote:
> We now have A342500 and A342501 (posets by maximum chain length),
> and apparently useful would also be the following 3 triangles (all values need TBC):
>
> ------
> The number of labeled connected posets enumerated by number of arcs
> (row sums A001927):
>
> 1: 1 (1)
> 2: 0 2 (2)
> 3: 0 0 12 (12)
> 4: 0 0 0 128 18 (146)
> 5: 0 0 0 0 2000 960 100 (3060)
> 6: 0 0 0 0 0 41472 43320 15000 1710 140 (101642)
> 7: 0 0 0 0 0 0 1075648 1985760 1453200 490560 90594 10080 770 (5106612)
>
> ------
> The number of labeled posets enumerated by number of arcs:
> (row sums A001035, unlabeled are A342447):
>
> 1: 1 (1)
> 2: 1 2 (3)
> 3: 1 6 12 (19)
> 4: 1 12 60 128 18 (219)
> 5: 1 20 180 880 2090 960 100 (4231)
> 6: 1 30 420 3480 17550 47772 43920 15000 1710 140 (130023)
> 7: 1 42 840 10360 84630 452004 1428868 2094960 1465170 491540 90594 10080 770 (6129859)
>
> ------
>
> The number of connected unlabeled posets enumerated by number of arcs:
> (row sums A000608):
> 1: 1 (1)
> 2: 0 1 (1)
> 3: 0 0 3 (3)
> 4: 0 0 0 8 2 (10)
> 5: 0 0 0 0 27 12 5 (44)
> 6: 0 0 0 0 0 91 87 45 12 3 (238)
> 7: 0 0 0 0 0 0 350 532 475 201 71 14 7 (1650)
>
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