The command IsAntichain(P, L) checks whether the elements in L represent an antichain in the partially ordered set P.

$\mathrm{with}\left(\mathrm{PartiallyOrderedSets}\right)\:$

$T≔\left\{3\,4\,5\,6\,7\,8\,9\right\}\:$

$V≔\varnothing \&colon;$$\mathrm{leq}≔\mathrm{`<=`}\&colon;$$\mathrm{empty\_poset}≔\mathrm{PartiallyOrderedSet}\left(V\&comma;\mathrm{leq}\right)$

$\mathrm{empty\_poset}≔\mathrm{<\; a\; poset\; with\; 0\; elements\; >}$

$\mathrm{IsAntichain}\left(\mathrm{empty\_poset}\&comma;\left[0\right]\right)$

$\mathrm{true}$

$S≔\left\{1\&comma;2\&comma;3\&comma;4\&comma;5\right\}\&colon;$$\mathrm{poset1}≔\mathrm{PartiallyOrderedSet}\left(S\&comma;\mathrm{leq}\right)$

$\mathrm{poset1}≔\mathrm{<\; a\; poset\; with\; 5\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset1}\right)$

$\mathrm{IsAntichain}\left(\mathrm{poset1}\&comma;\left[1\&comma;3\&comma;5\right]\right)$

$\mathrm{false}$

$\mathrm{IsAntichain}\left(\mathrm{poset1}\&comma;\left[3\right]\right)$

$\mathrm{true}$

$\mathrm{adjList5}≔\mathrm{map2}\left(\mathrm{map}\&comma;\mathrm{`+`}\&comma;\mathrm{Array}\left(\left[\left\{1\&comma;4\&comma;7\right\}\&comma;\left\{2\&comma;6\right\}\&comma;\left\{3\right\}\&comma;\left\{4\right\}\&comma;\left\{5\right\}\&comma;\left\{6\right\}\&comma;\left\{7\right\}\right]\right)\&comma;2\right)$

$\mathrm{adjList5}≔\left[\begin{array}{ccccccc}\left\{3\&comma;6\&comma;9\right\}& \left\{4\&comma;8\right\}& \left\{5\right\}& \left\{6\right\}& \left\{7\right\}& \left\{8\right\}& \left\{9\right\}\end{array}\right]$

$\mathrm{poset5}≔\mathrm{PartiallyOrderedSet}\left(\mathrm{convert}\left(T\&comma;'\mathrm{list}'\right)\&comma;\mathrm{adjList5}\right)$

$\mathrm{poset5}≔\mathrm{<\; a\; poset\; with\; 7\; elements\; >}$

$\mathrm{DrawGraph}\left(\mathrm{poset5}\right)$

$\mathrm{IsAntichain}\left(\mathrm{poset5}\&comma;\left[5\&comma;3\&comma;7\right]\right)$

$\mathrm{true}$

$\mathrm{IsAntichain}\left(\mathrm{poset5}\&comma;\left[4\&comma;8\right]\right)$

$\mathrm{false}$

Richard P. Stanley: Enumerative Combinatorics 1. 1997, Cambridge Studies in Advanced Mathematics. Vol. 49. Cambridge University Press.

The PartiallyOrderedSets[IsAntichain] command was introduced in Maple 2025.

For more information on Maple 2025 changes, see Updates in Maple 2025.