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%I #14 Aug 26 2016 16:55:01
%S 1,6,42,335,2956,28468,296540
%N Number of pointed multiset partitions of normal pointed multisets of weight n.
%C A pointed multiset k[1...k...n] with point k is normal if its entries [1...k...n] span an initial interval of positive integers. Pointed multiset partitions are triangles (or compositions) in the multiorder of pointed multisets.
%H Gus Wiseman, <a href="https://docs.google.com/document/d/1m0s6DGTBkDW9gvMuFmJHvy6oLGRAbQ7okAZcOPZawp0/pub">Comcategories and Multiorders</a>
%e The a(2) = 6 pointed multiset partitions are:
%e 1[1[11]],1[1[1]1[1]],
%e 1[1[12]],1[1[1]2[2]],
%e 2[2[12]],2[1[1]2[2]].
%e The a(3) = 42 pointed multiset partitions are:
%e 1[1[111]],1[1[1]1[11]],1[1[11]1[1]],1[1[1]1[1]1[1]],
%e 1[1[122]],1[1[1]2[22]],1[1[12]2[2]],1[1[1]2[2]2[2]],
%e 2[2[122]],2[1[1]2[22]],2[1[12]2[2]],2[2[2]2[12]],2[2[12]2[2]],2[1[1]2[2]2[2]],
%e 1[1[112]],1[1[1]1[12]],1[1[1]2[12]],1[1[11]2[2]],1[1[12]1[1]],1[1[1]1[1]2[2]],
%e 2[2[112]],2[1[1]2[12]],2[1[11]2[2]],2[1[1]1[1]2[2]],
%e 1[1[123]],1[1[1]2[23]],1[1[1]3[23]],1[1[12]3[3]],1[1[13]2[2]],1[1[1]2[2]3[3]],
%e 2[2[123]],2[1[1]2[23]],2[1[13]2[2]],2[2[2]3[13]],2[2[12]3[3]],2[1[1]2[2]3[3]],
%e 3[3[123]],3[1[1]3[23]],3[1[12]3[3]],3[2[2]3[13]],3[2[12]3[3]],3[1[1]2[2]3[3]].
%t ReplaceListRepeated[forms_List, rerules_List] :=
%t Union[Flatten[
%t FixedPointList[
%t Function[preforms,
%t Union[Flatten[ReplaceList[#, rerules] & /@ preforms, 1]]],
%t forms], 1]]
%t pointedPartitions[JIX[r_, b_List?OrderedQ]] /; MemberQ[b, r] :=
%t Cases[ReplaceListRepeated[{Z[Y[JIX[r, {r}]],
%t Y @@ DeleteCases[b, r, 1, 1]]}, {Z[Y[sof___, JIX[w_, t_]],
%t Y[for___, x_, aft___]] /; OrderedQ[{w, x}] :>
%t Z[Y[sof, JIX[w, t], JIX[x, {x}]], Y[for, aft]],
%t Z[Y[JIX[w_, t_], soa___], Y[for___, x_, aft___]] /;
%t OrderedQ[{x, w}] :>
%t Z[Y[JIX[x, {x}], JIX[w, t], soa], Y[for, aft]],
%t Z[Y[sof___, JIX[w_, {tof__}]], Y[for___, x_, aft___]] :>
%t Z[Y[sof, JIX[w, Sort[{tof, x}]]], Y[for, aft]],
%t Z[Y[JIX[w_, {tof__}], soa___], Y[for___, x_, aft___]] :>
%t Z[Y[JIX[w, Sort[{tof, x}]], soa], Y[for, aft]]}],
%t Z[Y[pts__], Y[]] :> JIX[r, {pts}]];
%t allnormpms[n_Integer] :=
%t Join @@ Function[s,
%t JIX[#, Array[Count[s, y_ /; y <= #] + 1 &, n]] & /@
%t Range[Length[s] + 1]] /@ Subsets[Range[n - 1] + 1];
%t Join @@ pointedPartitions /@ allnormpms[3] /.
%t JIX -> Apply(* to construct the example *)
%t Array[Plus @@ (Length[pointedPartitions[#]] & /@
%t allnormpms[#]) &, 7](* to compute the sequence *)
%Y Cf. A185298, A080108, A276024.
%K nonn,more
%O 1,2
%A _Gus Wiseman_, Sep 26 2015
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