A262671 Number of pointed multiset partitions of normal pointed multisets of weight n.

1, 6, 42, 335, 2956, 28468, 296540

Offset: 1

Author

Gus Wiseman, Sep 26 2015

Keywords

• nonn
• more

Comments

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.

Examples

			The a(2) = 6 pointed multiset partitions are:
1[1[11]],1[1[1]1[1]],
1[1[12]],1[1[1]2[2]],
2[2[12]],2[1[1]2[2]].
The a(3) = 42 pointed multiset partitions are:
1[1[111]],1[1[1]1[11]],1[1[11]1[1]],1[1[1]1[1]1[1]],
1[1[122]],1[1[1]2[22]],1[1[12]2[2]],1[1[1]2[2]2[2]],
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]],
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]],
2[2[112]],2[1[1]2[12]],2[1[11]2[2]],2[1[1]1[1]2[2]],
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]],
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]],
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]].
		

Links

• Gus Wiseman, Comcategories and Multiorders

Crossrefs

Cf. A185298, A080108, A276024.

Programs

Mathematica

ReplaceListRepeated[forms_List, rerules_List] :=
Union[Flatten[
   FixedPointList[
    Function[preforms,
     Union[Flatten[ReplaceList[#, rerules] & /@ preforms, 1]]],
    forms], 1]]
pointedPartitions[JIX[r_, b_List?OrderedQ]] /; MemberQ[b, r] :=
  Cases[ReplaceListRepeated[{Z[Y[JIX[r, {r}]],
      Y @@ DeleteCases[b, r, 1, 1]]}, {Z[Y[sof___, JIX[w_, t_]],
        Y[for___, x_, aft___]] /; OrderedQ[{w, x}] :>
      Z[Y[sof, JIX[w, t], JIX[x, {x}]], Y[for, aft]],
     Z[Y[JIX[w_, t_], soa___], Y[for___, x_, aft___]] /;
       OrderedQ[{x, w}] :>
      Z[Y[JIX[x, {x}], JIX[w, t], soa], Y[for, aft]],
     Z[Y[sof___, JIX[w_, {tof__}]], Y[for___, x_, aft___]] :>
      Z[Y[sof, JIX[w, Sort[{tof, x}]]], Y[for, aft]],
     Z[Y[JIX[w_, {tof__}], soa___], Y[for___, x_, aft___]] :>
      Z[Y[JIX[w, Sort[{tof, x}]], soa], Y[for, aft]]}],
   Z[Y[pts__], Y[]] :> JIX[r, {pts}]];
allnormpms[n_Integer] :=
  Join @@ Function[s,
     JIX[#, Array[Count[s, y_ /; y <= #] + 1 &, n]] & /@
      Range[Length[s] + 1]] /@ Subsets[Range[n - 1] + 1];
Join @@ pointedPartitions /@ allnormpms[3] /.
JIX -> Apply(* to construct the example *)
Array[Plus @@ (Length[pointedPartitions[#]] & /@
     allnormpms[#]) &, 7](* to compute the sequence *)