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Cell[CellGroupData[{

Cell["multipol", "Subsubsection"],

Cell["\<\
take it apart :   multipol [ par,  v ]
get the 'runs' or 'bin-sizes\" of all the partitions p of n (= length par, \
here=5), with  v-length( p ) appended to it,
then take the multinomial of that,
and set to zero all elements from partitions that are  (( either longer than \
v or ))  shorter than par,
and finally, clip of the elements preceding the rank of par (so, only 'lower \
down' p count).\
\>", "Text"],

Cell["\<\
multipol[par_List,v_Integer]:=multi[Tr[par],v,rankpartition[par],Length[par]];\
\
\>", "Input",
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  FontWeight->"Plain",
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Cell[CellGroupData[{

Cell["multi[5,4,1,3]", "Input"],

Cell[BoxData[
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Cell[CellGroupData[{

Cell["Append[runs[#],(*v=*)4 -Length[#]]&/@Partitions[5]", "Input"],

Cell[BoxData[
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Cell[BoxData[
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Cell[CellGroupData[{

Cell["\<\
If[Length[#]> (*v=*)4 ||Length[#]< (*le=*)3,0,1] * \
Multinomial@@Append[runs[#],(*v=*)4 -Length[#]]&/@Partitions[5]\
\>", "Input"],

Cell[BoxData[
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}, Open  ]],

Cell[CellGroupData[{

Cell["\<\
If[Length[#]< (*le=*)3,0,1] * Multinomial@@Append[runs[#],(*v=*)4 \
-Length[#]]&/@Partitions[5]\
\>", "Input"],

Cell[BoxData[
    \({0, 0, 0, 12, 12, 4, 0}\)], "Output"]
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Cell["simpler, more compact, but not much faster:", "Text"],

Cell["\<\
multipol2[par_List,v_Integer]:=If[Length[#]< Length[par] ,0,1] * \
Multinomial@@Append[runs[#],v-Length[#]]&/@(Drop[#,-1+Tr@Position[#,par]]&@\
Partitions[Tr[par]])\
\>", "Input"]
}, Closed]],

Cell[CellGroupData[{

Cell["how it hangs together", "Subsubsection"],

Cell[TextData[StyleBox["schur[p_]:=Block[{le=Length[p],n=Tr[p]},Together[\
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q->Sequence[]/. w[i_Integer, r__]-> i w[r]  /. x[_]^(e_:1) ->e ;\n\
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Cell[TextData[StyleBox[" stanley[(p_)?PartitionQ,t_Integer] :=  Times@@( \
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Cell[TextData[{
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Cell[TextData[StyleBox["SSYT[par_List,v_Integer]:=Block[{solspace},solspace=\
space[par,v];If[Length[solspace]===1,List/@Flatten[solspace,1], \
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Cell[TextData[StyleBox["sagan[par_List,v_Integer]:=Block[{freq},freq=Map[ \
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Cell["\<\
 Many sets of symmetric functions of degree n in v variables can be defined.
One way is to start from the (n+v-1 choose v-1) compositions of n into v \
parts:\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
Compositions[5,3]
{Length[%],Binomial[5+3-1,3-1]}
Tr[ (Times@@(Array[x,3]^#))&/@%% ]
tow[%]\
\>", "Input"],

Cell[BoxData[
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        2}, {2, 2, 1}, {2, 3, 0}, {3, 0, 2}, {3, 1, 1}, {3, 2, 0}, {4, 0, 
        1}, {4, 1, 0}, {5, 0, 0}}\)], "Output"],

Cell[BoxData[
    \({21, 21}\)], "Output"],

Cell[BoxData[
    \(x[1]\^5 + x[1]\^4\ x[2] + x[1]\^3\ x[2]\^2 + x[1]\^2\ x[2]\^3 + 
      x[1]\ x[2]\^4 + x[2]\^5 + x[1]\^4\ x[3] + x[1]\^3\ x[2]\ x[3] + 
      x[1]\^2\ x[2]\^2\ x[3] + x[1]\ x[2]\^3\ x[3] + x[2]\^4\ x[3] + 
      x[1]\^3\ x[3]\^2 + x[1]\^2\ x[2]\ x[3]\^2 + x[1]\ x[2]\^2\ x[3]\^2 + 
      x[2]\^3\ x[3]\^2 + x[1]\^2\ x[3]\^3 + x[1]\ x[2]\ x[3]\^3 + 
      x[2]\^2\ x[3]\^3 + x[1]\ x[3]\^4 + x[2]\ x[3]\^4 + x[3]\^5\)], "Output"],

Cell[BoxData[
    \(3\ w[5] + 6\ w[3, 2] + 6\ w[4, 1] + 3\ w[2, 2, 1] + 
      3\ w[3, 1, 1]\)], "Output"]
}, Open  ]],

Cell["\<\
Another way is to start from the partitions of n,  zero-padded to v parts,  \
and consider all their permutations\
\>", "Text"],

Cell[CellGroupData[{

Cell["\<\
PadRight[#,3]&/@Reverse[TransposePartition/@Partitions[5,3] ]
Permutations/@%\
\>", "Input"],

Cell[BoxData[
    \({{5, 0, 0}, {4, 1, 0}, {3, 2, 0}, {3, 1, 1}, {2, 2, 1}}\)], "Output"],

Cell[BoxData[
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          0}, {2, 0, 3}, {0, 3, 2}, {0, 2, 3}}, {{3, 1, 1}, {1, 3, 1}, {1, 1, 
          3}}, {{2, 2, 1}, {2, 1, 2}, {1, 2, 2}}}\)], "Output"]
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Cell[CellGroupData[{

Cell["\<\
Tr[ (Times@@(Array[x,3]^#))&/@Flatten[%,1] ]
tow[%]\
\>", "Input"],

Cell[BoxData[
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      x[1]\^2\ x[2]\^2\ x[3] + x[1]\ x[2]\^3\ x[3] + x[2]\^4\ x[3] + 
      x[1]\^3\ x[3]\^2 + x[1]\^2\ x[2]\ x[3]\^2 + x[1]\ x[2]\^2\ x[3]\^2 + 
      x[2]\^3\ x[3]\^2 + x[1]\^2\ x[3]\^3 + x[1]\ x[2]\ x[3]\^3 + 
      x[2]\^2\ x[3]\^3 + x[1]\ x[3]\^4 + x[2]\ x[3]\^4 + x[3]\^5\)], "Output"],

Cell[BoxData[
    \(3\ w[5] + 6\ w[3, 2] + 6\ w[4, 1] + 3\ w[2, 2, 1] + 
      3\ w[3, 1, 1]\)], "Output"]
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Cell[" the Schur poly's :", "Text"],

Cell[CellGroupData[{

Cell["\<\
toz/@(PadRight[#,3]&/@Reverse[TransposePartition/@Partitions[5,3] ]) \
//ColumnForm\
\>", "Input"],

Cell[BoxData[
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Compositions[3,5]
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Cell[" counting SSYT with content \[Mu].", "Subsubsection"],

Cell["\<\
http://unapologetic.wordpress.com/2011/02/18/more-kostka-numbers/\
\>", "Text"],

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contents[ list ] counts the occurrences of 1 ..n in the list \
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Cell[BoxData[
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among the SSYT of shape \[Mu], only the first has contents \[Mu]\
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Cell["\<\
zoo is the set of all SSYT (with largest element at most mx)  of any shape  \
with Tr[\[Mu]]  parts\
\>", "Text"],

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the count of SSYT with contents \[Mu] within zoo equals the rankpartition[\
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\>", "Text"],

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