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A342908
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Irregular triangular array of coefficients of the cd-index of the symmetric group S_n (or Boolean algebra B_n), n>=1.
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1
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1, 1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 4, 1, 4, 9, 9, 4, 12, 10, 12, 1, 5, 14, 19, 14, 5, 25, 35, 42, 18, 35, 25, 34, 1, 6, 20, 34, 34, 20, 6, 44, 84, 100, 72, 140, 100, 28, 72, 84, 44, 136, 112, 112, 136, 1, 7, 27, 55, 69, 55, 27, 7, 70, 168, 198, 196, 378, 268, 126, 324, 378, 198, 40, 126, 196, 168, 70, 364, 504, 504, 612, 256, 420, 504, 256, 504, 364, 496
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OFFSET
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1,6
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COMMENTS
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Equivalently, the cd-index of the face lattice of the (n-1)-dimensional simplex.
These polynomials encode the numbers given in A335845. The row lengths are A000045(n). The row sums are A000111(n).
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Vol I, second edition, page 54 and section 3.17.
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LINKS
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EXAMPLE
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1,
1,
1, 1,
1, 2, 2,
1, 3, 5, 3, 4,
1, 4, 9, 9, 4, 12, 10, 12,
1, 5, 14, 19, 14, 5, 25, 35, 42, 18, 35, 25, 34
The terms of the polynomials are ordered lexicographically. For example, row 5 represents the polynomial: c^4 + 3c^2d + 5cdc + 3dc^2 + 4d^2.
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MATHEMATICA
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Join[{{1}}, Table[h[list_]:=(-1)^(Length[list]+1)Apply[Multinomial, list]; g[S_]:=Abs[Total[Map[h, Map[Differences, Map[Prepend[#, 0]&, Map[Append[#, nn]&, Subsets[S]]]]]]]; rhs=Drop[Map[g, Subsets[Range[nn-1]]], -2^(nn-2)]; Clear[c, d]; fib=Reverse[Map[#/.{2->d, 1->c}&, Level[Map[Permutations, IntegerPartitions[nn-1, nn-1, {1, 2}]], {2}]]]; c:={{a}, {b}}; d:={{a, b}, {b, a}}; f[list1_, list2_]:=Level[Table[Table[Join[list1[[i]], list2[[k]]], {i, 1, Length[list1]}], {k, 1, Length[list2]}], {2}]; rr=Table[Map[Fold[f, #[[1]], Rest[#]]&, fib][[i]]->Subscript[x, i], {i, 1, Fibonacci[nn]}]; eqn[list_]:=Total[Select[Map[Fold[f, #[[1]], Rest[#]]&, fib], MemberQ[#, list]&]/.rr]==FromDigits[Part[rhs, Flatten[Position[charmon=Drop[Map[Table[If[MemberQ[#, i], b, a], {i, 1, nn-1}]&, Subsets[Range[nn-1]]], -2^(nn-2)], list]]]]; charmon=Drop[Map[Table[If[MemberQ[#, i], b, a], {i, 1, nn-1}]&, Subsets[Range[nn-1]]], -2^(nn-2)]; Table[Subscript[x, i], {i, 1, Fibonacci[nn]}]/.Flatten[Solve[Map[eqn[#]&, charmon]]], {nn, 2, 10}]]//Flatten
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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