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A093784
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Triangle T(n,k) read by rows in which n-th row gives the hook products of the partitions of n.
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4
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1, 2, 2, 3, 6, 6, 8, 8, 12, 24, 24, 20, 24, 24, 30, 30, 120, 120, 45, 72, 72, 80, 80, 144, 144, 144, 144, 720, 720, 144, 144, 240, 240, 252, 336, 336, 360, 360, 360, 360, 840, 840, 5040, 5040, 448, 576, 576, 630, 630, 720, 720, 960, 1152, 1152, 1440, 1440, 1920
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OFFSET
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1,2
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COMMENTS
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Row n consists of the numbers n!/A060240(n,k) written in reverse order.
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LINKS
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EXAMPLE
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1;
2, 2;
3, 6, 6;
8, 8, 12, 24, 24;
20, 24, 24, 30, 30, 120, 120;
45, 72, 72, 80, 80, 144, 144, 144, 144, 720, 720;
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MAPLE
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H:=proc(pa) local F, j, p, Q, i, col, a, A: F:=proc(x) local i, ct: ct:=0: for i from 1 to nops(x) do if x[i]>1 then ct:=ct+1 else fi od: ct; end: for j from 1 to nops(pa) do p[1][j]:=pa[j] od: Q[1]:=[seq(p[1][j], j=1..nops(pa))]: for i from 2 to pa[1] do for j from 1 to F(Q[i-1]) do p[i][j]:=Q[i-1][j]-1 od: Q[i]:=[seq(p[i][j], j=1..F(Q[i-1]))] od: for i from 1 to pa[1] do col[i]:=[seq(Q[i][j]+nops(Q[i])-j, j=1..nops(Q[i]))] od: a:=proc(i, j) if i<=nops(Q[j]) and j<=pa[1] then Q[j][i]+nops(Q[j])-i else 1 fi end: A:=matrix(nops(pa), pa[1], a): product(product(A[m, n], n=1..pa[1]), m=1..nops(pa)); end: with(combinat): rev:=proc(a) [seq(a[nops(a)+1-i], i=1..nops(a))] end: seq(sort([seq(H(rev(partition(s)[q])), q=1..numbpart(s))]), s=1..9);
# second Maple program:
h:= proc(l) local n; n:= nops(l); mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= n-> sort([g(n, n, [])])[]:
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MATHEMATICA
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h[l_List] := With[{n = Length[l]}, Product[Product[1+l[[i]]-j+Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[1, 1, {}] = {1}; g[n_, i_, l_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, Flatten @ Table[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Sort[g[n, n, {}]]; Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 29 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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