A093784 Triangle T(n,k) read by rows in which n-th row gives the hook products of the partitions of n.
1, 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
Offset: 0
Examples
Triangle T(n,k) begins: 1; 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; ...
Links
- Alois P. Heinz, Rows n = 0..36, flattened
Programs
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Maple
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, [])])[]: seq(T(n), n=0..10); # Alois P. Heinz, Jan 07 2013 -
Mathematica
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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