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

Author

Emeric Deutsch, May 17 2004

Keywords

• nonn
• look
• tabf

Comments

Row n consists of the numbers n!/A060240(n,k) written in reverse order.

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

Crossrefs

Cf. A060240, A093716, A093764.

Programs

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 *)