OFFSET
0,7
COMMENTS
Sum_{k>=1} T(n,k)^2 = n!. - R. J. Mathar, May 09 2013
From Emeric Deutsch, Oct 31 2014: (Start)
Number of entries in row n = A000041(n) = number of partitions of n.
Sum of entries in row n = A000085(n).
Largest (= last) entry in row n = A003040(n).
The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End)
REFERENCES
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New York.
LINKS
Alois P. Heinz, Rows n = 0..36, flattened
J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group Canad. J. Math, 6:316-324, 1954. See Theorem 1, p. 318.
EXAMPLE
Triangle begins:
1;
1;
1, 1;
1, 1, 2;
1, 1, 2, 3, 3;
1, 1, 4, 4, 5, 5, 6;
...
MAPLE
h:= proc(l) local n; n:= nops(l); add(i, i=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]}, Total[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[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, {}]]; T[1] = {1};
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)
PROG
(Magma) CharacterTable(SymmetricGroup(6)); // (say)
CROSSREFS
Maximal entry in each row gives A003040.
KEYWORD
AUTHOR
N. J. A. Sloane, Mar 21 2001
EXTENSIONS
More terms from Vladeta Jovovic, May 20 2003
STATUS
approved