A138785 Triangle read by rows: T(n,k) is the number of hook lengths equal to k among all hook lengths of all partitions of n (1 <= k <= n).

1, 2, 2, 4, 2, 3, 7, 6, 3, 4, 12, 8, 6, 4, 5, 19, 16, 12, 8, 5, 6, 30, 22, 18, 12, 10, 6, 7, 45, 38, 27, 24, 15, 12, 7, 8, 67, 52, 45, 32, 25, 18, 14, 8, 9, 97, 82, 63, 52, 40, 30, 21, 16, 9, 10, 139, 112, 93, 72, 60, 42, 35, 24, 18, 10, 11, 195, 166, 135, 112, 85, 72, 49, 40, 27, 20, 11, 12

Offset: 1

Author

Emeric Deutsch, May 16 2008

Keywords

• nonn
• tabl

Comments

T(n,k) is also the sum of all parts of size k in all partitions of n. - Omar E. Pol, Feb 24 2012

T(n,k) is also the sum of all k's that are divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. - Omar E. Pol, Feb 05 2021

Examples

			T(4,2) = 6 because for the partitions (4), (3,1), (2,2), (2,1,1), (1,1,1,1) of n=4 the hook length multi-sets are {4,3,2,1}, {4,2,1,1}, {3,2,2,1}, {4,1,2,1}, {4,3,2,1}, respectively, containing altogether six 2's.
Triangle starts:
   1;
   2,  2;
   4,  2,  3;
   7,  6,  3,  4;
  12,  8,  6,  4,  5;
  19, 16, 12,  8,  5,  6;
  30, 22, 18, 12, 10,  6,  7;
  45, 38, 27, 24, 15, 12,  7,  8;
  67, 52, 45, 32, 25, 18, 14,  8, 9;
  97, 82, 63, 52, 40, 30, 21, 16, 9, 10;
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened
• R. Bacher and L. Manivel, Hooks and powers of parts in partitions, Sem. Lotharingien de Combinatoire, 47, 2002, B47d.
• Guo-Niu Han, An explicit expansion formula for the powers of the Euler product in terms of partition hook lengths, arXiv:0804.1849v3 [math.CO] 9 May 2008 (p. 24).

Crossrefs

Row sums yield A066186.

Cf. A000041, A066186, A000070, A066183, A066633.

Programs

Maple

g:=sum(k*x^k*t^k/((1-x^k)*(product(1-x^m,m=1..50))),k=1..50): gser:= simplify(series(g,x=0,15)): for n to 12 do P[n]:= sort(coeff(gser,x,n)) end do: for n to 12 do seq(coeff(P[n],t,j),j=1..n) end do; # yields sequence in triangular form
# second program:
b:= proc(n, i) option remember; `if`(n=0, [1],
     `if`(i=1, [1, n], (p-> (g-> p(p(b(n, i-1), g),
      [0$i, g[1]]))(`if`(i>n, [0], b(n-i, i))))(
      (f, g)-> zip((x, y)-> x+y, f, g, 0))))
    end:
T:= n-> (l-> seq(l[i+1]*i, i=1..n))(b(n$2)):
seq(T(n), n=1..14);  # Alois P. Heinz, Mar 22 2012

Mathematica

max = 12; s = Series[Sum[k*t^k*x^k/((1 - x^k)*Product[1 - x^m, {m, 1, max}]), {k, 1, max}] , {x, 0, max}, {t, 0, max}] // Normal; t[n_, k_] := SeriesCoefficient[s, {x, 0, n}, {t, 0, k}]; Table[t[n, k], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014 *)
Table[Count[Flatten@IntegerPartitions@n, k]*k, {n, 12}, {k, n}] // Flatten (* Robert Price, Jun 15 2020 *)

Formula

T(n,1) = A000070(n-1).

Sum_{k=1..n} k*T(n,k) = A066183(n).

G.f.: Sum(k*t^k*x^k/[(1-x^k)*Product(1-x^m, m=1..infinity)], k=1..infinity).

T(n,k) = k*A066633(n,k).

T(n,k) = Sum_{j=1..n} A207383(j,k). - Omar E. Pol, May 02 2012