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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). 13
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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).

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

EXAMPLE

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;

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

p:= (f, g)-> zip((x, y)-> x+y, f, g, 0):

b:= proc(n, i) option remember; local g;

      if n=0 then [1] elif i=1 then [1, n]

    else g:= `if`(i>n, [0], b(n-i, i));

         p(p(b(n, i-1), g), [0$i, g[1]])

      fi

    end:

T:= proc(n) local l; l:= subsop (1=NULL, b(n, n));

       seq(l[i]*i, i=1..n)

    end:

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-Fran├žois Alcover, Jan 17 2014 *)

CROSSREFS

Row sums yield A066186.

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

Sequence in context: A043262 A130860 A161535 * A131817 A214430 A138232

Adjacent sequences:  A138782 A138783 A138784 * A138786 A138787 A138788

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, May 16 2008

STATUS

approved

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Last modified February 22 11:50 EST 2020. Contains 332135 sequences. (Running on oeis4.)