A131632 Triangle T(n,k) read by rows = number of partitions of n-set into k blocks with distinct sizes, k = 1..A003056(n).

1, 1, 1, 3, 1, 4, 1, 15, 1, 21, 60, 1, 63, 105, 1, 92, 448, 1, 255, 2016, 1, 385, 4980, 12600, 1, 1023, 15675, 27720, 1, 1585, 61644, 138600, 1, 4095, 155155, 643500, 1, 6475, 482573, 4408404, 1, 16383, 1733550, 12687675, 37837800, 1, 26332, 4549808, 60780720

Offset

1,4

Comments

Row sums = A007837.

Sum k! * T(n,k) = A032011.

Sum k * T(n,k) = A131623. - Geoffrey Critzer, Aug 30 2012.

T(n,k) is also the number of words w of length n over a k-ary alphabet {a1,a2,...,ak} with #(w,a1) > #(w,a2) > ... > #(w,ak) > 0, where #(w,x) counts the letters x in word w. T(5,2) = 15: aaaab, aaaba, aaabb, aabaa, aabab, aabba, abaaa, abaab, ababa, abbaa, baaaa, baaab, baaba, babaa, bbaaa. - Alois P. Heinz, Jun 21 2013

Links

Alois P. Heinz, Rows n = 1..500, flattened

Formula

E.g.f.: Product_{n>=1} (1+y*x^n/n!).

T(A000217(n),n) = A022915(n). - Alois P. Heinz, Jul 03 2018

Example

Triangle T(n,k)begins:
  1;
  1;
  1,     3;
  1,     4;
  1,    15;
  1,    21,      60;
  1,    63,     105;
  1,    92,     448;
  1,   255,    2016;
  1,   385,    4980,    12600;
  1,  1023,   15675,    27720;
  1,  1585,   61644,   138600;
  1,  4095,  155155,   643500;
  1,  6475,  482573,  4408404;
  1, 16383, 1733550, 12687675, 37837800;
  ...

Maple

b:= proc(n, i, t, v) option remember; `if`(t=1, 1/(n+v)!,
      add(b(n-j, j, t-1, v+1)/(j+v)!, j=i..n/t))
    end:
T:= (n, k)->`if`(k*(k+1)/2>n, 0, n!*b(n-k*(k+1)/2, 0, k, 1)):
seq(seq(T(n, k), k=1..floor(sqrt(2+2*n)-1/2)), n=1..20);
# Alois P. Heinz, Jun 21 2013
# second Maple program:
b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0,
      `if`(n=0, 1, b(n, i-1)+binomial(n, i)*
       expand(x*b(n-i, min(n-i, i-1)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Sep 27 2019

Mathematica

nn=10; p=Product[1+y x^i/i!, {i, 1, nn}]; Range[0, nn]! CoefficientList[ Series[p, {x, 0, nn}], {x, y}]//Grid  (* Geoffrey Critzer, Aug 30 2012 *)

Crossrefs

Columns k=1-10 give: A000012, A272514, A272515, A272516, A272517, A272518, A272519, A272520, A272521, A272522.

Cf. A000217, A003056, A007837, A022915, A032011, A131623, A226873, A226874, A327803.

Sequence in context: A262072 A321743 A341766 * A051348 A253828 A190445

Adjacent sequences: A131629 A131630 A131631 * A131633 A131634 A131635

Keyword

nonn,tabf

Author

Vladeta Jovovic, Sep 04 2007

Status

approved