A268187 Triangle read by rows: T(n,k) is the number of no-leg partitions of n having Durfee square of size k (n >= 1, 1 <= k <= floor(sqrt(n))). Also, number of no-arm partitions of n having Durfee square of size k.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 4, 1, 1, 4, 2, 1, 5, 3, 1, 5, 4, 1, 6, 5, 1, 6, 7, 1, 7, 8, 1, 1, 7, 10, 1, 1, 8, 12, 2, 1, 8, 14, 3, 1, 9, 16, 5, 1, 9, 19, 6, 1, 10, 21, 9, 1, 10, 24, 11, 1, 11, 27, 15, 1, 11, 30, 18, 1, 1, 12, 33, 23, 1

Offset: 1

Author

Emeric Deutsch, Jan 29 2016

Keywords

• nonn
• tabf
• look

Comments

Given a partition P, the partition formed by the cells situated below the Durfee square of P is called the leg of P. Similarly, the partition formed by the cells situated to the right of the Durfee square of P is called the arm of P.

Examples

			T(9,2) = 3 because we have [7,2], [6,3], and [5,4].
Triangle begins:
  1;
  1;
  1;
  1, 1;
  1, 1;
  1, 2;
  1, 2;
  1, 3;
  1, 3, 1;
  1, 4, 1;
  1, 4, 2;
  1, 5, 3;
  1, 5, 4;
  1, 6, 5;
  1, 6, 7;
  1, 7, 8, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..1000, flattened
• Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities

Crossrefs

Cf. A003114, A115994, A268188, A327710, A328346.

Programs

Maple

G := add(t^k*x^(k^2)/mul(1-x^i, i = 1 .. k), k = 0 .. 80): Gser := simplify(series(G, x = 0,40)): for n to 35 do P[n] := sort(coeff(Gser, x, n)) end do: for n to 35 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
T:= (n, k)-> b(n-k^2, k):
seq(seq(T(n, k), k=1..floor(sqrt(n))), n=1..30); # Alois P. Heinz, Jan 30 2016

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; T[n_, k_] := b[n-k^2, k]; Table[T[n, k], {n, 1, 30}, {k, 1, Floor[Sqrt[n]]}] // Flatten (* Jean-François Alcover, Dec 10 2016 after Alois P. Heinz *)

PARI

T(n, k) = polcoef(1/prod(j=1, k, 1-x^j+x*O(x^n)), n-k*k);
tabf(nn) = for(n=1, nn, for(k=1, sqrtint(n), print1(T(n, k), ", ")); print) \\ Seiichi Manyama, Oct 14 2019

Formula

G.f.: G(t,x) = Sum_{k>=0} ( t^k*x^(k^2)/Product_{i=1..k} (1-x^i) ).

Sum_{k>=0} T(n,k) = A003114(n).

Sum_{k>=1} k * T(n,k) = A268188(n).

Sum_{k>=0} k! * T(n,k) = A327710(n). - Alois P. Heinz, Feb 25 2020