A389656 Triangle read by rows: T(n,k) is the number of partitions of n into distinct parts containing the number of parts with rank k.

1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0

Offset

1

Comments

The rank of an integer partition is the size of the largest part minus the number of parts. All partitions of this kind have rank >= 0.

Links

Table of n, a(n) for n=1..91.

Formula

G.f.: Sum_{i>0} ( q^(i*(i+1)/2) * (q)_{i-1} * Sum_{m=1..i} (q^(m*(m-1)) * t^(m-1) / ((q)_{i-m}*(q)_{m-1}*(q*t)_{m-1}) ) ) where (a)_k = Product_{i>=0..k-1} (1-a*q^i).

Example

Triangle begins:
      k=0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19
  n=1  [1]
  n=2  [0, 0]
  n=3  [1, 0, 0]
  n=4  [0, 0, 0, 0]
  n=5  [0, 1, 0, 0, 0]
  n=6  [1, 0, 1, 0, 0, 0]
  n=7  [0, 0, 0, 1, 0, 0, 0]
  n=8  [0, 1, 0, 0, 1, 0, 0, 0]
  n=9  [0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=10 [1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=11 [0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=12 [0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=13 [0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=14 [0, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=15 [1, 0, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=16 [0, 0, 1, 1, 2, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=17 [0, 1, 1, 1, 1, 2, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=18 [0, 1, 2, 2, 1, 2, 2, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=19 [0, 1, 1, 3, 2, 1, 2, 2, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  n=20 [0, 1, 1, 2, 4, 2, 2, 2, 2, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0]
  ...
T(9,1) = 1 counts: (4,3,2).
T(9,2) = 1 counts: (5,3,1).
T(9,5) = 1 counts: (7,2).

Prog

(PARI)
qs(a, q, n) = {prod(k=0, n-1, 1-a*q^k)}
A_qt(rowmax) = { my(N = rowmax+1, q='q+O('q^N), g = sum(i=1, floor(1/2+sqrt(2*N)), q^(i*(i+1)/2) * qs(q, q, i-1) * sum(m=1, i, t^(m-1) * q^(m*(m-1)) / ( qs(q, q, i-m)*qs(q, q, m-1)* qs(t*q, q, m-1)) ) ) ); vector(N-1, n, my(r =Vecrev(polcoeff(g, n))); if(#r<n, concat(r, vector(n-#r, i, 0)), r))}

Crossrefs

Cf. A000009, A010054 (column k=0), A063995, A240855 (row sums), A240861, A389344.

Sequence in context: A154282 A359552 A224877 * A178600 A373977 A359170

Adjacent sequences: A389653 A389654 A389655 * A389657 A389658 A389659

Keyword

nonn,easy,tabl

Author

John Tyler Rascoe, Oct 09 2025

Status

approved