A337584 Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) that match the multiplicity multiset of some partition of n (n >= 1, 1 <= k <= n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 2, 4, 3, 2, 1, 1, 1, 1, 3, 2, 4, 3, 2, 1, 1, 1, 2, 2, 4, 4, 4, 3, 2, 1, 1, 1, 1, 2, 3, 5, 3, 5, 3, 2, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 1, 2, 3, 5, 5, 8, 5, 5, 3, 2, 1, 1, 1, 2, 2, 4, 5, 7, 8, 8, 5, 5, 3

Offset

1,8

Comments

The relevant partitions of n have exactly k parts.

Column k is k-periodic from n = k*(k+1)/2.

Links

Alois P. Heinz, Rows n = 1..125 (first 71 rows from Álvar Ibeas)

Álvar Ibeas, First 30 rows

Formula

If k > (2*n+1)/3, T(n, k) = A088887(n - k).

If n >= k*(k+1)/2, T(n, k) = Sum_{d | gcd(n, k)} A000837(k/d).

T(n, k) = A000041(k) iff k|n and n >= k*(k+1)/2.

Example

There is no partition of 5 with multiplicity multiset (3) or (1, 1, 1).
Indeed, both (2 = A008284(5, 3)) partitions of 5 into 3 parts (namely, (3, 1, 1) and (2, 2, 1)) have multiplicities (2, 1). Therefore, T(5, 3) = 1.
Triangle begins:
  k:  1 2 3 4 5 6 7 8 9 10
      --------------------
n=1:  1
n=2:  1 1
n=3:  1 1 1
n=4:  1 2 1 1
n=5:  1 1 1 1 1
n=6:  1 2 3 2 1 1
n=7:  1 1 2 2 2 1 1
n=8:  1 2 2 4 3 2 1 1
n=9:  1 1 3 2 4 3 2 1 1
n=10: 1 2 2 4 4 4 3 2 1 1

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n=0, {[]}, {[n]}),
     {b(n, i-1)[], seq(map(x-> sort([x[], j]), b(n-i*j, i-1))[], j=1..n/i)})
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(add(x^add(i, i=t), t=b(n$2))):
seq(T(n), n=1..20);  # Alois P. Heinz, Aug 17 2021

Crossrefs

Cf. A000041, A008284, A088887 (row sums).

T(2n,n) gives A344680.

Sequence in context: A261794 A328929 A098744 * A273975 A025429 A325561

Adjacent sequences: A337581 A337582 A337583 * A337585 A337586 A337587

Keyword

nonn,tabl

Author

Álvar Ibeas, Sep 02 2020

Status

approved