A364911 Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.

1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12

Offset

0,5

Comments

Also the number of ways to write any number up to n as a positive linear combination of a strict integer partition of k.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)

Formula

G.f.: A(x,y) = (1/(1 - x)) * Product_{k>=1} (1 - y^k + y^k/(1 - x^k)). - Andrew Howroyd, Jan 11 2024

Example

Triangle begins:
  1
  1  1
  1  2  1
  1  3  1  2
  1  4  2  3  2
  1  5  2  5  3  3
  1  6  3  8  4  4  4
  1  7  3 11  6  6  6  5
  1  8  4 14  9  8 10  7  6
  1  9  4 19 11 11 14 11  9  8
  1 10  5 23 14 15 21 15 14 11 10
  1 11  5 28 17 19 28 22 20 17 15 12
  1 12  6 34 21 22 40 28 28 24 24 17 15
  1 13  6 40 25 27 50 38 37 34 35 27 22 18
  1 14  7 46 29 32 65 49 50 43 51 38 35 26 22
  1 15  7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
    .    1           2     3         4       5
         1+1         2+2   1+2       1+3     1+4
         1+1+1             1+1+2     1+1+3   2+3
         1+1+1+1           1+1+1+2
         1+1+1+1+1         1+2+2
Row n = 5 counts the following positive linear combinations:
  .  1*1  1*2  1*3      1*4      1*5
     2*1  2*2  1*2+1*1  1*3+1*1  1*3+1*2
     3*1       1*2+2*1  1*3+2*1  1*4+1*1
     4*1       1*2+3*1
     5*1       2*2+1*1

Mathematica

Table[Length[Select[Array[IntegerPartitions, n+1, 0, Join], Total[Union[#]]==k&]], {n, 0, 9}, {k, 0, n}]

Prog

(PARI) T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024

Crossrefs

Column n = k is A000009.

Column k = 0 is A000012.

Column k = 1 is A000027.

Row sums are A000070.

Column k = 2 is A008619.

Columns are partial sums of columns of A116861.

Column k = 3 appears to be the partial sums of A137719.

Diagonal n = 2k is A364910.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A114638 counts partitions where (length) = (sum of distinct parts).

A116608 counts partitions by number of distinct parts.

A364350 counts combination-free strict partitions, complement A364839.

Cf. A002865, A066328, A179009, A236912, A237113, A237667, A364912, A364913, A364915, A364916, A365002, A365004.

Sequence in context: A327856 A175488 A309914 * A115758 A228351 A124734

Adjacent sequences: A364908 A364909 A364910 * A364912 A364913 A364914

Keyword

nonn,tabl

Author

Gus Wiseman, Aug 27 2023

Status

approved