A354234 Triangle read by rows where T(n,k) is the number of integer partitions of n with at least one part divisible by k.

1, 2, 1, 3, 1, 1, 5, 3, 1, 1, 7, 4, 2, 1, 1, 11, 7, 4, 2, 1, 1, 15, 10, 6, 3, 2, 1, 1, 22, 16, 9, 6, 3, 2, 1, 1, 30, 22, 14, 8, 5, 3, 2, 1, 1, 42, 32, 20, 13, 8, 5, 3, 2, 1, 1, 56, 44, 29, 18, 12, 7, 5, 3, 2, 1, 1, 77, 62, 41, 27, 17, 12, 7, 5, 3, 2, 1, 1

Offset: 1

Author

Gus Wiseman, May 22 2022

Keywords

• nonn
• tabl

Comments

Also partitions of n with at least one part appearing k or more times. It would be interesting to have a bijective proof of this.

Examples

			Triangle begins:
   1
   2  1
   3  1  1
   5  3  1  1
   7  4  2  1  1
  11  7  4  2  1  1
  15 10  6  3  2  1  1
  22 16  9  6  3  2  1  1
  30 22 14  8  5  3  2  1  1
  42 32 20 13  8  5  3  2  1  1
  56 44 29 18 12  7  5  3  2  1  1
  77 62 41 27 17 12  7  5  3  2  1  1
For example, row n = 5 counts the following partitions:
  (5)      (32)    (32)   (41)  (5)
  (32)     (41)    (311)
  (41)     (221)
  (221)    (2111)
  (311)
  (2111)
  (11111)
At least one part appearing k or more times:
  (5)      (221)    (2111)   (11111)  (11111)
  (32)     (311)    (11111)
  (41)     (2111)
  (221)    (11111)
  (311)
  (2111)
  (11111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Crossrefs

The complement is counted by A061199.

Differences of consecutive terms are A091602.

Column k = 1 is A000041.

Column k = 2 is A047967, ranked by A013929 and A324929.

Column k = 3 is A295341, ranked by A046099 and A354235.

Column k = 4 is A295342.

A000041 counts integer partitions, strict A000009.

A047966 counts uniform partitions.

Cf. A002033, A006918, A064410, A117485, A238394, A238395, A325534.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],MemberQ[#/k,_?IntegerQ]&]],{n,1,15},{k,1,n}]
- or -
Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]>=k&]],{n,1,15},{k,1,n}]

PARI

\\ here P(k,n) is partitions with no part divisible by k as g.f.
P(k,n)={1/prod(i=1, n, 1 - if(i%k, x^i) + O(x*x^n))}
M(n,m=n)={my(p=P(n+1,n)); Mat(vector(m, k, Col(p-P(k,n), -n) ))}
{ my(A=M(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Jan 19 2023