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

1,2

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.

Links

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

Example

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)

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}]

Prog

(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

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.

Sequence in context: A106740 A178534 A110619 * A191861 A350200 A129761

Adjacent sequences: A354231 A354232 A354233 * A354235 A354236 A354237

Keyword

nonn,tabl

Author

Gus Wiseman, May 22 2022

Status

approved