A364673 Number of (necessarily strict) integer partitions of n containing all of their own first differences.

1, 1, 1, 2, 1, 1, 3, 2, 1, 2, 2, 2, 5, 2, 2, 4, 2, 3, 6, 4, 4, 8, 4, 4, 10, 8, 7, 8, 13, 9, 15, 12, 13, 17, 20, 15, 31, 24, 27, 32, 33, 32, 50, 42, 45, 53, 61, 61, 85, 76, 86, 101, 108, 118, 137, 141, 147, 179, 184, 196, 222, 244, 257, 295, 324, 348, 380, 433

Offset

0,4

Links

John Tyler Rascoe, Table of n, a(n) for n = 0..200

Example

The partition y = (12,6,3,2,1) has differences (6,3,1,1), and {1,3,6} is a subset of {1,2,3,6,12}, so y is counted under a(24).
The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:
  (1)  (3)    (6)      (12)       (15)         (18)         (21)
       (2,1)  (4,2)    (8,4)      (10,5)       (12,6)       (14,7)
              (3,2,1)  (6,4,2)    (8,4,2,1)    (9,6,3)      (12,6,3)
                       (5,4,2,1)  (5,4,3,2,1)  (6,5,4,2,1)  (8,6,4,2,1)
                       (6,3,2,1)               (7,5,3,2,1)  (9,5,4,2,1)
                                               (8,4,3,2,1)  (9,6,3,2,1)
                                                            (10,5,3,2,1)
                                                            (6,5,4,3,2,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&SubsetQ[#, -Differences[#]]&]], {n, 0, 30}]

Prog

(Python)
from collections import Counter
def A364673_list(maxn):
    count = Counter()
    for i in range(maxn//3):
        A, f, i = [[(i+1, )]], 0, 0
        while f == 0:
            A.append([])
            for j in A[i]:
                for k in j:
                    x = j + (j[-1] + k, )
                    y = sum(x)
                    if y <= maxn:
                        A[i+1].append(x)
                        count.update({y})
            if len(A[i+1]) < 1: f += 1
            i += 1
    return [count[z]+1 for z in range(maxn+1)] # John Tyler Rascoe, Mar 09 2024

Crossrefs

Containing all differences: A007862.

Containing no differences: A364464, strict complement A364536.

Containing at least one difference: A364467, complement A363260.

For subsets of {1..n} we have A364671, complement A364672.

A non-strict version is A364674.

For submultisets instead of subsets we have A364675.

A000041 counts integer partitions, strict A000009.

A008284 counts partitions by length, strict A008289.

A236912 counts sum-free partitions w/o re-used parts, complement A237113.

A325325 counts partitions with distinct first differences.

Cf. A002865, A025065, A196723, A229816, A237667, A320347, A363225, A364272, A364345, A364463, A364537, A370386.

Sequence in context: A165162 A125106 A229874 * A330439 A243611 A273102

Adjacent sequences: A364670 A364671 A364672 * A364674 A364675 A364676

Keyword

nonn

Author

Gus Wiseman, Aug 03 2023

Status

approved