A364673 - OEIS

%I #15 Mar 09 2024 19:18:48

%S 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,

%T 13,17,20,15,31,24,27,32,33,32,50,42,45,53,61,61,85,76,86,101,108,118,

%U 137,141,147,179,184,196,222,244,257,295,324,348,380,433

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

%H John Tyler Rascoe, <a href="/A364673/b364673.txt">Table of n, a(n) for n = 0..200</a>

%e 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).

%e The a(n) partitions for n = 1, 3, 6, 12, 15, 18, 21:

%e   (1)  (3)    (6)      (12)       (15)         (18)         (21)

%e        (2,1)  (4,2)    (8,4)      (10,5)       (12,6)       (14,7)

%e               (3,2,1)  (6,4,2)    (8,4,2,1)    (9,6,3)      (12,6,3)

%e                        (5,4,2,1)  (5,4,3,2,1)  (6,5,4,2,1)  (8,6,4,2,1)

%e                        (6,3,2,1)               (7,5,3,2,1)  (9,5,4,2,1)

%e                                                (8,4,3,2,1)  (9,6,3,2,1)

%e                                                             (10,5,3,2,1)

%e                                                             (6,5,4,3,2,1)

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

%o (Python)

%o from collections import Counter

%o def A364673_list(maxn):

%o     count = Counter()

%o     for i in range(maxn//3):

%o         A,f,i = [[(i+1, )]],0,0

%o         while f == 0:

%o             A.append([])

%o             for j in A[i]:

%o                 for k in j:

%o                     x = j + (j[-1] + k, )

%o                     y = sum(x)

%o                     if y <= maxn:

%o                         A[i+1].append(x)

%o                         count.update({y})

%o             if len(A[i+1]) < 1: f += 1

%o             i += 1

%o     return [count[z]+1 for z in range(maxn+1)] # _John Tyler Rascoe_, Mar 09 2024

%Y Containing all differences: A007862.

%Y Containing no differences: A364464, strict complement A364536.

%Y Containing at least one difference: A364467, complement A363260.

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

%Y A non-strict version is A364674.

%Y For submultisets instead of subsets we have A364675.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

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

%Y A325325 counts partitions with distinct first differences.

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

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 03 2023