cp's OEIS Frontend

A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.

%I A364675 #8 Aug 10 2023 07:11:48
%S A364675 1,1,2,3,4,4,7,7,10,12,15,15,26,25,35,45,55,60,86,94,126,150,186,216,
%T A364675 288,328,407,493,610,699,896,1030,1269,1500,1816,2130,2620,3029,3654,
%U A364675 4300,5165,5984,7222,8368,9976,11637,13771,15960,18978,21896,25815,29915
%N A364675 Number of integer partitions of n whose nonzero first differences are a submultiset of the parts.
%C A364675 Conjecture: For subsets of {1..n} instead of partitions of n we have A101925.
%C A364675 Conjecture: The strict version is A154402.
%e A364675 The partition y = (3,2,1,1) has first differences (1,1,0), and (1,1) is a submultiset of y, so y is counted under a(7).
%e A364675 The a(1) = 1 through a(8) = 10 partitions:
%e A364675   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A364675        (11)  (21)   (22)    (221)    (33)      (421)      (44)
%e A364675              (111)  (211)   (2111)   (42)      (2221)     (422)
%e A364675                     (1111)  (11111)  (222)     (3211)     (2222)
%e A364675                                      (2211)    (22111)    (4211)
%e A364675                                      (21111)   (211111)   (22211)
%e A364675                                      (111111)  (1111111)  (32111)
%e A364675                                                           (221111)
%e A364675                                                           (2111111)
%e A364675                                                           (11111111)
%t A364675 submultQ[cap_,fat_] := And@@Function[i,Count[fat,i] >= Count[cap,i]] /@ Union[List@@cap];
%t A364675 Table[Length[Select[IntegerPartitions[n], submultQ[Differences[Union[#]],#]&]], {n,0,30}]
%Y A364675 For subsets of {1..n} we appear to have A101925, A364671, A364672.
%Y A364675 The strict case (no differences of 0) appears to be A154402.
%Y A364675 Starting with the distinct parts gives A342337.
%Y A364675 For disjoint multisets: A363260, subsets A364463, strict A364464.
%Y A364675 For overlapping multisets: A364467, ranks A364537, strict A364536.
%Y A364675 For subsets instead of submultisets we have A364673.
%Y A364675 A000041 counts integer partitions, strict A000009.
%Y A364675 A008284 counts partitions by length, strict A008289.
%Y A364675 A236912 counts sum-free partitions, complement A237113.
%Y A364675 A325325 counts partitions with distinct first differences.
%Y A364675 Cf. A002865, A007862, A108917, A229816, A237667, A237668, A320347, A363225, A364272, A364345, A364466.
%K A364675 nonn
%O A364675 0,3
%A A364675 _Gus Wiseman_, Aug 04 2023