cp's OEIS Frontend

A336106 Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.

%I A336106 #8 Jul 09 2020 22:43:18
%S A336106 1,1,1,2,3,5,7,11,15,23,30,44,58,82,105,146,186,252,318,423,530,695,
%T A336106 863,1116,1380,1763,2164,2738,3345,4192,5096,6334,7665,9459,11395,
%U A336106 13968,16765,20425,24418,29588,35251,42496,50460,60547,71669,85628
%N A336106 Number of integer partitions of n whose greatest part is at most one more than the sum of the other parts.
%C A336106 Also the number of separable strong multisets of length n covering an initial interval of positive integers. A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.
%e A336106 The a(1) = 1 through a(8) = 15 partitions:
%e A336106   (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A336106              (111)  (211)   (221)    (222)     (322)      (332)
%e A336106                     (1111)  (311)    (321)     (331)      (422)
%e A336106                             (2111)   (2211)    (421)      (431)
%e A336106                             (11111)  (3111)    (2221)     (2222)
%e A336106                                      (21111)   (3211)     (3221)
%e A336106                                      (111111)  (4111)     (3311)
%e A336106                                                (22111)    (4211)
%e A336106                                                (31111)    (22211)
%e A336106                                                (211111)   (32111)
%e A336106                                                (1111111)  (41111)
%e A336106                                                           (221111)
%e A336106                                                           (311111)
%e A336106                                                           (2111111)
%e A336106                                                           (11111111)
%t A336106 Table[Length[Select[IntegerPartitions[n],2*Max@@#<=1+n&]],{n,0,15}]
%Y A336106 The inseparable version is A025065.
%Y A336106 The Heinz numbers of these partitions are A335127.
%Y A336106 The non-strong version is A336103.
%Y A336106 Sequences covering an initial interval are A000670.
%Y A336106 Anti-run compositions are A003242.
%Y A336106 Anti-run patterns are A005649.
%Y A336106 Separable partitions are A325534.
%Y A336106 Inseparable partitions are A325535.
%Y A336106 Separable factorizations are A335434.
%Y A336106 Heinz numbers of separable partitions are A335433.
%Y A336106 Cf. A049610, A106351, A292884, A335126, A335433, A335452, A335548, A336102.
%K A336106 nonn
%O A336106 0,4
%A A336106 _Gus Wiseman_, Jul 09 2020