A238495 - OEIS

%I #21 Jul 15 2023 14:02:33

%S 1,2,3,4,7,9,14,19,27,36,51,66,90,118,156,201,264,336,434,550,700,880,

%T 1112,1385,1733,2149,2666,3283,4049,4956,6072,7398,9009,10922,13237,

%U 15970,19261,23147,27790,33260,39776,47425,56497,67133,79685,94371,111653

%N Number of partitions p of n such that min(p) + (number of parts of p) is not a part of p.

%C Also the number of integer partitions of n + 1 with median > 1, or with no more 1's than non-1 parts. - _Gus Wiseman_, Jul 10 2023

%F From _Gus Wiseman_, Jul 11 2023: (Start)

%F a(n>2) = A000041(n) - A096373(n-2).

%F a(n>1) = A000041(n-2) + A002865(n+1).

%F a(n) = A000041(n+1) - A027336(n).

%F (End)

%e a(6) = 9 counts all the 11 partitions of 6 except 42 and 411.

%e From _Gus Wiseman_, Jul 10 2023 (Start)

%e The a(2) = 1 through a(8) = 14 partitions:

%e   (2)  (3)   (4)   (5)    (6)     (7)     (8)

%e        (21)  (22)  (32)   (33)    (43)    (44)

%e              (31)  (41)   (42)    (52)    (53)

%e                    (221)  (51)    (61)    (62)

%e                           (222)   (322)   (71)

%e                           (321)   (331)   (332)

%e                           (2211)  (421)   (422)

%e                                   (2221)  (431)

%e                                   (3211)  (521)

%e                                           (2222)

%e                                           (3221)

%e                                           (3311)

%e                                           (4211)

%e                                           (22211)

%e (End)

%t Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Length[p] + Min[p]]], {n, 50}]

%t Table[Length[Select[IntegerPartitions[n+1],Median[#]>1&]],{n,30}] (* _Gus Wiseman_, Jul 10 2023 *)

%Y Cf. A096373.

%Y For mean instead of median we have A000065, ranks A057716.

%Y The complement is counted by A027336, ranks A364056.

%Y Rows sums of A359893 if we remove the first column.

%Y These partitions have ranks A364058.

%Y A000041 counts integer partitions.

%Y A008284 counts partitions by length, A058398 by mean.

%Y A025065 counts partitions with low mean 1, ranks A363949.

%Y A124943 counts partitions by low median, high A124944.

%Y A241131 counts partitions with low mode 1, ranks A360015.

%Y Cf. A000070, A000975, A002865, A110618, A237984, A363488.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Feb 27 2014

%E Formula corrected by _Gus Wiseman_, Jul 11 2023