cp's OEIS Frontend

A316496 Number of totally strong integer partitions of n.

%I A316496 #43 Mar 08 2020 21:48:18
%S A316496 1,1,2,3,4,5,8,8,12,13,18,20,27,27,38,41,52,56,73,77,99,105,129,145,
%T A316496 176,186,229,253,300,329,395,427,504,555,648,716,836,905,1065,1173,
%U A316496 1340,1475,1703,1860,2140,2349,2671,2944,3365,3666,4167,4582,5160,5668
%N A316496 Number of totally strong integer partitions of n.
%C A316496 An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.
%e A316496 The a(1) = 1 through a(8) = 12 totally strong partitions:
%e A316496   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A316496        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A316496              (111)  (31)    (41)     (42)      (52)       (53)
%e A316496                     (1111)  (221)    (51)      (61)       (62)
%e A316496                             (11111)  (222)     (331)      (71)
%e A316496                                      (321)     (421)      (332)
%e A316496                                      (2211)    (2221)     (431)
%e A316496                                      (111111)  (1111111)  (521)
%e A316496                                                           (2222)
%e A316496                                                           (3311)
%e A316496                                                           (22211)
%e A316496                                                           (11111111)
%e A316496 For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.
%t A316496 totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
%t A316496 Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]
%Y A316496 Cf. A181819, A182850, A182857, A304660, A305563, A316597.
%Y A316496 The Heinz numbers of these partitions are A316529.
%Y A316496 The version for compositions is A332274.
%Y A316496 The dual version is A332275.
%Y A316496 The version for reversed partitions is (also) A332275.
%Y A316496 The narrowly normal version is A332297.
%Y A316496 The alternating version is A332339 (see also A317256).
%Y A316496 Partitions with weakly decreasing run-lengths are A100882.
%Y A316496 Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.
%K A316496 nonn
%O A316496 0,3
%A A316496 _Gus Wiseman_, Jul 29 2018
%E A316496 Updated with corrected terminology by _Gus Wiseman_, Mar 07 2020