This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A333192 #10 May 18 2020 19:11:03
%S A333192 1,1,2,2,4,5,7,10,14,16,24,31,37,51,67,76,103,129,158,199,242,293,370,
%T A333192 450,538,652,799,953,1147,1376,1635,1956,2322,2757,3271,3845,4539,
%U A333192 5336,6282,7366,8589,10046,11735,13647,15858,18442,21354,24716,28630,32985
%N A333192 Number of compositions of n with strictly increasing run-lengths.
%C A333192 A composition of n is a finite sequence of positive integers summing to n.
%H A333192 Giovanni Resta, <a href="/A333192/b333192.txt">Table of n, a(n) for n = 0..1000</a>
%e A333192 The a(1) = 1 through a(8) = 14 compositions:
%e A333192 (1) (2) (3) (4) (5) (6) (7) (8)
%e A333192 (11) (111) (22) (122) (33) (133) (44)
%e A333192 (211) (311) (222) (322) (233)
%e A333192 (1111) (2111) (411) (511) (422)
%e A333192 (11111) (3111) (1222) (611)
%e A333192 (21111) (4111) (2222)
%e A333192 (111111) (22111) (5111)
%e A333192 (31111) (11222)
%e A333192 (211111) (41111)
%e A333192 (1111111) (122111)
%e A333192 (221111)
%e A333192 (311111)
%e A333192 (2111111)
%e A333192 (11111111)
%e A333192 For example, the composition (1,2,2,1,1,1) has run-lengths (1,2,3), so is counted under a(8).
%t A333192 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Less@@Length/@Split[#]&]],{n,0,15}]
%t A333192 b[n_, lst_, v_] := b[n, lst, v] = If[n == 0, 1, If[n <= lst, 0, Sum[If[k == v, 0, b[n - k pz, pz, k]], {pz, lst + 1, n}, {k, Floor[n/pz]}]]]; a[n_] := b[n, 0, 0]; a /@ Range[0, 50] (* _Giovanni Resta_, May 18 2020 *)
%Y A333192 The case of partitions is A100471.
%Y A333192 The non-strict version is A332836.
%Y A333192 Strictly increasing compositions are A000009.
%Y A333192 Unimodal compositions are A001523.
%Y A333192 Strict compositions are A032020.
%Y A333192 Partitions with strictly increasing run-lengths are A100471.
%Y A333192 Partitions with strictly decreasing run-lengths are A100881.
%Y A333192 Compositions with equal run-lengths are A329738.
%Y A333192 Compositions whose run-lengths are unimodal are A332726.
%Y A333192 Compositions with strictly increasing or decreasing run-lengths are A333191.
%Y A333192 Numbers with strictly increasing prime multiplicities are A334965.
%Y A333192 Cf. A072706, A098859, A100882, A100883, A304686, A329744, A329766, A332726, A332833, A332834, A332835, A333147, A333149, A333190.
%K A333192 nonn
%O A333192 0,3
%A A333192 _Gus Wiseman_, May 17 2020
%E A333192 Terms a(26) and beyond from _Giovanni Resta_, May 18 2020