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 A375133 #13 Aug 21 2024 18:49:45
%S A375133 1,1,1,2,3,4,5,8,10,14,17,23,29,38,47,60,74,93,113,141,171,211,253,
%T A375133 309,370,447,532,639,758,904,1066,1265,1487,1754,2053,2411,2813,3289,
%U A375133 3823,4454,5161,5990,6920,8005,9223,10634,12218,14048,16101,18462,21107
%N A375133 Number of integer partitions of n whose maximal anti-runs have distinct maxima.
%C A375133 An anti-run is a sequence with no adjacent equal parts.
%C A375133 These are partitions with no part appearing more than twice and greatest part appearing only once.
%C A375133 Also the number of reversed integer partitions of n whose maximal anti-runs have distinct maxima.
%H A375133 John Tyler Rascoe, <a href="/A375133/b375133.txt">Table of n, a(n) for n = 0..300</a>
%F A375133 G.f.: Sum_{i>=0} (x^i * Product_{j=1..i-1} (1-x^(3*j))/(1-x^j)). - _John Tyler Rascoe_, Aug 21 2024
%e A375133 The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with maxima (6,5,3), so y is counted under a(29).
%e A375133 The a(0) = 1 through a(9) = 14 partitions:
%e A375133 () (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e A375133 (21) (31) (32) (42) (43) (53) (54)
%e A375133 (211) (41) (51) (52) (62) (63)
%e A375133 (311) (321) (61) (71) (72)
%e A375133 (411) (322) (422) (81)
%e A375133 (421) (431) (432)
%e A375133 (511) (521) (522)
%e A375133 (3211) (611) (531)
%e A375133 (3221) (621)
%e A375133 (4211) (711)
%e A375133 (4221)
%e A375133 (4311)
%e A375133 (5211)
%e A375133 (32211)
%t A375133 Table[Length[Select[IntegerPartitions[n], UnsameQ@@Max/@Split[#,UnsameQ]&]],{n,0,30}]
%o A375133 (PARI)
%o A375133 A_x(N) = {my(x='x+O('x^N), f=sum(i=0,N,(x^i)*prod(j=1,i-1,(1-x^(3*j))/(1-x^j)))); Vec(f)}
%o A375133 A_x(51) \\ _John Tyler Rascoe_, Aug 21 2024
%Y A375133 Includes all strict partitions A000009.
%Y A375133 For identical instead of distinct see: A034296, A115029, A374760, A374759.
%Y A375133 For compositions instead of partitions we have A374761.
%Y A375133 For minima instead of maxima we have A375134, ranks A375398.
%Y A375133 The complement is counted by A375401, ranks A375403.
%Y A375133 These partitions are ranked by A375402, for compositions A374767.
%Y A375133 The complement for minima instead of maxima is A375404, ranks A375399.
%Y A375133 A000041 counts integer partitions.
%Y A375133 A003242 counts anti-run compositions, ranks A333489.
%Y A375133 A011782 counts integer compositions.
%Y A375133 A055887 counts sequences of partitions with total sum n.
%Y A375133 A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
%Y A375133 Cf. A141199, A279790, A358830, A358833, A358836, A358905, A374704, A374757, A374758, A375136, A375400.
%K A375133 nonn
%O A375133 0,4
%A A375133 _Gus Wiseman_, Aug 14 2024