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 A375134 #12 Aug 21 2024 18:49:30
%S A375134 1,1,1,2,2,4,4,6,8,11,12,18,21,28,33,43,52,66,78,98,116,145,171,209,
%T A375134 247,300,352,424,499,595,695,826,963,1138,1322,1553,1802,2106,2435,
%U A375134 2835,3271,3795,4365,5046,5792,6673,7641,8778,10030,11490,13099,14968,17030
%N A375134 Number of integer partitions of n whose maximal anti-runs have distinct minima.
%C A375134 These are partitions with no part appearing more than twice and with the least part appearing only once.
%C A375134 Also the number of reversed integer partitions of n whose maximal anti-runs have distinct minima.
%H A375134 John Tyler Rascoe, <a href="/A375134/b375134.txt">Table of n, a(n) for n = 0..300</a>
%F A375134 G.f.: 1 + Sum_{i>0} (x^i * Product_{j>i} (1-x^(3*j))/(1-x^j)). - _John Tyler Rascoe_, Aug 21 2024
%e A375134 The partition y = (6,5,5,4,3,3,2,1) has maximal anti-runs ((6,5),(5,4,3),(3,2,1)), with minima (5,3,1), so y is counted under a(29).
%e A375134 The a(1) = 1 through a(9) = 11 partitions:
%e A375134 (1) (2) (3) (4) (5) (6) (7) (8) (9)
%e A375134 (12) (13) (14) (15) (16) (17) (18)
%e A375134 (23) (24) (25) (26) (27)
%e A375134 (122) (123) (34) (35) (36)
%e A375134 (124) (125) (45)
%e A375134 (133) (134) (126)
%e A375134 (233) (135)
%e A375134 (1223) (144)
%e A375134 (234)
%e A375134 (1224)
%e A375134 (1233)
%t A375134 Table[Length[Select[IntegerPartitions[n], UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]
%o A375134 (PARI)
%o A375134 A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,(x^i)*prod(j=i+1,N-i,(1-x^(3*j))/(1-x^j)))); Vec(f)}
%o A375134 A_x(51) \\ _John Tyler Rascoe_, Aug 21 2024
%Y A375134 Includes all strict partitions A000009.
%Y A375134 For identical instead of distinct leaders we have A115029.
%Y A375134 A version for compositions instead of partitions is A374518, ranks A374638.
%Y A375134 For minima instead of maxima we have A375133, ranks A375402.
%Y A375134 These partitions have ranks A375398.
%Y A375134 The complement is counted by A375404, ranks A375399.
%Y A375134 A000041 counts integer partitions.
%Y A375134 A003242 counts anti-run compositions, ranks A333489.
%Y A375134 A011782 counts integer compositions.
%Y A375134 A055887 counts sequences of partitions with total sum n.
%Y A375134 A375128 lists minima of maximal anti-runs of prime indices, sums A374706.
%Y A375134 Cf. A034296, A141199, A358830, A358836, A358905, A374704, A374757, A374758, A374761, A375136, A375400, A375401.
%K A375134 nonn
%O A375134 0,4
%A A375134 _Gus Wiseman_, Aug 14 2024