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Number of integer partitions of n whose maximal anti-runs have distinct minima.
13

%I #12 Aug 21 2024 18:49:30

%S 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 247,300,352,424,499,595,695,826,963,1138,1322,1553,1802,2106,2435,

%U 2835,3271,3795,4365,5046,5792,6673,7641,8778,10030,11490,13099,14968,17030

%N Number of integer partitions of n whose maximal anti-runs have distinct minima.

%C These are partitions with no part appearing more than twice and with the least part appearing only once.

%C Also the number of reversed integer partitions of n whose maximal anti-runs have distinct minima.

%H John Tyler Rascoe, <a href="/A375134/b375134.txt">Table of n, a(n) for n = 0..300</a>

%F 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 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 The a(1) = 1 through a(9) = 11 partitions:

%e (1) (2) (3) (4) (5) (6) (7) (8) (9)

%e (12) (13) (14) (15) (16) (17) (18)

%e (23) (24) (25) (26) (27)

%e (122) (123) (34) (35) (36)

%e (124) (125) (45)

%e (133) (134) (126)

%e (233) (135)

%e (1223) (144)

%e (234)

%e (1224)

%e (1233)

%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@Min/@Split[#,UnsameQ]&]],{n,0,30}]

%o (PARI)

%o 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 A_x(51) \\ _John Tyler Rascoe_, Aug 21 2024

%Y Includes all strict partitions A000009.

%Y For identical instead of distinct leaders we have A115029.

%Y A version for compositions instead of partitions is A374518, ranks A374638.

%Y For minima instead of maxima we have A375133, ranks A375402.

%Y These partitions have ranks A375398.

%Y The complement is counted by A375404, ranks A375399.

%Y A000041 counts integer partitions.

%Y A003242 counts anti-run compositions, ranks A333489.

%Y A011782 counts integer compositions.

%Y A055887 counts sequences of partitions with total sum n.

%Y A375128 lists minima of maximal anti-runs of prime indices, sums A374706.

%Y Cf. A034296, A141199, A358830, A358836, A358905, A374704, A374757, A374758, A374761, A375136, A375400, A375401.

%K nonn

%O 0,4

%A _Gus Wiseman_, Aug 14 2024