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A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.

%I A385574 #11 Jul 23 2025 00:17:48
%S A385574 1,1,1,1,2,3,2,4,5,6,10,11,13,17,20,30,36,44,55,70,86,98,128,156,190,
%T A385574 235,288,351,409,499,603,722,863,1025,1227,1461,1757,2061,2444,2892,
%U A385574 3406,3996,4708,5497,6430,7595,8835,10294,12027,13971,16252,18887,21878
%N A385574 Number of integer partitions of n with the same number of adjacent equal parts as adjacent unequal parts.
%C A385574 These are also integer partitions of n with the same number of distinct parts as maximal anti-runs of parts.
%H A385574 Christian Sievers, <a href="/A385574/b385574.txt">Table of n, a(n) for n = 0..1000</a>
%F A385574 For a partition p, let s(p) be its sum, e(p) the number of equal adjacent pairs, and d(p) the number of distinct adjacent pairs. Then Sum_{p partition} x^s(p) * t^(e(p)-d(p)) = (Product_{i>=1} (1 + x^i/(t*(1-t*x^i))) - 1) * t + 1, so a(n) is the coefficient of x^n*t^0 of this expression. - _Christian Sievers_, Jul 18 2025
%e A385574 The partition (5,3,2,1,1,1,1) has 4 runs ((5),(3),(2),(1,1,1,1)) and 4 anti-runs ((5,3,2,1),(1),(1),(1)) so is counted under a(14).
%e A385574 The a(1) = 1 through a(10) = 10 reversed partitions (A = 10):
%e A385574   (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (A)
%e A385574                  (112)  (113)  (114)  (115)  (116)    (117)    (118)
%e A385574                         (122)         (133)  (224)    (144)    (226)
%e A385574                                       (223)  (233)    (225)    (244)
%e A385574                                              (11123)  (11124)  (334)
%e A385574                                                       (11223)  (11125)
%e A385574                                                                (11134)
%e A385574                                                                (11224)
%e A385574                                                                (11233)
%e A385574                                                                (12223)
%t A385574 Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union[#]]==Length[Split[#,#2!=#1&]]&]],{n,0,30}]
%o A385574 (PARI) lista(n)=Vec(polcoef((prod(i=1,n,1+x^i/(t*(1-t*x^i))+O(x*x^n))-1)*t+1,0,t)) \\ _Christian Sievers_, Jul 18 2025
%Y A385574 The RHS is counted by A116608, rank statistic A297155.
%Y A385574 The LHS is counted by A133121, rank statistic A046660.
%Y A385574 For related inequalities see A212165, A212168, A361204.
%Y A385574 For subsets instead of partitions see A217615, A385572, A385575.
%Y A385574 These partitions are ranked by A385576.
%Y A385574 A000041 counts integer partitions, strict A000009.
%Y A385574 A007690 counts partitions with no singletons, complement A183558.
%Y A385574 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A385574 A034839 counts subsets by number maximal runs, for partitions A384881, strict A116674.
%Y A385574 A047993 counts partitions with max part = length (A106529).
%Y A385574 A098859 counts Wilf partitions (complement A336866), compositions A242882.
%Y A385574 A268193 counts partitions by maximal anti-runs, strict A384905, subsets A384893.
%Y A385574 A355394 counts partitions with neighbors, complement A356236.
%Y A385574 Cf. A000071, A003114, A008284, A010027, A047966, A210034, A325324, A325325, A356606, A384882, A384885.
%K A385574 nonn
%O A385574 0,5
%A A385574 _Gus Wiseman_, Jul 04 2025