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A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).

%I A384892 #13 Jun 22 2025 16:29:04
%S A384892 1,1,2,4,13,54,314,2120,16700,148333,1468512,16019532,190899736,
%T A384892 2467007774,34361896102,513137616840,8178130784179,138547156531410,
%U A384892 2486151753462260,47106033220679060,939765362754015750,19690321886243848784,432292066866187743954
%N A384892 Number of permutations of {1..n} with all equal lengths of maximal runs (increasing by 1).
%H A384892 Christian Sievers, <a href="/A384892/b384892.txt">Table of n, a(n) for n = 0..450</a>
%F A384892 a(n) = Sum_{d|n} A000255(d-1). - _Christian Sievers_, Jun 22 2025
%e A384892 The permutation (1,2,5,6,3,4,7,8) has maximal runs ((1,2),(5,6),(3,4),(7,8)), with lengths (2,2,2,2), so is counted under a(8).
%e A384892 The a(0) = 1 through a(4) = 13 permutations:
%e A384892   ()  (1)  (12)  (123)  (1234)
%e A384892            (21)  (132)  (1324)
%e A384892                  (213)  (1432)
%e A384892                  (321)  (2143)
%e A384892                         (2413)
%e A384892                         (2431)
%e A384892                         (3142)
%e A384892                         (3214)
%e A384892                         (3241)
%e A384892                         (3412)
%e A384892                         (4132)
%e A384892                         (4213)
%e A384892                         (4321)
%t A384892 Table[Length[Select[Permutations[Range[n]],SameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]
%o A384892 (PARI) a(n)=if(n,sumdiv(n,d,sum(i=0,d-1,(-1)^i*(d-i)!*binomial(d-1,i))),1) \\ _Christian Sievers_, Jun 22 2025
%Y A384892 For subsets instead of permutations we have A243815, for anti-runs A384889.
%Y A384892 For strict partitions and distinct lengths we have A384178, anti-runs A384880.
%Y A384892 For integer partitions and distinct lengths we have A384884, anti-runs A384885.
%Y A384892 For distinct lengths we have A384891, for anti-runs A384907.
%Y A384892 For partitions we have A384904, strict A384886.
%Y A384892 A010027 counts permutations by maximal anti-runs, for runs A123513.
%Y A384892 A034839 counts subsets by number of maximal runs, for strict partitions A116674.
%Y A384892 A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
%Y A384892 A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.
%Y A384892 Cf. A000255, A044813, A047993, A242882, A325325, A329739, A351202, A384175, A384176, A384177.
%K A384892 nonn
%O A384892 0,3
%A A384892 _Gus Wiseman_, Jun 19 2025
%E A384892 a(11) and beyond from _Christian Sievers_, Jun 22 2025