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A351204 Number of integer partitions of n such that every permutation has all distinct runs.

%I A351204 #11 Feb 16 2022 01:50:08
%S A351204 1,1,2,3,4,5,8,9,11,14,18,20,25,28,34,41,47,53,64,72,84,98,113,128,
%T A351204 148,169,194,223,255,289,333,377,428,488,554,629,715,807,913,1033,
%U A351204 1166,1313,1483,1667,1875,2111,2369,2655,2977,3332,3729,4170,4657,5195,5797,6459
%N A351204 Number of integer partitions of n such that every permutation has all distinct runs.
%C A351204 Partitions enumerated by this sequence include those in which all parts are either the same or distinct as well as partitions with an even number of parts all of which except one are the same. - _Andrew Howroyd_, Feb 15 2022
%H A351204 Andrew Howroyd, <a href="/A351204/b351204.txt">Table of n, a(n) for n = 0..1000</a>
%H A351204 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e A351204 The a(1) = 1 through a(8) = 11 partitions:
%e A351204   (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A351204        (11)  (21)   (22)    (32)     (33)      (43)       (44)
%e A351204              (111)  (31)    (41)     (42)      (52)       (53)
%e A351204                     (1111)  (2111)   (51)      (61)       (62)
%e A351204                             (11111)  (222)     (421)      (71)
%e A351204                                      (321)     (2221)     (431)
%e A351204                                      (3111)    (4111)     (521)
%e A351204                                      (111111)  (211111)   (2222)
%e A351204                                                (1111111)  (5111)
%e A351204                                                           (311111)
%e A351204                                                           (11111111)
%t A351204 Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],!UnsameQ@@Split[#]&]=={}&]],{n,0,15}]
%o A351204 (PARI) \\ here Q(n) is A000009.
%o A351204 Q(n)={polcoef(prod(k=1, n, 1 + x^k + O(x*x^n)), n)}
%o A351204 a(n)={Q(n) + if(n, numdiv(n) - 1) + sum(k=1, (n-1)\3, sum(j=3, (n-1)\k, j%2==1 && n-k*j<>k))} \\ _Andrew Howroyd_, Feb 15 2022
%Y A351204 The version for run-lengths instead of runs is A000005.
%Y A351204 The version for normal multisets is 2^(n-1) - A283353(n-3).
%Y A351204 The complement is counted by A351203, ranked by A351201.
%Y A351204 A005811 counts runs in binary expansion.
%Y A351204 A044813 lists numbers whose binary expansion has distinct run-lengths.
%Y A351204 A059966 counts Lyndon compositions, necklaces A008965, aperiodic A000740.
%Y A351204 A098859 counts partitions with distinct multiplicities, ordered A242882.
%Y A351204 A238130 and A238279 count compositions by number of runs.
%Y A351204 A297770 counts distinct runs in binary expansion.
%Y A351204 A003242 counts anti-run compositions.
%Y A351204 Counting words with all distinct runs:
%Y A351204 - A351013 = compositions, for run-lengths A329739, ranked by A351290.
%Y A351204 - A351016 = binary words, for run-lengths A351017.
%Y A351204 - A351018 = binary expansions, for run-lengths A032020, ranked by A175413.
%Y A351204 - A351200 = patterns, for run-lengths A351292.
%Y A351204 - A351202 = permutations of prime factors.
%Y A351204 Cf. A000041, A035363, A047993, A116608, A144300, A329746, A351291.
%K A351204 nonn
%O A351204 0,3
%A A351204 _Gus Wiseman_, Feb 15 2022
%E A351204 Terms a(26) and beyond from _Andrew Howroyd_, Feb 15 2022