A374762 - OEIS

%I #12 Jul 31 2024 17:27:34

%S 1,1,1,3,4,6,11,18,27,41,64,98,151,229,339,504,746,1097,1618,2372,

%T 3451,5009,7233,10394,14905,21316,30396,43246,61369,86830,122529,

%U 172457,242092,339062,473850,660829,919822,1277935,1772174,2453151,3389762,4675660,6438248

%N Number of integer compositions of n whose leaders of strictly decreasing runs are strictly increasing.

%C The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

%C Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the maxima are strictly decreasing. The weakly decreasing version is A374764.

%H Andrew Howroyd, <a href="/A374762/b374762.txt">Table of n, a(n) for n = 0..1000</a>

%H Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.

%F G.f.: Product_{k>=1} (1 + x^k*Product_{j=1..k-1} (1 + x^j)). - _Andrew Howroyd_, Jul 31 2024

%e The a(0) = 1 through a(7) = 18 compositions:

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

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

%e                 (21)  (31)   (23)   (24)   (25)

%e                       (121)  (32)   (42)   (34)

%e                              (41)   (51)   (43)

%e                              (131)  (123)  (52)

%e                                     (132)  (61)

%e                                     (141)  (124)

%e                                     (213)  (142)

%e                                     (231)  (151)

%e                                     (321)  (214)

%e                                            (232)

%e                                            (241)

%e                                            (421)

%e                                            (1213)

%e                                            (1231)

%e                                            (1321)

%e                                            (2131)

%t Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,Greater]&]],{n,0,15}]

%o (PARI) seq(n) = Vec(prod(k=1, n, 1 + x^k*prod(j=1, min(n-k,k-1), 1 + x^j, 1 + O(x^(n-k+1))))) \\ _Andrew Howroyd_, Jul 31 2024

%Y For partitions instead of compositions we have A000009.

%Y The weak version appears to be A188900.

%Y The opposite version is A374689.

%Y Other types of runs (instead of strictly decreasing):

%Y - For leaders of identical runs we have A000041.

%Y - For leaders of weakly increasing runs we have A374634.

%Y - For leaders of anti-runs we have A374679.

%Y Other types of run-leaders (instead of strictly increasing):

%Y - For identical leaders we have A374760, ranks A374759.

%Y - For distinct leaders we have A374761, ranks A374767.

%Y - For strictly decreasing leaders we have A374763.

%Y - For weakly increasing leaders we have A374764.

%Y - For weakly decreasing leaders we have A374765.

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

%Y A011782 counts compositions.

%Y A238130, A238279, A333755 count compositions by number of runs.

%Y A274174 counts contiguous compositions, ranks A374249.

%Y A373949 counts compositions by run-compressed sum, opposite A373951.

%Y A374700 counts compositions by sum of leaders of strictly increasing runs.

%Y Cf. A106356, A188920, A189076, A238343, A261982, A333213, A374518, A374631, A374632, A374687, A374742, A374743.

%K nonn

%O 0,4

%A _Gus Wiseman_, Jul 29 2024

%E a(24) onwards from _Andrew Howroyd_, Jul 31 2024