cp's OEIS Frontend

A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.

%I A374688 #16 Sep 16 2024 08:43:28
%S A374688 1,1,1,2,2,4,5,7,11,16,21,31,45,63,87,122,170,238,328,449,616,844,
%T A374688 1151,1565,2121,2861,3855,5183,6953,9299,12407,16513,21935,29078,
%U A374688 38468,50793,66935,88037,115577,151473,198175,258852,337560,439507,571355,741631
%N A374688 Number of integer compositions of n whose leaders of strictly increasing runs are themselves strictly increasing.
%C A374688 The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.
%C A374688 Also the number of ways to choose a strict integer partition of each part of an integer composition of n (A304969) such that the minima are strictly decreasing.
%H A374688 Christian Sievers, <a href="/A374688/b374688.txt">Table of n, a(n) for n = 0..500</a>
%H A374688 Gus Wiseman, <a href="/A374629/a374629.txt">Sequences counting and ranking compositions by their leaders (for six types of runs)</a>.
%e A374688 The a(0) = 1 through a(9) = 16 compositions:
%e A374688   ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
%e A374688                 (12)  (13)  (14)   (15)   (16)   (17)    (18)
%e A374688                             (23)   (24)   (25)   (26)    (27)
%e A374688                             (122)  (123)  (34)   (35)    (36)
%e A374688                                    (132)  (124)  (125)   (45)
%e A374688                                           (133)  (134)   (126)
%e A374688                                           (142)  (143)   (135)
%e A374688                                                  (152)   (144)
%e A374688                                                  (233)   (153)
%e A374688                                                  (1223)  (162)
%e A374688                                                  (1232)  (234)
%e A374688                                                          (243)
%e A374688                                                          (1224)
%e A374688                                                          (1233)
%e A374688                                                          (1242)
%e A374688                                                          (1323)
%t A374688 Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Less@@First/@Split[#,Less]&]],{n,0,15}]
%Y A374688 The weak version is A374635.
%Y A374688 Ranked by positions of strictly increasing rows in A374683 (sums A374684).
%Y A374688 The opposite version is A374763.
%Y A374688 Types of runs (instead of strictly increasing):
%Y A374688 - For leaders of identical runs we have A000041.
%Y A374688 - For leaders of anti-runs we have A374679.
%Y A374688 - For leaders of weakly increasing runs we have A374634.
%Y A374688 - For leaders of strictly decreasing runs we have A374762.
%Y A374688 Types of run-leaders (instead of strictly increasing):
%Y A374688 - For identical leaders we have A374686, ranks A374685.
%Y A374688 - For distinct leaders we have A374687, ranks A374698.
%Y A374688 - For strictly decreasing leaders we have A374689.
%Y A374688 - For weakly increasing leaders we have A374690.
%Y A374688 - For weakly decreasing leaders we have A374697.
%Y A374688 A003242 counts anti-run compositions, ranks A333489.
%Y A374688 A011782 counts compositions.
%Y A374688 A238130, A238279, A333755 count compositions by number of runs.
%Y A374688 A373949 counts compositions by run-compressed sum, opposite A373951.
%Y A374688 A374700 counts compositions by sum of leaders of strictly increasing runs.
%Y A374688 Cf. A000009, A106356, A188920, A189076, A238343, A261982, A333213, A374632.
%K A374688 nonn
%O A374688 0,4
%A A374688 _Gus Wiseman_, Jul 27 2024
%E A374688 a(26) and beyond from _Christian Sievers_, Aug 08 2024