A374680 -id:A374680 - cp's OEIS frontend

A188920 a(n) is the limiting term of the n-th column of the triangle in A188919.

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 169, 274, 434, 686, 1069, 1660, 2548, 3897, 5906, 8911, 13352, 19917, 29532, 43605, 64056, 93715, 136499, 198059, 286233, 412199, 591455, 845851, 1205687, 1713286, 2427177, 3428611, 4829563, 6784550, 9505840, 13284849

Offset: 0

N. J. A. Sloane, Apr 13 2011

nonn

Also the number of integer compositions of n whose reverse avoids 12-1 and 23-1.

Theorem: The reverse of a composition avoids 12-1 and 23-1 iff its leaders of maximal weakly increasing runs, obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each, are strictly decreasing. For example, the composition y = (4,5,3,2,2,3,1,3,5) has reverse (5,3,1,3,2,2,3,5,4), which avoids 12-1 and 23-1, while the maximal weakly increasing runs of y are ((4,5),(3),(2,2,3),(1,3,5)), with leaders (4,3,2,1), which are strictly decreasing, as required. - Gus Wiseman, Aug 20 2024

			From _Gus Wiseman_, Aug 20 2024: (Start)
The a(0) = 1 through a(6) = 22 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (13)    (14)     (15)
                 (21)   (22)    (23)     (24)
                 (111)  (31)    (32)     (33)
                        (112)   (41)     (42)
                        (211)   (113)    (51)
                        (1111)  (122)    (114)
                                (212)    (123)
                                (221)    (132)
                                (311)    (213)
                                (1112)   (222)
                                (2111)   (312)
                                (11111)  (321)
                                         (411)
                                         (1113)
                                         (1122)
                                         (2112)
                                         (2211)
                                         (3111)
                                         (11112)
                                         (21111)
                                         (111111)
(End)

John Tyler Rascoe, Table of n, a(n) for n = 0..200
A. M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv preprint arXiv:1108.2642 [math.CO], 2011-2012.
Wikipedia, Permutation pattern.
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For leaders of identical runs we have A000041.
Matching 23-1 only gives A189076.
An opposite version is A358836.
For identical leaders we have A374631, ranks A374633.
For distinct leaders we have A374632, ranks A374768.
For weakly increasing leaders we have A374635.
For non-weakly decreasing leaders we have A374636, ranks A375137.
For leaders of anti-runs we have A374680.
For leaders of strictly increasing runs we have A374689.
The complement is counted by A375140, ranks A375295, reverse A375296.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
Cf. A056823, A102726, A188900, A188919, A189077, A238343, A333213, A335480/A335482, A374679, A374688.

b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u - j, o + j - 1]*x^(o + j - 1), {j, 1, u}] + Sum[If[u == 0, b[u + j - 1, o - j]*x^(o - j), 0], {j, 1, o}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[0, n]];
Take[T[40], 40] (* Jean-François Alcover, Sep 15 2018, after Alois P. Heinz in A188919 *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Greater@@First/@Split[Reverse[#],LessEqual]&]],{n,0,15}] (* Gus Wiseman, Aug 20 2024 *)
- or -
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], !MatchQ[#,{_,y_,z_,_,x_,_}/;x<=yGus Wiseman, Aug 20 2024 *)

B_x(i,N) = {my(x='x+O('x^N), f=(x^i)/(1-x^i)*prod(j=i+1,N-i,1/(1-x^j))); f}
A_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N, B_x(i,N)*prod(j=1,i-1,1+B_x(j,N)))); Vec(f)}
A_x(60) \\ John Tyler Rascoe, Aug 23 2024

a(n) = 2^(n-1) - A375140(n).

G.f.: 1 + Sum_{i>0} (B(i,x) * Product_{j=1..i-1} (1 + B(j,x))) where B(i,x) = (x^i)/(1-x^i) * Product_{j>i} (1/(1-x^j)). - John Tyler Rascoe, Aug 23 2024

More terms from Andrew Baxter, May 17 2011

a(30)-a(39) from Alois P. Heinz, Nov 14 2015

A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 17, 32, 58, 112, 201, 371, 694, 1276, 2342, 4330, 7958, 14613, 26866, 49303, 90369, 165646, 303342, 555056, 1015069, 1855230

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(6) = 17 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                      (211)  (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                             (311)  (213)
                                    (231)
                                    (312)
                                    (321)
                                    (411)
                                    (1212)
                                    (1221)
                                    (2112)
                                    (2121)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

These compositions have ranks A374638.
The complement is counted by A374678.
For partitions instead of compositions we have A375133.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A274174, ranks A374249.
- For leaders of weakly increasing runs we have A374632, ranks A374768.
- For leaders of strictly increasing runs we have A374687, ranks A374698.
- For leaders of weakly decreasing runs we have A374743, ranks A374701.
- For leaders of strictly decreasing runs we have A374761, ranks A374767.
Other types of run-leaders (instead of distinct):
- For identical leaders we have A374517.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
Cf. A188920, A233564, A238343, A333213, A333381, A373949, A374515.

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

A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 25, 46, 85, 160, 301, 561, 1056, 1984, 3730, 7037, 13273, 25056, 47382, 89666, 169833, 322038, 611128, 1160660, 2206219, 4196730, 7988731, 15217557, 29005987, 55321015, 105570219, 201569648, 385059094, 735929616, 1407145439, 2691681402

Offset: 0

Gus Wiseman, Aug 01 2024

nonn

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

John Tyler Rascoe, Table of n, a(n) for n = 0..100
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296 or A115029.
These compositions have ranks A374519.
The complement is counted by A374640.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000005 for n > 0, ranks A272919.
- For leaders of weakly increasing runs we have A374631, ranks A374633.
- For leaders of strictly increasing runs we have A374686, ranks A374685.
- For leaders of weakly decreasing runs we have A374742, ranks A374741.
- For leaders of strictly decreasing runs we have A374760, ranks A374759.
Other types of run-leaders (instead of identical):
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A189076, A238343, A333213, A373949, A374515, A374518.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],SameQ@@First/@Split[#,UnsameQ]&]],{n,0,15}]

C_x(N) = {my(g =1/(1 - sum(k=1, N, x^k/(1+x^k))));g}
A_x(i,N) = {my(x='x+O('x^N), f=(x^i)*(C_x(N)*(x^i)+x^i+1)/(1+x^i)^2);f}
B_x(i,j,N) = {my(x='x+O('x^N), f=C_x(N)*x^(i+j)/((1+x^i)*(1+x^j)));f}
D_x(N) = {my(x='x+O('x^N), f=1+sum(i=1,N,-1+sum(j=0,N-i, A_x(i,N)^j)*(1-B_x(i,i,N)+sum(k=1,N-i,B_x(i,k,N)))));Vec(f)}
D_x(30) \\ John Tyler Rascoe, Aug 16 2024

G.f.: 1 + Sum_{i>0} (-1 + Sum_{j>=0} (A(i,x)^j)*(1 + Sum_{k>0, k<>i} (B(i,k,x)))) where A(i,x) = (x^i)*(C(x)*(x^i) + x^i + 1)/(1+x^i)^2, B(i,k,x) = C(x)*x^(i+k)/((1+x^i)*(1+x^k)), and C(x) is the g.f. for A003242. - John Tyler Rascoe, Aug 16 2024

a(26) onwards from John Tyler Rascoe, Aug 16 2024

A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

Original entry on oeis.org

1, 1, 1, 3, 4, 8, 15, 24, 45, 84, 142, 256, 464, 817, 1464, 2621, 4649, 8299, 14819, 26389, 47033, 83833, 149325, 266011, 473867, 843853

Offset: 0

Gus Wiseman, Aug 01 2024

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(6) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)
                (12)  (13)   (14)   (15)
                (21)  (31)   (23)   (24)
                      (121)  (32)   (42)
                             (41)   (51)
                             (122)  (123)
                             (131)  (132)
                             (212)  (141)
                                    (213)
                                    (231)
                                    (312)
                                    (321)
                                    (1212)
                                    (1221)
                                    (2121)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For distinct but not necessarily increasing leaders we have A374518.
For partitions instead of compositions we have A375134.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A374634.
- For leaders of strictly increasing runs we have A374688.
- For leaders of strictly decreasing runs we have A374762.
Other types of run-leaders (instead of strictly increasing):
- For identical leaders we have A374517.
- For distinct leaders we have A374518.
- For weakly increasing leaders we have A374681.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs.
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A188920, A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706, A375133.

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

A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 3, 3, 6, 10, 13, 21, 32, 48, 66, 101, 144, 207, 298, 415, 592, 833, 1163, 1615, 2247, 3088, 4259, 5845, 7977, 10862, 14752, 19969, 26941, 36310, 48725, 65279, 87228, 116274, 154660, 205305, 271879, 359400, 474157, 624257, 820450, 1076357, 1409598

Offset: 0

Gus Wiseman, Jul 27 2024

nonn

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.

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. The weakly decreasing version is A374697.

			The a(0) = 1 through a(8) = 21 compositions:
  ()  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)
                (12)  (13)  (14)   (15)   (16)   (17)
                (21)  (31)  (23)   (24)   (25)   (26)
                            (32)   (42)   (34)   (35)
                            (41)   (51)   (43)   (53)
                            (212)  (123)  (52)   (62)
                                   (213)  (61)   (71)
                                   (231)  (124)  (125)
                                   (312)  (214)  (134)
                                   (321)  (241)  (215)
                                          (313)  (251)
                                          (412)  (314)
                                          (421)  (323)
                                                 (341)
                                                 (413)
                                                 (431)
                                                 (512)
                                                 (521)
                                                 (2123)
                                                 (2312)
                                                 (3212)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The weak version appears to be A189076.
Ranked by positions of strictly decreasing rows in A374683.
The opposite version is A374762.
Types of runs (instead of strictly increasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374680.
- For leaders of weakly increasing runs we have A188920.
- For leaders of weakly decreasing runs we have A374746.
- For leaders of strictly decreasing runs we have A374763.
Types of run-leaders (instead of strictly decreasing):
- For identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For weakly increasing leaders we have A374690.
- For weakly decreasing leaders we have A374697.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf. A000009, A106356, A238343, A261982, A304969, A333213, A374632, A374635, A374640, A374679.

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

C_x(N) = {my(x='x+O('x^N), h=prod(i=1,N, 1+(x^i)*prod(j=i+1,N, 1+x^j))); Vec(h)}
C_x(50) \\ John Tyler Rascoe, Jul 29 2024

G.f.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - John Tyler Rascoe, Jul 29 2024

a(26) onwards from John Tyler Rascoe, Jul 29 2024

A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 27, 50, 96, 185, 353, 672, 1289, 2466, 4722, 9052, 17342, 33244, 63767, 122325, 234727, 450553, 864975, 1660951, 3190089, 6128033

Offset: 0

Gus Wiseman, Aug 01 2024

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 14 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (122)
                                (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (1211)
                                (11111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296.
Other types of runs (instead of anti-):
- For leaders of constant runs we have A000041.
- For leaders of weakly decreasing runs we have A188900.
- For leaders of weakly increasing runs we have A374635.
- For leaders of strictly increasing runs we have A374690.
- For leaders of strictly decreasing runs we have A374764.
Other types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For strictly increasing leaders we have A374679.
- For weakly decreasing leaders we have A374682.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A189076, A238343, A333213, A333381, A373949, A374515, A374632, A374678, A374700, A374706.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 15, 30, 59, 114, 222, 434, 844, 1641, 3189, 6192, 12020, 23320, 45213, 87624, 169744, 328684, 636221, 1231067, 2381269, 4604713, 8901664

Offset: 0

Gus Wiseman, Aug 01 2024

The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.

			The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (131)
                        (1111)  (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For reversed partitions instead of compositions we have A115029.
The complement is A374699.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A189076, complement A374636.
- For leaders of weakly decreasing runs we have A374747.
- For leaders of strictly decreasing runs we have A374765.
- For leaders of strictly increasing runs we have A374697.
Other types of run-leaders (instead of weakly decreasing):
- For identical leaders we have A374517, ranks A374519.
- For distinct leaders we have A374518, ranks A374638.
- For weakly increasing leaders we have A374681.
- For strictly increasing leaders we have A374679.
- For strictly decreasing leaders we have A374680.
A003242 counts anti-runs, ranks A333489.
A106356 counts compositions by number of maximal anti-runs.
A238279 counts compositions by number of maximal runs
A238424 counts partitions whose first differences are an anti-run.
A274174 counts contiguous compositions, ranks A374249.
Cf. A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706.

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,UnsameQ]&]],{n,0,15}]

A374746 Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 18, 31, 51, 86, 143, 241, 397, 657, 1082, 1771, 2889, 4697, 7605, 12269, 19720, 31580, 50412, 80205, 127208, 201149, 317171, 498717, 782076, 1223230, 1908381, 2969950, 4610949, 7141972, 11037276, 17019617, 26188490, 40213388, 61624824

Offset: 0

Gus Wiseman, Jul 26 2024

nonn

The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.

			The a(0) = 1 through a(7) = 18 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)
           (11)  (21)   (22)    (32)     (33)      (43)
                 (111)  (31)    (41)     (42)      (52)
                        (211)   (221)    (51)      (61)
                        (1111)  (311)    (222)     (322)
                                (2111)   (312)     (331)
                                (11111)  (321)     (412)
                                         (411)     (421)
                                         (2211)    (511)
                                         (3111)    (2221)
                                         (21111)   (3112)
                                         (111111)  (3121)
                                                   (3211)
                                                   (4111)
                                                   (22111)
                                                   (31111)
                                                   (211111)
                                                   (1111111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Ranked by positions of strictly decreasing rows in A374740, opp. A374629.
Types of runs (instead of weakly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we have A188920.
- For leaders of anti-runs we have A374680.
- For leaders of strictly increasing runs we have A374689.
- For leaders of strictly decreasing runs we have A374763.
Types of run-leaders (instead of strictly decreasing):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742.
- For distinct leaders we have A374743, ranks A374701.
- For strictly increasing leaders we have opposite A374634.
- For weakly decreasing leaders we have A374747.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A335456 counts patterns matched by compositions.
A373949 counts compositions by run-compressed sum, opposite A373951.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.

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

seq(n)={my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q/(1-x^k); p /= 1 - x^k; q *= (1 - x^k/(1-x^k) + x^k*p)/(1-x^k) );  Vec(r + x^(n\2+1)*q/(1-x))} \\ Andrew Howroyd, Dec 30 2024

G.f.: Sum_{k>=0} x^k*Q(k,x)/(1 - x^k) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 - x^k/(1 - x^k) + x^k*Product_{j=1..k} (1 - x^j))/(1 - x^k) for k > 0. - Andrew Howroyd, Dec 30 2024

a(24)-a(39) from Alois P. Heinz, Jul 26 2024

A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 32, 47, 71, 106, 156, 227, 328, 473, 683, 986, 1421, 2040, 2916, 4149, 5882, 8314, 11727, 16515, 23221, 32593, 45655, 63810, 88979, 123789, 171838, 238055, 329187, 454451, 626412, 862164, 1184917, 1626124, 2228324, 3048982, 4165640, 5682847

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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.

			The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
The a(0) = 1 through a(8) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
                (21)  (31)   (32)   (42)   (43)    (53)
                      (211)  (41)   (51)   (52)    (62)
                             (311)  (312)  (61)    (71)
                                    (321)  (322)   (413)
                                    (411)  (412)   (422)
                                           (421)   (431)
                                           (511)   (512)
                                           (3121)  (521)
                                           (3211)  (611)
                                                   (3212)
                                                   (3221)
                                                   (4121)
                                                   (4211)
                                                   (31211)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

The opposite version is A374688.
The weak version is A374747.
For partitions instead of compositions we have A375133.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of weakly increasing runs we appear to have A188920.
- For leaders of anti-runs we have A374680.
- For leaders of strictly increasing runs we have A374689.
- For leaders of weakly decreasing runs we have A374746.
Other types of run-leaders (instead of strictly decreasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For weakly increasing leaders we have A374764.
- For weakly decreasing leaders we have A374765.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.

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

seq(n)={ my(A=O(x*x^n), p=1+A, q=p, r=p); for(k=1, n\2, r += x^k*q; p *= 1 + x^k; q *= 1 + x^k*p); Vec(r + x^(n\2+1)*q/(1-x)) } \\ Andrew Howroyd, Dec 30 2024

G.f.: Sum_{k>=0} x^k*Q(k,x) where Q(0,x) = 1 and Q(k,x) = Q(k-1,x) * (1 + x^k*Product_{j=1..k} (1 + x^j)) for k > 0. - Andrew Howroyd, Dec 30 2024

a(24) onwards from Andrew Howroyd, Dec 30 2024

A374764 Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 23, 40, 69, 118, 199, 333, 553, 911, 1492, 2428, 3928, 6323, 10129, 16151, 25646, 40560, 63905, 100332, 156995, 244877, 380803, 590479, 913100, 1408309, 2166671, 3325445, 5092283, 7780751, 11863546, 18052080, 27415291, 41556849, 62879053, 94975305, 143213145

Offset: 0

Gus Wiseman, Jul 30 2024

nonn

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.

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 weakly increasing [but weakly decreasing works too]. The strictly increasing version is A374762.

			The composition (1,1,2,1) has strictly decreasing runs ((1),(1),(2,1)) with leaders (1,1,2) so is counted under a(5).
The composition (1,2,1,1) has strictly decreasing runs ((1),(2,1),(1)) with leaders (1,2,1) so is not counted under a(5).
The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (1111)  (122)
                                (131)
                                (212)
                                (221)
                                (1112)
                                (1121)
                                (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

For partitions instead of compositions we have A034296.
For strictly increasing leaders we have A374688.
The opposite version is A374697.
Other types of runs (instead of strictly decreasing):
- For leaders of identical runs we have A000041.
- For leaders of anti-runs we have A374681.
- For leaders of weakly increasing runs we have A374635.
- For leaders of strictly increasing runs we have A374690.
- For leaders of weakly decreasing runs we have A188900.
Other types of run-leaders (instead of weakly increasing):
- For identical leaders we have A374760, ranks A374759.
- For distinct leaders we have A374761, ranks A374767.
- For strictly increasing leaders we have A374762.
- For weakly decreasing leaders we have A374765.
- For strictly decreasing leaders we have A374763.
A003242 counts anti-run compositions, ranks A333489.
A011782 counts compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335548 counts non-contiguous compositions, ranks A374253.
A373949 counts compositions by run-compressed sum, opposite A373951.
Cf. A106356, A188920, A238343, A261982, A333213, A374687, A374679, A374680, A374742, A374743, A374747.

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

seq(n) = Vec(1/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

G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j=1..k-1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024

a(24) onwards from Andrew Howroyd, Jul 31 2024

cp's OEIS Frontend

A188920 a(n) is the limiting term of the n-th column of the triangle in A188919.

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A374518 Number of integer compositions of n whose leaders of anti-runs are distinct.

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A374517 Number of integer compositions of n whose leaders of anti-runs are identical.

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A374679 Number of integer compositions of n whose leaders of anti-runs are strictly increasing.

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A374689 Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.

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A374681 Number of integer compositions of n whose leaders of anti-runs are weakly increasing.

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A374682 Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.

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