Displaying 1-10 of 11 results found.
a(n) is the limiting term of the n-th column of the triangle in A188919.
+10
59
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
COMMENTS
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
FORMULA
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
EXAMPLE
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)
MATHEMATICA
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]];
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<=y<z]&]], {n, 0, 15}] (* Gus Wiseman, Aug 20 2024 *)
PROG
(PARI)
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)}
CROSSREFS
For leaders of identical runs we have A000041. For weakly increasing leaders we have A374635.
For leaders of anti-runs we have A374680. For leaders of strictly increasing runs we have A374689.
Number of integer compositions of n whose leaders of anti-runs are distinct.
+10
34
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], UnsameQ@@First/@Split[#, UnsameQ]&]], {n, 0, 15}]
CROSSREFS
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 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. 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.
Number of integer compositions of n whose leaders of anti-runs are identical.
+10
29
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
COMMENTS
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.
FORMULA
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
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], SameQ@@First/@Split[#, UnsameQ]&]], {n, 0, 15}]
PROG
(PARI)
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)}
CROSSREFS
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. 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.
Number of integer compositions of n whose leaders of anti-runs are strictly increasing.
+10
18
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Less@@First/@Split[#, UnsameQ]&]], {n, 0, 15}]
CROSSREFS
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. 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, A238343, A333213, A333381, A373949, A374515, A374632, A374635, A374678, A374700, A374706, A375133.
Number of integer compositions of n whose leaders of strictly increasing runs are strictly decreasing.
+10
15
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
COMMENTS
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.
FORMULA
G.f.: Product_{i>0} (1 + (x^i)*Product_{j>i} (1 + x^j)). - John Tyler Rascoe, Jul 29 2024
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Greater@@First/@Split[#, Less]&]], {n, 0, 15}]
PROG
(PARI)
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)}
CROSSREFS
The weak version appears to be A189076. Ranked by positions of strictly decreasing rows in A374683. 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 strictly increasing leaders we have A374688. - For weakly increasing leaders we have A374690. - For weakly decreasing leaders we have A374697. A335456 counts patterns matched by compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Number of integer compositions of n whose leaders of anti-runs are weakly increasing.
+10
11
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], LessEqual@@First/@Split[#, UnsameQ]&]], {n, 0, 15}]
CROSSREFS
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 strictly increasing leaders we have A374679. - For weakly decreasing leaders we have A374682. - For strictly decreasing leaders we have A374680. 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.
Number of integer compositions of n whose leaders of anti-runs are weakly decreasing.
+10
11
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], GreaterEqual@@First/@Split[#, UnsameQ]&]], {n, 0, 15}]
CROSSREFS
For reversed partitions instead of compositions we have A115029. 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 weakly increasing leaders we have A374681. - For strictly increasing leaders we have A374679. - For strictly decreasing leaders we have A374680. 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.
Number of integer compositions of n whose leaders of strictly decreasing runs are weakly increasing.
+10
11
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
COMMENTS
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.
FORMULA
G.f.: 1/(Product_{k>=1} (1 - x^k*Product_{j=1..k-1} (1 + x^j))). - Andrew Howroyd, Jul 31 2024
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], LessEqual@@First/@Split[#, Greater]&]], {n, 0, 15}]
PROG
(PARI) 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
CROSSREFS
For partitions instead of compositions we have A034296. For strictly increasing leaders we have A374688. 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 strictly increasing leaders we have A374762. - For weakly decreasing leaders we have A374765. - For strictly decreasing leaders we have A374763. Cf. A106356, A188920, A238343, A261982, A333213, A374687, A374679, A374680, A374742, A374743, A374747.
Number of integer compositions of n whose leaders of weakly decreasing runs are strictly decreasing.
+10
10
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
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Greater@@First/@Split[#, GreaterEqual]&]], {n, 0, 15}]
CROSSREFS
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 strictly increasing leaders we have opposite A374634. - For weakly decreasing leaders we have A374747. A335456 counts patterns matched by compositions.
A374748 counts compositions by sum of leaders of weakly decreasing runs.
Cf. A000009, A003242, A106356, A189076, A238343, A261982, A333213, A358836, A374632, A374635, A374741.
Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.
+10
10
1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 32, 47, 71, 106, 156, 227, 328, 473, 683, 986, 1421, 2040, 2916, 4149
COMMENTS
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.
EXAMPLE
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)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Greater@@First/@Split[#, Greater]&]], {n, 0, 15}]
CROSSREFS
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 strictly increasing leaders we have A374762. - For weakly increasing leaders we have A374764. - For weakly decreasing leaders we have A374765. Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.
Search completed in 0.017 seconds
| 