A374690
Number of integer compositions of n whose leaders of strictly increasing runs are weakly increasing.
Original entry on oeis.org
1, 1, 2, 3, 6, 10, 19, 34, 63, 115, 211, 387, 710, 1302, 2385, 4372, 8009, 14671, 26867, 49196, 90069, 164884, 301812, 552406, 1011004, 1850209, 3385861, 6195832, 11337470, 20745337, 37959030, 69454669, 127081111, 232517129, 425426211, 778376479, 1424137721
Offset: 0
The composition (1,1,3,2,3,2) has strictly increasing runs ((1),(1,3),(2,3),(2)), with leaders (1,1,2,2), so is counted under a(12).
The a(0) = 1 through a(6) = 19 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (12) (13) (14) (15)
(111) (22) (23) (24)
(112) (113) (33)
(121) (122) (114)
(1111) (131) (123)
(1112) (132)
(1121) (141)
(1211) (222)
(11111) (1113)
(1122)
(1131)
(1212)
(1311)
(11112)
(11121)
(11211)
(12111)
(111111)
Ranked by positions of weakly increasing 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
A374681.
- For leaders of weakly increasing runs we have
A374635.
- For leaders of weakly decreasing runs we have
A188900.
- For leaders of strictly decreasing runs we have
A374764.
Types of run-leaders (instead of weakly increasing):
- For strictly increasing leaders we have
A374688.
- For strictly decreasing leaders we have
A374689.
- For weakly decreasing leaders we have
A374697.
A335456 counts patterns matched by compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf.
A000009,
A106356,
A188920,
A189076,
A238343,
A261982,
A333213,
A374629,
A374630,
A374632,
A374679.
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Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],LessEqual@@First/@Split[#,Less]&]],{n,0,15}]
A374765
Number of integer compositions of n whose leaders of strictly decreasing runs are weakly decreasing.
Original entry on oeis.org
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 141, 225, 357, 565, 891, 1399, 2191, 3420, 5321, 8256, 12774, 19711, 30339, 46584, 71359, 109066, 166340, 253163, 384539, 582972, 882166, 1332538, 2009377, 3024969, 4546562, 6822926, 10223632, 15297051, 22855872, 34103117
Offset: 0
The composition (3,1,2,2,1) has strictly decreasing runs ((3,1),(2),(2,1)), with leaders (3,2,2), so is counted under a(9).
The a(0) = 1 through a(6) = 13 compositions:
() (1) (2) (3) (4) (5) (6)
(11) (21) (22) (32) (33)
(111) (31) (41) (42)
(211) (212) (51)
(1111) (221) (222)
(311) (312)
(2111) (321)
(11111) (411)
(2121)
(2211)
(3111)
(21111)
(111111)
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
A189076.
- For leaders of anti-runs we have
A374682.
- For leaders of strictly increasing runs we have
A374697.
- For leaders of weakly decreasing runs we have
A374747.
Other types of run-leaders (instead of weakly decreasing):
- For strictly increasing leaders we have
A374762.
- For strictly decreasing leaders we have
A374763.
- For weakly increasing leaders we have
A374764.
Cf.
A106356,
A188900,
A188920,
A238343,
A261982,
A333213,
A374635,
A374636,
A374689,
A374742,
A374743,
A375133.
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Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],GreaterEqual@@First/@Split[#,Greater]&]],{n,0,15}]
-
dfs(m, r, u) = 1 + sum(s=r, min(m, u), dfs(m-s, s, s)*x^s + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)*prod(i=t+1, s-1, 1+x^i)));
lista(nn) = Vec(dfs(nn, 1, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 13 2025
A375135
Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613, 27378579, 55801506, 113517749, 230544752, 467519136, 946815630, 1915199736, 3869892105, 7812086380, 15756526347
Offset: 0
The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
. . . . . (122) (132) (133) (143)
(1122) (142) (152)
(1221) (1132) (233)
(1222) (1133)
(1321) (1142)
(2122) (1223)
(11122) (1232)
(11221) (1322)
(12211) (1331)
(1421)
(2132)
(3122)
(11132)
(11222)
(11321)
(12122)
(12212)
(12221)
(13211)
(21122)
(21221)
(111122)
(111221)
(112211)
(122111)
For leaders of constant runs we have
A056823.
For leaders of weakly increasing runs we have
A374636, complement
A189076?
The complement is counted by
A374697.
A335456 counts patterns matched by compositions.
A374700 counts compositions by sum of leaders of strictly increasing runs.
-
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], !GreaterEqual@@First/@Split[#,Less]&]],{n,0,15}]
A376263
Number of strict integer compositions of n whose leaders of increasing runs are increasing.
Original entry on oeis.org
1, 1, 1, 2, 2, 3, 5, 6, 8, 11, 18, 21, 30, 38, 52, 77, 96, 126, 167, 217, 278, 402, 488, 647, 822, 1073, 1340, 1747, 2324, 2890, 3695, 4690, 5924, 7469, 9407, 11718, 15405, 18794, 23777, 29507, 37188, 45720, 57404, 70358, 87596, 110672, 135329, 167018, 206761, 254200, 311920
Offset: 0
The a(1) = 1 through a(9) = 11 compositions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7) (1,8)
(2,3) (2,4) (2,5) (2,6) (2,7)
(1,2,3) (3,4) (3,5) (3,6)
(1,3,2) (1,2,4) (1,2,5) (4,5)
(1,4,2) (1,3,4) (1,2,6)
(1,4,3) (1,3,5)
(1,5,2) (1,5,3)
(1,6,2)
(2,3,4)
(2,4,3)
For less-greater or greater-less we have
A294617.
A374700 counts compositions by sum of leaders of strictly increasing runs.
Cf.
A000110,
A008289,
A056823,
A106356,
A188920,
A238343,
A261982,
A274174,
A333213,
A374634,
A374683,
A374698,
A374763.
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Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], UnsameQ@@#&&Less@@First/@Split[#,Less]&]],{n,0,15}]
-
\\ here Q(n) gives n-th row of A008289.
Q(n)={Vecrev(polcoef(prod(k=1, n, 1 + y*x^k, 1 + O(x*x^n)), n)/y)}
a(n)={if(n==0, 1, my(r=Q(n), s=Vec(serlaplace(exp(exp(x+O(x^#r))- 1)))); sum(k=1, #r, r[k]*s[k]))} \\ Andrew Howroyd, Sep 18 2024
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