A335456 -id:A335456 - cp's OEIS frontend

A335838 Number of normal patterns contiguously matched by integer partitions of n.

1, 2, 5, 9, 18, 31, 54, 89, 145, 225, 349, 524, 778, 1137, 1645, 2330, 3293, 4586, 6341, 8676, 11794, 15880, 21292, 28298, 37419, 49163, 64301, 83576, 108191, 139326, 178699, 228183, 290286, 367760, 464374, 584146, 732481, 915468, 1140773, 1417115, 1755578

Offset: 0

Gus Wiseman, Jun 27 2020

nonn

We define a (normal) pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to contiguously match a pattern P if there is a contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) contiguously matches (1,1,2) and (2,1,1) but not (2,1,2), (1,2,1), (1,2,2), or (2,2,1).

			The patterns contiguously matched by (3,2,2,1) are: (), (1), (1,1), (2,1), (2,1,1), (2,2,1), (3,2,2,1). Note that (3,2,1) is not contiguously matched. See A335837 for a larger example.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for compositions in standard order is A335474.
The version for compositions is A335457.
The not necessarily contiguous version is A335837.
Patterns are counted by A000670 and ranked by A333217.
Patterns contiguously matched by prime indices are counted by A335516.
Contiguous divisors are counted by A335519.
Minimal patterns avoided by prime indices are counted by A335550.
Cf. A056986, A108917, A333257, A335454, A335456, A335458, A335549.

mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Sum[Length[Union[mstype/@ReplaceList[y,{_,s___,_}:>{s}]]],{y,IntegerPartitions[n]}],{n,0,8}]

More terms from Jinyuan Wang, Jun 27 2020

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

A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 24, 43, 76, 136, 242, 431, 764, 1353, 2387, 4202, 7376, 12918, 22567, 39338, 68421, 118765, 205743, 355756, 614038, 1058023, 1820029, 3125916, 5360659, 9179700, 15697559, 26807303, 45720739, 77881393, 132505599, 225182047, 382252310, 648187055

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 composition y = (3,2,1,2,2,1,2,5,1,1,1) has weakly decreasing runs ((3,2,1),(2,2,1),(2),(5,1,1,1)), with leaders (3,2,2,5), which are not weakly decreasing, so y is not counted under a(21).
The a(0) = 1 through a(6) = 14 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)
                                         (2112)
                                         (2121)
                                         (2211)
                                         (3111)
                                         (21111)
                                         (111111)

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

Ranked by positions of weakly decreasing rows in A374740, opposite A374629.
Types of runs (instead of weakly 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 strictly decreasing runs we have A374765.
Types of run-leaders (instead of weakly decreasing):
- For weakly increasing leaders we appear to have A188900.
- For identical leaders we have A374742, ranks A374744.
- For distinct leaders we have A374743, ranks A374701.
- For strictly increasing leaders we have opposite A374634.
- For strictly decreasing leaders we have A374746.
A011782 counts compositions.
A124765 counts weakly decreasing runs in standard 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, A188920, A238343, A261982, A333213, A374630, A374635, A374636, A374741.

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

dfs(m, r, u) = 1 + sum(s=r+1, min(m, u), x^s/(1-x^s) + sum(t=1, min(s-1, m-s), dfs(m-s-t, t, s)*x^(s+t)/prod(i=t, s, 1-x^i)));
lista(nn) = Vec(dfs(nn, 0, nn) + O(x^(1+nn))); \\ Jinyuan Wang, Feb 14 2025

More terms from Jinyuan Wang, Feb 14 2025

A335455 Number of compositions of n with some part appearing more than twice.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 11, 30, 69, 142, 334, 740, 1526, 3273, 6840, 14251, 29029, 59729, 122009, 248070, 500649, 1012570, 2040238, 4107008, 8257466, 16562283, 33229788, 66621205, 133478437, 267326999, 535146239, 1071183438, 2143604313, 4289194948, 8581463248

Offset: 0

Gus Wiseman, Jun 15 2020

nonn

Also the number of compositions of n matching the pattern (1,1,1).

A composition of n is a finite sequence of positive integers summing to n.

			The a(3) = 1 through a(6) = 11 compositions:
  (111)  (1111)  (1112)   (222)
                 (1121)   (1113)
                 (1211)   (1131)
                 (2111)   (1311)
                 (11111)  (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The case of partitions is A000726.
The avoiding version is A232432.
The (1,1)-matching version is A261982.
The version for patterns is A335508.
The version for prime indices is A335510.
These compositions are ranked by A335512.
Compositions are counted by A011782.
Combinatory separations are counted by A269134.
Normal patterns matched by compositions are counted by A335456.
Cf. A000740, A034691, A056986, A080599, A106356, A181796, A232464, A238279, A333755.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Max@@Length/@Split[Sort[#]]>=3&]],{n,0,10}]

a(n > 0) = 2^(n - 1) - A232432(n).

A335466 Numbers k such that the k-th composition in standard order (A066099) matches (1,2,1).

Original entry on oeis.org

13, 25, 27, 29, 45, 49, 51, 53, 54, 55, 57, 59, 61, 77, 82, 89, 91, 93, 97, 99, 101, 102, 103, 105, 107, 108, 109, 110, 111, 113, 115, 117, 118, 119, 121, 123, 125, 141, 153, 155, 157, 162, 165, 166, 173, 177, 178, 179, 181, 182, 183, 185, 187, 189, 193, 195

Offset: 1

Gus Wiseman, Jun 15 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The sequence of terms together with the corresponding compositions begins:
  13: (1,2,1)
  25: (1,3,1)
  27: (1,2,1,1)
  29: (1,1,2,1)
  45: (2,1,2,1)
  49: (1,4,1)
  51: (1,3,1,1)
  53: (1,2,2,1)
  54: (1,2,1,2)
  55: (1,2,1,1,1)
  57: (1,1,3,1)
  59: (1,1,2,1,1)
  61: (1,1,1,2,1)
  77: (3,1,2,1)
  82: (2,3,2)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335467 is the avoiding version.
The (2,1,2)-matching version is A335468.
These compositions are counted by A335470.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335446, A335456, A335458, A335509.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],MatchQ[stc[#],{_,x_,_,y_,_,x_,_}/;x

A335509 Number of patterns of length n matching the pattern (1,1,2).

Original entry on oeis.org

0, 0, 0, 1, 15, 181, 2163, 27133, 364395, 5272861, 82289163, 1383131773, 24978057195, 483269202781, 9987505786443, 219821796033853, 5137810967933355, 127169580176271901, 3324712113052429323, 91585136315240091133, 2652142325158529483115, 80562824634615270041821

Offset: 0

Gus Wiseman, Jun 18 2020

nonn

Also the number of (1,2,1)-matching patterns of length n.
Also the number of (2,1,2)-matching patterns of length n.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(3) = 1 through a(4) = 15 patterns:
  (1,1,2)  (1,1,1,2)
           (1,1,2,1)
           (1,1,2,2)
           (1,1,2,3)
           (1,1,3,2)
           (1,2,1,2)
           (1,2,1,3)
           (1,2,2,3)
           (1,3,1,2)
           (2,1,1,2)
           (2,1,1,3)
           (2,1,2,3)
           (2,2,1,3)
           (2,2,3,1)
           (3,1,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A001710 is the avoiding version.
Compositions matching this pattern are counted by A335470 and ranked by A335476.
Permutations of prime indices matching this pattern are counted by A335446.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Patterns matching (1,2,3) are counted by A335515.
Cf. A034691, A056986, A238279, A292884, A333755, A335456, A335457.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MatchQ[#,{_,x_,_,x_,_,y_,_}/;x

seq(n)={Vec(serlaplace(1/(2-exp(x + O(x*x^n))) - (2-2*x+x^2)/(2*(1-x)^2)), -(n+1))} \\ Andrew Howroyd, Dec 31 2020

E.g.f.: 1/(2-exp(x)) - (2-2*x+x^2)/(2*(1-x)^2). - Andrew Howroyd, Dec 31 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 31 2020

A335520 Number of (1,2,3)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 30, 60, 120, 210, 180, 480:
  (123)  (1123)  (11123)  (1234)  (11223)  (1111123)
         (1213)  (11213)  (1243)  (11232)  (1111213)
         (1231)  (11231)  (1324)  (12123)  (1111231)
                 (12113)  (1342)  (12132)  (1112113)
                 (12131)  (1423)  (12213)  (1112131)
                 (12311)  (2134)  (12231)  (1112311)
                          (2314)  (12312)  (1121113)
                          (2341)  (12321)  (1121131)
                          (3124)  (21123)  (1121311)
                          (4123)  (21213)  (1123111)
                                  (21231)  (1211113)
                                           (1211131)
                                           (1211311)
                                           (1213111)
                                           (1231111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of nonzero terms are A000977.
These permutations are ranked by A335479.
These compositions are counted by A335514.
Patterns matching this pattern are counted by A335515.
The complement A335521 is the avoiding version.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

For n > 0, a(n) + A335521(n) = A008480(n).

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

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.

			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)

Christian Sievers, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

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 identical leaders we have A374686, ranks A374685.
- For distinct leaders we have A374687, ranks A374698.
- For strictly increasing leaders we have A374688.
- For strictly decreasing leaders we have A374689.
- 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, A188920, A189076, A238343, A261982, A333213, A374629, A374630, A374632, A374679.

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

a(26) and beyond from Christian Sievers, Aug 08 2024

A002051 Steffensen's bracket function [n,2].

Original entry on oeis.org

0, 0, 1, 9, 67, 525, 4651, 47229, 545707, 7087005, 102247051, 1622631549, 28091565547, 526858344285, 10641342962251, 230283190961469, 5315654681948587, 130370767029070365, 3385534663256714251, 92801587319328148989, 2677687796244383678827, 81124824998504072833245, 2574844419803190382447051

Offset: 1

N. J. A. Sloane

a(n) is the number of ways to arrange the blocks of the partitions of {1,2,...,n} in an undirected cycle of length 3 or more, see A000629. - Geoffrey Critzer, Nov 23 2012
From Gus Wiseman, Jun 24 2020: (Start)
Also the number of (1,2)-matching length-n sequences covering an initial interval of positive integers. For example, the a(2) = 1 and a(3) = 9 sequences are:
  (1,2)  (1,1,2)
         (1,2,1)
         (1,2,2)
         (1,2,3)
         (1,3,2)
         (2,1,2)
         (2,1,3)
         (2,3,1)
         (3,1,2)
Missing from this list are:
  (1,1)  (1,1,1)
  (2,1)  (2,1,1)
         (2,2,1)
         (3,2,1)
(End)

			a(4) = 9. There are 6 partitions of {1,2,3,4} into exactly three blocks and one way to put them in an undirected cycle of length three. There is one partition of {1,2,3,4} into four blocks and 3 ways to make an undirected cycle of length four. 6 + 3 = 9. - _Geoffrey Critzer_, Nov 23 2012

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Steffensen, J. F. Interpolation. 2d ed. Chelsea Publishing Co., New York, N. Y., 1950. ix+248 pp. MR0036799 (12,164d)

T. D. Noe, Table of n, a(n) for n = 1..100
G. J. Simmons, Letter to N. J. Sloane, May 29 1974
J. F. Steffensen, On a class of polynomials and their application to actuarial problems, Skandinavisk Aktuarietidskrift, Vol. 11, pp. 75-97, 1928.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

A diagonal of the triangular array in A241168.
(1,2)-avoiding patterns are counted by A011782.
(1,1)-matching patterns are counted by A019472.
(1,2)-matching permutations are counted by A033312.
(1,2)-matching compositions are counted by A056823.
(1,2)-matching permutations of prime indices are counted by A335447.
(1,2)-matching compositions are ranked by A335485.
Patterns are counted by A000670 and ranked by A333217.
Patterns matched by compositions are counted by A335456.
Cf. A056986, A335454, A335465, A335486, A335515.

a[n_] := Sum[ k!*StirlingS2[n-1, k], {k, 0, n-1}] - 2^(n-2); Table[a[n], {n, 3, 17}] (* Jean-François Alcover, Nov 18 2011, after Manfred Goebel *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!GreaterEqual@@#&]],{n,0,5}] (* Gus Wiseman, Jun 24 2020 *)

a(n) = sum(s=2, n-1, stirling(n,s+1,2)*s!/2); \\ Michel Marcus, Jun 24 2020

[n,2] = Sum_{s=2..n-1} Stirling2(n,s+1)*s!/2 (cf. A241168).
a(1)=0; for n >= 2, a(n) = A000670(n-1) - 2^(n-2). - Manfred Goebel (mkgoebel(AT)essex.ac.uk), Feb 20 2000; formula adjusted by N. J. A. Sloane, Apr 22 2014. For example, a(5) = 67 = A000670(4)-2^3 = 75-8 = 67.
E.g.f.: (1 - exp(2*x) - 2*log(2 - exp(x)))/4 = B(A(x)) where A(x) = exp(x)-1 and B(x) = (log(1/(1-x))- x - x^2/2)/2. - Geoffrey Critzer, Nov 23 2012

Entry revised by N. J. A. Sloane, Apr 22 2014

A335469 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (2,1,2).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Jun 16 2020

nonn

First differs from A374701 in having 150, corresponding to (3,2,1,2). - Gus Wiseman, Sep 18 2024
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			See A335468 for an example of the complement.

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.
Gus Wiseman, Statistics, classes, and transformations of standard compositions

The complement A335468 is the matching version.
The (1,2,1)-avoiding version is A335467.
These compositions are counted by A335473.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134 and ranked by A334030.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A001710, A034691, A056986, A108917, A114994, A238279, A333224, A333257, A335450, A335456, A335458.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
Select[Range[0,100],!MatchQ[stc[#],{_,x_,_,y_,_,x_,_}/;x>y]&]

cp's OEIS Frontend

A335838 Number of normal patterns contiguously matched by integer partitions of n.

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

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A374747 Number of integer compositions of n whose leaders of weakly decreasing runs are themselves weakly decreasing.

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A335455 Number of compositions of n with some part appearing more than twice.

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A335466 Numbers k such that the k-th composition in standard order (A066099) matches (1,2,1).

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A335509 Number of patterns of length n matching the pattern (1,1,2).

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A335520 Number of (1,2,3)-matching permutations of the prime indices of n.

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