A107429 -id:A107429 - cp's OEIS frontend

A188575 Number of non-complete compositions of n.

0, 1, 1, 4, 8, 14, 31, 63, 129, 248, 509, 1011, 2044, 4089, 8167, 16360, 32725, 65482, 131017, 262176, 524167, 1048678, 2096985, 4194358, 8387802, 16776408, 33550943, 67101615, 134199983, 268399122, 536793004, 1073590077, 2147187353, 4294419287, 8588940438

Offset: 1

N. J. A. Sloane, Apr 04 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000
M. Archibald and A. Knopfmacher, The largest missing value in a composition of an integer, Discrete Math., 311 (2011), 723-731.
Alois P. Heinz, Plot of (a(n)-2^(n-2))/2^(n-2) for n = 60..1000

Cf. A107429, A188576, A188577, A251729.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, t!, 0),
      `if`(i<1 or n 2^(n-1) -add(b(n, i, 0), i=1..n):
seq(a(n), n=1..40);  # Alois P. Heinz, Dec 06 2014

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, t!, 0], If[i<1 || nJean-François Alcover, Mar 09 2015, after Alois P. Heinz *)

a(n) = 2^(n-1) - A107429(n) ~ 2^(n-2). - Alois P. Heinz, Dec 06 2014

More terms from Alois P. Heinz, Dec 06 2014

A332272 Number of narrowly recursively normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 14, 18, 23, 30, 37, 46, 52, 70, 80, 100, 116, 146, 171, 203, 236, 290, 332, 401, 458, 547, 626, 744, 851, 1004, 1157, 1353, 1553, 1821, 2110, 2434, 2810, 3250, 3741, 4304, 4949, 5661, 6510, 7450, 8501, 9657, 11078, 12506, 14329, 16185

Offset: 0

Gus Wiseman, Mar 08 2020

nonn

A sequence is narrowly recursively normal if either it is constant (narrow) or its run-lengths are a narrowly recursively normal sequence covering an initial interval of positive integers (normal).

			The a(6) = 8 partitions are (6), (51), (42), (411), (33), (321), (222), (111111). Missing from this list are (3111), (2211), (21111).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (111111)  (511)      (431)
                                               (3211)     (521)
                                               (1111111)  (611)
                                                          (2222)
                                                          (3221)
                                                          (4211)
                                                          (11111111)

The strict instead of narrow version is A330937.
The normal case is A332277.
The widely normal case is A332277(n) - 1 for n > 1.
The wide version is A332295(n) - 1.
Cf. A000009, A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332576.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[Length[qtn]<=1,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

For n > 1, a(n) = A317491(n) + A000005(n) - 2.

A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 14, 14, 5, 0, 1, 30, 45, 32, 8, 0, 1, 62, 124, 131, 65, 13, 0, 1, 126, 315, 438, 323, 128, 21, 0, 1, 254, 762, 1305, 1270, 747, 243, 34, 0, 1, 510, 1785, 3612, 4346, 3370, 1629, 452, 55

Offset: 0

Gus Wiseman, Mar 02 2020

In other words, parts of subsequent, non-successive blocks are increasing.

			Triangle begins:
    1
    0    1
    0    1    2
    0    1    6    3
    0    1   14   14    5
    0    1   30   45   32    8
    0    1   62  124  131   65   13
    0    1  126  315  438  323  128   21
    0    1  254  762 1305 1270  747  243   34
    ...
Row n = 4 counts the following ordered set partitions:
  {1234}  {1}{234}  {1}{2}{34}  {1}{2}{3}{4}
          {12}{34}  {1}{23}{4}  {1}{2}{4}{3}
          {123}{4}  {12}{3}{4}  {1}{3}{2}{4}
          {124}{3}  {1}{24}{3}  {2}{1}{3}{4}
          {13}{24}  {12}{4}{3}  {2}{1}{4}{3}
          {134}{2}  {1}{3}{24}
          {14}{23}  {13}{2}{4}
          {2}{134}  {1}{34}{2}
          {23}{14}  {1}{4}{23}
          {234}{1}  {2}{1}{34}
          {24}{13}  {2}{13}{4}
          {3}{124}  {2}{14}{3}
          {34}{12}  {23}{1}{4}
          {4}{123}  {3}{12}{4}

An apparently related triangle is A056242.
Column k = n - 1 is A332724.
Row sums are A332872, which appears to be A007052 shifted right once.
Ordered set-partitions are A000670.
Unimodal compositions are A001523.
Non-unimodal normal sequences are A328509.
Cf. A072704, A097805, A107429, A227038, A332280, A332283, A332288, A332577.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[Join@@Permutations/@sps[Range[n]],Length[#]==k&&!MatchQ[#,{_,{_,a_,_},,{_,b_,_},_}/;a>b]&]],{n,0,5},{k,0,n}]

A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 0, 2, 3, 3, 6, 12, 12, 42, 114, 210, 60, 360, 720, 1320, 1590, 3690, 6450, 16110, 33120, 59940, 61320, 112980, 171780, 387240, 803880, 769440, 1773240, 2823240, 5790960, 9916200, 19502280, 28244160, 56881440, 130548600, 279578880, 320554080, 541323720

Offset: 0

Gus Wiseman, Nov 21 2019

nonn

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

			The a(1) = 1 through a(8) = 12 compositions (empty column not shown):
  (1)  (12)  (112)  (122)  (123)  (1123)  (1223)
       (21)  (121)  (212)  (132)  (1132)  (1232)
             (211)  (221)  (213)  (1213)  (1322)
                           (231)  (1231)  (2123)
                           (312)  (1312)  (2132)
                           (321)  (1321)  (2213)
                                  (2113)  (2231)
                                  (2131)  (2312)
                                  (2311)  (2321)
                                  (3112)  (3122)
                                  (3121)  (3212)
                                  (3211)  (3221)

Looking at run-lengths instead of multiplicities gives A329749.
The non-complete version is A329741.
Complete compositions are A107429.
Cf. A008965, A098504, A244164, A274174, A329740, A329744, A329766.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[Sort[#]]]&]],{n,0,10}]

a(21)-a(38) from Alois P. Heinz, Jul 06 2020

A332274 Number of totally strong compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 7, 11, 22, 33, 56, 93, 162, 264, 454, 765, 1307, 2237, 3849, 6611, 11472, 19831, 34446, 59865, 104293, 181561, 316924

Offset: 0

Gus Wiseman, Feb 11 2020

A sequence is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong sequence.
A composition of n is a finite sequence of positive integers with sum n.
Also the number of totally co-strong compositions of n.

			The a(1) = 1 through a(5) = 11 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (121)   (41)
                    (211)   (122)
                    (1111)  (131)
                            (212)
                            (311)
                            (2111)
                            (11111)

The case of partitions is A316496.
The co-strong case is A332274 (this sequence).
The case of reversed partitions is A332275.
The alternating version is A332338.
Cf. A100883, A107429, A317245, A317256, A317491, A329744, A332272, A332279, A332289, A332292, A332336, A332337, A332339, A332340.

tni[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],tni[Length/@Split[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tni]],{n,0,15}]

A333148 Number of compositions of n whose non-adjacent parts are weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 7, 12, 19, 30, 46, 69, 102, 149, 214, 304, 428, 596, 823, 1127, 1532, 2068, 2774, 3697, 4900, 6460, 8474, 11061, 14375, 18600, 23970, 30770, 39354, 50153, 63702, 80646, 101783, 128076, 160701, 201076, 250933, 312346, 387832, 480409, 593716, 732105, 900810, 1106063, 1355336, 1657517, 2023207, 2464987, 2997834, 3639464

Offset: 0

Gus Wiseman, May 16 2020

nonn

			The a(1) = 1 through a(6) = 19 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)
       (11)  (12)   (13)    (14)     (15)
             (21)   (22)    (23)     (24)
             (111)  (31)    (32)     (33)
                    (121)   (41)     (42)
                    (211)   (131)    (51)
                    (1111)  (212)    (141)
                            (221)    (222)
                            (311)    (231)
                            (1211)   (312)
                            (2111)   (321)
                            (11111)  (411)
                                     (1311)
                                     (2121)
                                     (2211)
                                     (3111)
                                     (12111)
                                     (21111)
                                     (111111)
For example, (2,3,1,2) is such a composition, because the non-adjacent pairs of parts are (2,1), (2,2), (3,2), all of which are weakly decreasing.

Alois P. Heinz, Table of n, a(n) for n = 0..700

Unimodal compositions are A001523.
The case of normal sequences appears to be A028859.
A version for ordered set partitions is A332872.
The case of strict compositions is A333150.
The version for strictly decreasing parts is A333193.
Standard composition numbers (A066099) of these compositions are A334966.
Cf. A056242, A059204, A072706, A107429, A115981, A329398, A332578, A332669, A332673, A332724, A332834.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{_,x_,,y_,_}/;y>x]&]],{n,0,15}]

def a333148(n): return number_of_partitions(n) + sum( Partitions(m, max_part=l, length=k).cardinality() * Partitions(n-m-l^2, min_length=k+2*l).cardinality() for l in range(1, (n+1).isqrt()) for m in range((n-l^2-2*l)*l//(l+1)+1) for k in range(ceil(m/l), min(m,n-m-l^2-2*l)+1) ) # Max Alekseyev, Oct 31 2024

See Sage code for the formula. - Max Alekseyev, Oct 31 2024

Edited and terms a(21)-a(51) added by Max Alekseyev, Oct 30 2024

A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

Original entry on oeis.org

1, 3, 9, 15, 27, 45, 75, 81, 105, 135, 225, 243, 315, 375, 405, 525, 675, 729, 735, 945, 1125, 1155, 1215, 1575, 1875, 2025, 2187, 2205, 2625, 2835, 3375, 3465, 3645, 3675, 4725, 5145, 5625, 5775, 6075, 6561, 6615, 7875, 8085, 8505, 9375, 10125, 10395, 10935

Offset: 1

Gus Wiseman, Apr 20 2021

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    3: {2}
    9: {2,2}
   15: {2,3}
   27: {2,2,2}
   45: {2,2,3}
   75: {2,3,3}
   81: {2,2,2,2}
  105: {2,3,4}
  135: {2,2,2,3}
  225: {2,2,3,3}
  243: {2,2,2,2,2}
  315: {2,2,3,4}
  375: {2,3,3,3}
  405: {2,2,2,2,3}
  525: {2,3,3,4}
  675: {2,2,2,3,3}
  729: {2,2,2,2,2,2}
  735: {2,3,4,4}
  945: {2,2,2,3,4}

The version starting at 1 is A055932.
The partitions with these Heinz numbers are counted by A264396.
Positions of 1's in A339662.
A000009 counts partitions covering an initial interval.
A000070 counts partitions with a selected part.
A016945 lists numbers with smallest prime index 2.
A034296 counts gap-free (or flat) partitions.
A056239 adds up prime indices, row sums of A112798.
A073491 lists numbers with gap-free prime indices.
A107428 counts gap-free compositions (initial: A107429).
A286469 and A286470 give greatest difference for Heinz numbers.
A325240 lists numbers with smallest prime multiplicity 2.
A342050/A342051 have prime indices with odd/even least gap.
Cf. A001223, A001522, A006128, A007052, A124010, A257989, A257993, A264401, A317090, A317589, A339737.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Select[Range[100],normQ[primeMS[#]-1]&]

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]

A373306 Sum over all complete compositions of n of the element multiset size.

Original entry on oeis.org

0, 1, 2, 7, 13, 30, 73, 157, 345, 743, 1650, 3517, 7593, 16120, 34294, 72683, 153475, 323293, 679231, 1423721, 2977692, 6218395, 12959249, 26970243, 56037071, 116280086, 240953162, 498719275, 1031029386, 2129266321, 4392871427, 9054428894, 18645998093

Offset: 0

Alois P. Heinz, May 31 2024

nonn

A complete composition of n has element set [k] with k<=n (without gaps).

			a(1) = 1: 1.
a(2) = 2: 11.
a(3) = 7 = 2 + 2 + 3: 12, 21, 111.
a(4) = 13 = 3 + 3 + 3 + 4: 112, 121, 211, 1111.
a(5) = 30 = 3*3 + 4*4 + 5: 122, 212, 221, 1112, 1121, 1211, 2111, 11111.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A107429, A373118, A373305.

b:= proc(n, i, t) option remember; `if`(n=0, `if`(i=0, [t!, 0], 0),
     `if`(i<1 or n p+[0, p[1]]*j)(
        b(n-i*j, i-1, t+j)/j!), j=1..n/i)))
    end:
a:= n-> add(b(n, k, 0)[2], k=0..floor((sqrt(1+8*n)-1)/2)):
seq(a(n), n=0..32);

b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[i == 0, {t!, 0}, {0, 0}], If[i < 1 || n < i*(i + 1)/2, {0, 0}, Sum[Function[p, p + {0, p[[1]]}*j][b[n - i*j, i - 1, t + j]/j!], {j, 1, n/i}]]];
a[n_] := Sum[b[n, k, 0][[2]], {k, 0, Floor[(Sqrt[1 + 8*n] - 1)/2]}];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Jun 08 2024, after Alois P. Heinz *)

G.f.: Sum_{k>0} d/dy C({1..k},x,y)|y = 1 where C({s},x,y) = Sum_{i in {s}} (C({s}-{i},x,y)*y*x^i)/(1 - Sum_{i in {s}} (y*x^i)) with C({},x,y) = 1. - John Tyler Rascoe, Jun 18 2024

A374147 Number of complete Carlitz compositions of n.

Original entry on oeis.org

1, 0, 2, 1, 1, 8, 7, 9, 20, 49, 72, 115, 202, 349, 695, 1171, 2009, 3530, 6203, 10818, 19320, 33961, 59449, 104349, 183370, 321635, 564081, 992513, 1741441, 3057547, 5363570, 9410785, 16516575, 28967505, 50798456, 89106542, 156276871, 274037619, 480437247, 842350671, 1476760717, 2588651452, 4537418431, 7952741429, 13938276465

Offset: 1

John Tyler Rascoe, Jun 28 2024

nonn

These are integer compositions such that no adjacent parts are equal and their set of parts covers an initial interval.

			a(7) = 7 counts: (1,2,1,3), (1,2,3,1), (1,3,2,1), (1,3,1,2), (2,1,3,1), (3,2,1,2), (1,2,1,2,1).

Cf. A003242, A107428, A107429.

Ca_x(s,N)={my(x='x+O('x^N), g=if(#s <1,1, sum(i=1,#s, (Ca_x(s[^i],N) * x^(s[i])/(1+x^(s[i]))))/(1-sum(i=1,#s, (x^(s[i]))/(1+x^(s[i])))))); return(g)}
B_x(N)={my(x='x+O('x^N), j=1, h=0); while((j*(j+1))/2 <= N, h += Ca_x([1..j],N+1); j+=1); my(a = Vec(h)); vector(N,i,a[i])}
B_x(45)

G.f.: Sum_{k>0} Ca({1..k},x) where Ca({s},x) = Sum_{i in {s}} ( (Ca({s}-{i},x)*x^i)/(1 + x^i) )/(1 - Sum_{i in {s}} ( (x^i)/(1 + x^i) )) is the g.f. for Carlitz compositions such that their set of parts equals {s} with Ca({},x) = 1.

cp's OEIS Frontend

A188575 Number of non-complete compositions of n.

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A332272 Number of narrowly recursively normal integer partitions of n.

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A332673 Triangle read by rows where T(n,k) is the number of length-k ordered set partitions of {1..n} whose non-adjacent blocks are pairwise increasing.

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A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

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A332274 Number of totally strong compositions of n.

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A333148 Number of compositions of n whose non-adjacent parts are weakly decreasing.

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A339886 Numbers whose prime indices cover an interval of positive integers starting with 2.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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