A238861 -id:A238861 - cp's OEIS frontend

A219282 Number of superdiagonal bargraphs with area n.

1, 1, 1, 2, 3, 4, 6, 9, 13, 18, 25, 35, 49, 68, 93, 126, 170, 229, 308, 413, 551, 731, 965, 1269, 1664, 2177, 2842, 3701, 4806, 6222, 8031, 10337, 13272, 17003, 21740, 27745, 35343, 44936, 57021, 72213, 91274, 115149, 145010, 182309, 228841, 286819, 358964, 448614, 559857, 697694

Offset: 0

Joerg Arndt, Dec 04 2012

nonn

Number of compositions n = p(1) + p(2) + ... + p(m) such that p(k) >= k (superdiagonal compositions), see example. - Joerg Arndt, Dec 19 2012

Number of (n-2)-bit binary strings in which the runs of ones are successively (1, 11, 111, 1111, ...), as in for example 00101100111011110011111000... To turn such a string into a composition, add 'X0 to the start of the empty string and the mark ' to the end, replace the runs 1, 11, 111,... with '01, '011, '0111, ... then consider the distances between the marks. - Andrew Woods, Jan 02 2015

			From _Joerg Arndt_, Dec 19 2012: (Start)
The a(9) = 18 compositions 9 = p(1) + p(2) + ... + p(m) such that p(k) >= k are
[ 1]  [ 1 2 6 ]
[ 2]  [ 1 3 5 ]
[ 3]  [ 1 4 4 ]
[ 4]  [ 1 5 3 ]
[ 5]  [ 1 8 ]
[ 6]  [ 2 2 5 ]
[ 7]  [ 2 3 4 ]
[ 8]  [ 2 4 3 ]
[ 9]  [ 2 7 ]
[10]  [ 3 2 4 ]
[11]  [ 3 3 3 ]
[12]  [ 3 6 ]
[13]  [ 4 2 3 ]
[14]  [ 4 5 ]
[15]  [ 5 4 ]
[16]  [ 6 3 ]
[17]  [ 7 2 ]
[18]  [ 9 ]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 201 terms from Vincenzo Librandi)
Margaret Archibald, Aubrey Blecher, Arnold Knopfmacher, and Stephan Wagner, Subdiagonal and superdiagonal compositions, Art Disc. Appl. Math. (2024). See p. 10.
Emeric Deutsch, Emanuele Munarini, and Simone Rinaldi, Skew Dyck paths, area, and superdiagonal bargraphs, Journal of Statistical Planning and Inference, Vol. 140, Issue 6, June 2010, pp. 1550-1562.

Cf. A063978 (compositions such that p(k) >= k-1 for k >= 2).
Cf. A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A098131 (compositions with smallest part >= number of parts; g.f. Sum_{k>=0} x^(k^2)/(1-x)^k).
Cf. A143862 (compositions with every part divisible by number of parts; g.f. Sum_{k>=0} x^(k^2) / (1 - x^k)^k).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Row sums of A305556.

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1), j=i..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 28 2014

nmax = 50; CoefficientList[Series[Sum[x^(k*(k+1)/2) / (1-x)^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 05 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 24 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+1)/2) / (1-q)^n );
v=Vec(gf)

G.f.: Sum_{n>=0} q^(n*(n+1)/2) / (1-q)^n.

a(n) = Sum_{k=0..floor((sqrt(8*n+1)-3)/2)} C(n-1-C(k+1,2),k), for n >= 1.

A238874 Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 4, 5, 7, 10, 14, 19, 25, 33, 44, 59, 79, 105, 138, 180, 234, 304, 395, 513, 665, 859, 1105, 1416, 1809, 2306, 2935, 3731, 4737, 6005, 7598, 9593, 12085, 15192, 19061, 23875, 29861, 37299, 46532, 57978, 72145, 89650, 111243, 137837, 170545, 210725, 260034, 320492, 394557, 485213, 596074, 731508

Offset: 0

Joerg Arndt, Mar 23 2014

nonn

			The a(13) = 25 such composition of 13 are:
01:  [ 2 3 8 ]
02:  [ 2 4 7 ]
03:  [ 2 5 6 ]
04:  [ 2 6 5 ]
05:  [ 2 7 4 ]
06:  [ 2 11 ]
07:  [ 3 3 7 ]
08:  [ 3 4 6 ]
09:  [ 3 5 5 ]
10:  [ 3 6 4 ]
11:  [ 3 10 ]
12:  [ 4 3 6 ]
13:  [ 4 4 5 ]
14:  [ 4 5 4 ]
15:  [ 4 9 ]
16:  [ 5 3 5 ]
17:  [ 5 4 4 ]
18:  [ 5 8 ]
19:  [ 6 3 4 ]
20:  [ 6 7 ]
21:  [ 7 6 ]
22:  [ 8 5 ]
23:  [ 9 4 ]
24:  [ 10 3 ]
25:  [ 13 ]

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

Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

b:= proc(n, i) option remember; `if`(n=0, 1,
      add(b(n-j, i+1), j=i..n))
    end:
a:= n-> b(n, 2):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 24 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[b[n-j, i+1], {j, i, n}]]; a[n_] := b[n, 2]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf=sum(n=0,N, q^(n*(n+3)/2) / (1-q)^n );
v=Vec(gf) \\ Joerg Arndt, Mar 30 2014

G.f.: Sum_{n>=0} q^(n*(n+3)/2) / (1-q)^n. - Joerg Arndt, Mar 30 2014

A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 14, 25, 46, 87, 167, 324, 634, 1248, 2466, 4887, 9706, 19308, 38455, 76659, 152925, 305232, 609488, 1217429, 2432399, 4860881, 9715511, 19421029, 38826059, 77626471, 155211785, 310357462, 620608652, 1241046343, 2481817484, 4963191718, 9925669171, 19850186856, 39698516655, 79394037319

Offset: 0

Mats Granvik, Dec 11 2010

nonn

a(0)=1, otherwise row sums of A179748.
For n>=1 cumulative sums of A008930.
a(n) is proportional to A048651*A000079. The error (a(n)-A048651*A000079) divided by sequence A186425 tends to the golden ratio A001622. This can be seen when using about 1000 decimals of the constant A048651 = 0.2887880950866024212... - [Mats Granvik, Jan 01 2015]
From Gus Wiseman, Mar 31 2022: (Start)
Also the number of integer compositions of n with exactly one part on or above the diagonal. For example, the a(1) = 1 through a(5) = 8 compositions are:
  (1)  (2)   (3)    (4)     (5)
       (11)  (21)   (31)    (41)
             (111)  (112)   (212)
                    (211)   (311)
                    (1111)  (1112)
                            (1121)
                            (2111)
                            (11111)
(End)

			From _Joerg Arndt_, Mar 24 2014: (Start)
The a(7) = 25 such compositions are:
01:  [ 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 2 1 ]
04:  [ 1 1 1 1 3 ]
05:  [ 1 1 1 2 1 1 ]
06:  [ 1 1 1 2 2 ]
07:  [ 1 1 1 3 1 ]
08:  [ 1 1 1 4 ]
09:  [ 1 1 2 1 1 1 ]
10:  [ 1 1 2 1 2 ]
11:  [ 1 1 2 2 1 ]
12:  [ 1 1 2 3 ]
13:  [ 1 1 3 1 1 ]
14:  [ 1 1 3 2 ]
15:  [ 1 2 1 1 1 1 ]
16:  [ 1 2 1 1 2 ]
17:  [ 1 2 1 2 1 ]
18:  [ 1 2 1 3 ]
19:  [ 1 2 2 1 1 ]
20:  [ 1 2 2 2 ]
21:  [ 1 2 3 1 ]
22:  [ 2 2 3 ]
23:  [ 2 3 2 ]
24:  [ 3 4 ]
25:  [ 7 ]
(End)

Cf. A238859 (compositions with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
The version for partitions is A001477, strong A002620.
The version for permutations is A057427, strong A000295.
The opposite version is A238874, first column of A352522.
The version for fixed points is A240736, nonfixed A352520.
The strong version is A351983, column k=1 of A352524.
This is column k = 1 of A352525.
A238349 counts compositions by fixed points, first col A238351.
A352517 counts weak excedances of standard compositions.
Cf. A008930, A010054, A088218, A098825, A114088, A219282, A238352, A319005, A350839, A352488, A352489, A352514, A352515, A352516.

A179748 := proc(n,k) option remember; if k= 1 then 1; elif k> n then 0 ; else add( procname(n-i,k-1),i=1..k-1) ; end if; end proc:
A177510 := proc(n) add(A179748(n,k),k=1..n) ;end proc:
seq(A177510(n),n=1..20) ; # R. J. Mathar, Dec 14 2010

Clear[t, nn]; nn = 39; t[n_, 1] = 1; t[n_, k_] := t[n, k] = If[n >= k, Sum[t[n - i, k - 1], {i, 1, k - 1}], 0]; Table[Sum[t[n, k], {k, 1, n}], {n, 1, nn}] (* Mats Granvik, Jan 01 2015 *)
pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]]; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pdw[#]==1&]],{n,0,10}] (* Gus Wiseman, Mar 31 2022 *)

N=66; q='q+O('q^N); Vec( 1 + q/(1-q) * sum(n=0, N, q^n * prod(k=1, n, (1-q^k)/(1-q) ) ) ) \\ Joerg Arndt, Mar 24 2014

@CachedFunction
def T(n, k): # A179748
    if n == 0:  return int(k==0);
    if k == 1:  return int(n>=1);
    return sum( T(n-i, k-1) for i in [1..k-1] );
# to display triangle A179748 including column zero = [1,0,0,0,...]:
#for n in [0..10]: print([ T(n,k) for k in [0..n] ])
def a(n): return sum( T(n,k) for k in [0..n] )
print([a(n) for n in [0..66]])
# Joerg Arndt, Mar 24 2014

G.f.: 1 + q/(1-q) * sum(n>=0, q^n * prod(k=1..n, (1-q^k)/(1-q) ) ). [Joerg Arndt, Mar 24 2014]

New name and a(0) = 1 prepended, Joerg Arndt, Mar 24 2014

A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that pi >= i.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 7, 9, 11, 14, 19, 25, 31, 38, 46, 59, 73, 92, 112, 135, 162, 196, 237, 289, 349, 417, 496, 587, 691, 820, 970, 1151, 1357, 1598, 1870, 2183, 2537, 2952, 3433, 3997, 4644, 5393, 6248, 7220, 8318, 9566, 10981, 12605, 14457, 16582, 19002, 21767, 24886, 28424, 32396, 36873, 41901, 47579, 53974, 61221

Offset: 0

Joerg Arndt, Mar 23 2014

nonn

			The a(13) = 31 such partitions of 13 are:
  01:  [ 1 2 3 7 ]
  02:  [ 1 2 4 6 ]
  03:  [ 1 2 5 5 ]
  04:  [ 1 2 10 ]
  05:  [ 1 3 3 6 ]
  06:  [ 1 3 4 5 ]
  07:  [ 1 3 9 ]
  08:  [ 1 4 4 4 ]
  09:  [ 1 4 8 ]
  10:  [ 1 5 7 ]
  11:  [ 1 6 6 ]
  12:  [ 1 12 ]
  13:  [ 2 2 3 6 ]
  14:  [ 2 2 4 5 ]
  15:  [ 2 2 9 ]
  16:  [ 2 3 3 5 ]
  17:  [ 2 3 4 4 ]
  18:  [ 2 3 8 ]
  19:  [ 2 4 7 ]
  20:  [ 2 5 6 ]
  21:  [ 2 11 ]
  22:  [ 3 3 3 4 ]
  23:  [ 3 3 7 ]
  24:  [ 3 4 6 ]
  25:  [ 3 5 5 ]
  26:  [ 3 10 ]
  27:  [ 4 4 5 ]
  28:  [ 4 9 ]
  29:  [ 5 8 ]
  30:  [ 6 7 ]
  31:  [ 13 ]

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 132 terms from Joerg Arndt)
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 5.

Cf. A219282 (superdiagonal compositions), A238394 (strictly superdiagonal partitions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A238875 (subdiagonal partitions), A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).

A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 47, 61, 80, 103, 138, 175, 224, 283, 362, 455, 577, 721, 898, 1111, 1380, 1701, 2106, 2577, 3156, 3844, 4680, 5671, 6879, 8312, 10034, 12060, 14478, 17319, 20715, 24703, 29442, 35004, 41578, 49247, 58278, 68796, 81132, 95502, 112320, 131877, 154705, 181158, 211908, 247475

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

Partitions with subdiagonal growth (A238876) with first part = 1.

			The a(11) = 26 such partitions of 11 are:
  01:  [ 1 1 1 1 1 1 1 1 1 1 1 ]
  02:  [ 1 1 1 1 1 1 1 1 1 2 ]
  03:  [ 1 1 1 1 1 1 1 1 3 ]
  04:  [ 1 1 1 1 1 1 1 2 2 ]
  05:  [ 1 1 1 1 1 1 1 4 ]
  06:  [ 1 1 1 1 1 1 2 3 ]
  07:  [ 1 1 1 1 1 1 5 ]
  08:  [ 1 1 1 1 1 2 2 2 ]
  09:  [ 1 1 1 1 1 2 4 ]
  10:  [ 1 1 1 1 1 3 3 ]
  11:  [ 1 1 1 1 1 6 ]
  12:  [ 1 1 1 1 2 2 3 ]
  13:  [ 1 1 1 1 2 5 ]
  14:  [ 1 1 1 1 3 4 ]
  15:  [ 1 1 1 2 2 2 2 ]
  16:  [ 1 1 1 2 2 4 ]
  17:  [ 1 1 1 2 3 3 ]
  18:  [ 1 1 1 3 5 ]
  19:  [ 1 1 1 4 4 ]
  20:  [ 1 1 2 2 2 3 ]
  21:  [ 1 1 2 2 5 ]
  22:  [ 1 1 2 3 4 ]
  23:  [ 1 1 3 3 3 ]
  24:  [ 1 2 2 2 2 2 ]
  25:  [ 1 2 2 2 4 ]
  26:  [ 1 2 2 3 3 ]

Andrew Howroyd, Table of n, a(n) for n = 0..1000
M. Archibald, A. Blecher, S. Elizalde, and A. Knopfmacher, Subdiagonal and superdiagonal partitions, Afr. Mat. 36, 77 (2025). See p. 9.

Cf. A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A129176 and A227543.

\\ here b: nr parts; k: max part, b+w-1: partition sum.
seq(n)={my(M=matrix(n,1), v=vector(n+1)); M[1,1]=v[1]=v[2]=1; for(b=2, n, M=matrix(n-b+1,b,w,k, if(w>=k, sum(j=1, min(b-1,k), M[w+1-k,j]))); v+=concat(vector(b),vecsum(Vec(M))~)); v} \\ Andrew Howroyd, Jan 19 2024

N=55;
VP=vector(N+1);  VP[1] =VP[2] = 1;  \\ one-based; memoization
P(n) = VP[n+1];
for (n=2, N, VP[n+1] = sum( i=0, n-1, P(i) * P(n-1 -i) * x^((i+1)*(n-1-i)) ) );
x='x+O('x^N);
A(x) = sum(n=0, N, x^n * P(n) );
Vec(A(x)) \\ Joerg Arndt, Jan 23 2024

G.f.: Sum_{n>=0} x^n * P(n) where P(n) is the row polynomial of the n-th row of A129176. This works because A129176(j,k) is also the number of subdiagonal partitions of j+k with j parts. - John Tyler Rascoe, Jan 20 2024

A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 26, 51, 99, 195, 383, 759, 1504, 2988, 5944, 11840, 23602, 47084, 93975, 187647, 374812, 748857, 1496487, 2991017, 5978900, 11952780, 23897506, 47782081, 95543378, 191053334, 382052880, 764019152, 1527898772, 3055572646, 6110782652, 12220980359

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

			There are a(6) = 26 such compositions of 6:
01:  [ 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 2 ]
03:  [ 1 1 1 2 1 ]
04:  [ 1 1 1 3 ]
05:  [ 1 1 2 1 1 ]
06:  [ 1 1 2 2 ]
07:  [ 1 1 3 1 ]
08:  [ 1 2 1 1 1 ]
09:  [ 1 2 1 2 ]
10:  [ 1 2 2 1 ]
11:  [ 1 2 3 ]
12:  [ 2 1 1 1 1 ]
13:  [ 2 1 1 2 ]
14:  [ 2 1 2 1 ]
15:  [ 2 1 3 ]
16:  [ 2 2 1 1 ]
17:  [ 2 2 2 ]
18:  [ 2 3 1 ]
19:  [ 3 1 1 1 ]
20:  [ 3 1 2 ]
21:  [ 3 2 1 ]
22:  [ 3 3 ]
23:  [ 4 1 1 ]
24:  [ 4 2 ]
25:  [ 5 1 ]
26:  [ 6 ]

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

Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i=0, add(b(n-j, j+1), j=1..n),
       add(b(n-j, i+1), j=1..min(n,i))))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Mar 25 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 0, Sum[b[n-j, j+1], {j, 1, n}], Sum[ b[n-j, i+1], {j, 1, Min[n, i]}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 13 2015, after Alois P. Heinz *)

a(n) ~ c * 2^n, where c = 1/2 - QPochhammer(1/2)/2 = 0.3556059524566987893605501390353846099555440475796571079426294669... - Vaclav Kotesovec, May 01 2014, updated Mar 17 2024

A238860 Partitions with superdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 >= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 6, 9, 11, 15, 18, 23, 26, 35, 43, 53, 64, 79, 91, 113, 135, 166, 197, 237, 277, 331, 387, 459, 541, 646, 754, 888, 1032, 1204, 1395, 1626, 1882, 2196, 2542, 2952, 3404, 3934, 4507, 5182, 5935, 6812, 7800, 8947, 10225, 11709, 13354, 15231, 17314, 19685, 22316, 25323, 28686, 32524, 36817, 41695

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

			There are a(13) = 23 such partitions of 13:
01:  [ 1 2 3 7 ]
02:  [ 1 2 4 6 ]
03:  [ 1 2 5 5 ]
04:  [ 1 2 10 ]
05:  [ 1 3 3 6 ]
06:  [ 1 3 4 5 ]
07:  [ 1 3 9 ]
08:  [ 1 4 4 4 ]
09:  [ 1 4 8 ]
10:  [ 1 5 7 ]
11:  [ 1 6 6 ]
12:  [ 1 12 ]
13:  [ 2 3 8 ]
14:  [ 2 4 7 ]
15:  [ 2 5 6 ]
16:  [ 2 11 ]
17:  [ 3 4 6 ]
18:  [ 3 5 5 ]
19:  [ 3 10 ]
20:  [ 4 9 ]
21:  [ 5 8 ]
22:  [ 6 7 ]
23:  [ 13 ]

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A238859 (compositions of n with subdiagonal growth), A238876 (partitions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A008930 (subdiagonal compositions), A238875 (subdiagonal partitions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

A238876 Partitions with subdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 <= i.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 24, 34, 46, 58, 76, 97, 126, 166, 209, 262, 333, 422, 529, 667, 833, 1024, 1268, 1567, 1934, 2385, 2911, 3549, 4319, 5237, 6340, 7675, 9274, 11164, 13404, 16046, 19173, 22889, 27278, 32458, 38574, 45750, 54140, 63976, 75449, 88848, 104503, 122773, 144077, 168860, 197609, 230916, 269494

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

The partitions are represented as weakly increasing lists of parts.

The number of such partitions that start with part p0 = 1 are given in A238875.

			The a(9) = 20 such partitions are:
01:  [ 1 1 1 1 1 1 1 1 1 ]
02:  [ 1 1 1 1 1 1 1 2 ]
03:  [ 1 1 1 1 1 1 3 ]
04:  [ 1 1 1 1 1 2 2 ]
05:  [ 1 1 1 1 1 4 ]
06:  [ 1 1 1 1 2 3 ]
07:  [ 1 1 1 1 5 ]
08:  [ 1 1 1 2 2 2 ]
09:  [ 1 1 1 2 4 ]
10:  [ 1 1 1 3 3 ]
11:  [ 1 1 2 2 3 ]
12:  [ 1 1 3 4 ]
13:  [ 1 2 2 2 2 ]
14:  [ 1 2 2 4 ]
15:  [ 1 2 3 3 ]
16:  [ 2 2 2 3 ]
17:  [ 2 3 4 ]
18:  [ 3 3 3 ]
19:  [ 4 5 ]
20:  [ 9 ]

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A238859 (compositions with subdiagonal growth), A001227 (partitions into distinct parts with subdiagonal growth).
Cf. A238860 (partitions with superdiagonal growth), A238861 (compositions with superdiagonal growth), A000009 (partitions into distinct parts have superdiagonal growth by definition).
Cf. A008930 (subdiagonal compositions), A010054 (subdiagonal partitions into distinct parts).
Cf. A219282 (superdiagonal compositions), A238873 (superdiagonal partitions), A238394 (strictly superdiagonal partitions), A238874 (strictly superdiagonal compositions), A025147 (strictly superdiagonal partitions into distinct parts).

cp's OEIS Frontend

A219282 Number of superdiagonal bargraphs with area n.

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A238874 Strictly superdiagonal compositions: compositions (p1, p2, p3, ...) of n such that pi > i.

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A177510 Number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i and pi >= p0.

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A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that pi >= i.

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A238875 Subdiagonal partitions: number of partitions (p1, p2, p3, ...) of n with pi <= i.

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A238859 Compositions with subdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 <= i.

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A238860 Partitions with superdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 >= i.

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A238876 Partitions with subdiagonal growth: number of partitions (p0, p1, p2, ...) of n with pi - p0 <= i.

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