A238875 -id:A238875 - 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

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).

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).

A238861 Compositions with superdiagonal growth: number of compositions (p0, p1, p2, ...) of n with pi - p0 >= i.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 10, 13, 18, 24, 32, 41, 55, 72, 95, 125, 164, 212, 275, 355, 459, 592, 763, 980, 1257, 1605, 2044, 2598, 3298, 4179, 5290, 6685, 8435, 10623, 13353, 16751, 20978, 26228, 32746, 40831, 50850, 63247, 78569, 97475, 120770, 149429, 184641, 227853, 280832, 345722, 425134, 522232, 640847, 785604

Offset: 0

Joerg Arndt, Mar 24 2014

nonn

			There are a(12) = 24 such compositions of 12:
01:  [ 1 2 3 6 ]
02:  [ 1 2 4 5 ]
03:  [ 1 2 5 4 ]
04:  [ 1 2 9 ]
05:  [ 1 3 3 5 ]
06:  [ 1 3 4 4 ]
07:  [ 1 3 8 ]
08:  [ 1 4 3 4 ]
09:  [ 1 4 7 ]
10:  [ 1 5 6 ]
11:  [ 1 6 5 ]
12:  [ 1 7 4 ]
13:  [ 1 8 3 ]
14:  [ 1 11 ]
15:  [ 2 3 7 ]
16:  [ 2 4 6 ]
17:  [ 2 5 5 ]
18:  [ 2 6 4 ]
19:  [ 2 10 ]
20:  [ 3 4 5 ]
21:  [ 3 9 ]
22:  [ 4 8 ]
23:  [ 5 7 ]
24:  [ 12 ]

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Cf. A238860 (partitions 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).

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=i..n)))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Mar 26 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, i, n}]]]; a[n_] := b[n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 18 2015, after Alois P. Heinz *)

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

G.f.: 1 + sum(n>=1, q^(n*(n+1)/2) / ( (1-q)^(n-1) * (1-q^n) ) ). [Joerg Arndt, Mar 30 2014]

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).

A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 9, 1, 1, 12, 2, 1, 16, 5, 1, 20, 9, 1, 25, 16, 1, 30, 25, 1, 36, 39, 1, 1, 42, 56, 2, 1, 49, 80, 5, 1, 56, 109, 10, 1, 64, 147, 19, 1, 72, 192, 32, 1, 81, 249, 54, 1, 90, 315, 84, 1, 100, 396, 129, 1, 1, 110, 489, 190, 2, 1, 121, 600, 275, 5

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1   1
   1   2
   1   4
   1   6
   1   9   1
   1  12   2
   1  16   5
   1  20   9
   1  25  16
   1  30  25
   1  36  39   1
   1  42  56   2
   1  49  80   5
   1  56 109  10
For example, row n = 7 counts the following partitions:
  (1111111)  (7)       (43)
             (52)      (331)
             (61)
             (322)
             (421)
             (511)
             (2221)
             (3211)
             (4111)
             (22111)
             (31111)
             (211111)

Row sums are A000041.
Row lengths are A000194, reversed A003056.
Column k = 1 is A002620, reversed A238875.
Column k = 2 is A097701.
The version for permutations is A008292, opposite A123125.
The weak version is A115720/A115994, rank statistic A257990.
The version for compositions is A352524, weak A352525.
The version for reversed partitions is A353319.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
Cf. A000701, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

partsabove[y_]:=Length[Select[Range[Length[y]],#

A353319 Irregular triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k excedances (parts above the diagonal), all zeros removed.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 4, 2, 1, 5, 4, 2, 7, 6, 2, 10, 6, 6, 15, 7, 7, 1, 18, 14, 7, 3, 26, 15, 11, 4, 35, 17, 19, 6, 47, 24, 19, 11, 61, 33, 22, 18, 1, 80, 44, 28, 20, 4, 103, 54, 42, 25, 7, 138, 60, 57, 31, 11, 175, 85, 58, 52, 15, 224, 112, 66, 64, 24

Offset: 1

Gus Wiseman, May 21 2022

			Triangle begins:
   1
   1  1
   2  1
   2  3
   4  2  1
   5  4  2
   7  6  2
  10  6  6
  15  7  7  1
  18 14  7  3
  26 15 11  4
  35 17 19  6
  47 24 19 11
  61 33 22 18  1
  80 44 28 20  4
For example, row n = 9 counts the following reversed partitions:
  (1134)       (9)     (27)   (234)
  (1224)       (18)    (36)
  (1233)       (117)   (45)
  (11115)      (126)   (135)
  (11124)      (1116)  (144)
  (11133)      (1125)  (225)
  (11223)      (2223)  (333)
  (12222)
  (111114)
  (111123)
  (111222)
  (1111113)
  (1111122)
  (11111112)
  (111111111)

Row sums are A000041.
Row lengths are A003056.
The version for permutations is A008292, opposite A123125.
The weak unreversed version is A115720/A115994, rank statistic A257990.
For fixed points instead of excedances we have A238352, rank stat A352822.
Column k = 0 is A238875.
The version for compositions is A352524, weak A352525.
The version for unreversed partitions is A353318.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
Cf. A000701, A002620, A006918, A008290, A008930, A114088, A177510, A219282, A238874, A300788, A352522.

partsabove[y_]:=Length[Select[Range[Length[y]],#

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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A238873 Number of superdiagonal partitions: partitions (p1, p2, p3, ...) of n such that 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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A238861 Compositions with superdiagonal growth: number of compositions (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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A353318 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k excedances (parts above the diagonal), zeros omitted.

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