A238874 -id:A238874 - cp's OEIS frontend

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

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).

A351983 Number of integer compositions of n with exactly one part above the diagonal.

Original entry on oeis.org

0, 0, 1, 2, 5, 9, 18, 35, 67, 131, 257, 505, 996, 1973, 3915, 7781, 15486, 30855, 61527, 122764, 245069, 489412, 977673, 1953515, 3904108, 7803545, 15599618, 31187269, 62355347, 124679883, 249310255, 498540890, 996953659, 1993701032, 3987069747, 7973603891

Offset: 0

Gus Wiseman, Apr 02 2022

nonn

			The a(2) = 1 through a(6) = 18 compositions:
  (2)  (3)   (4)    (5)     (6)
       (21)  (13)   (14)    (15)
             (22)   (32)    (42)
             (31)   (41)    (51)
             (211)  (131)   (114)
                    (212)   (132)
                    (221)   (141)
                    (311)   (213)
                    (2111)  (222)
                            (312)
                            (321)
                            (411)
                            (1311)
                            (2112)
                            (2121)
                            (2211)
                            (3111)
                            (21111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for permutations is A000295, weak A057427.
The version for partitions is A002620, weak A001477.
The weak version is A177510.
The version for fixed points is A240736, nonfixed A352520.
This is column k = 1 of A352524; column k = 0 is A008930.
A238349 counts compositions by fixed points, first column A238351.
A352521 counts compositions by strong nonexcedances, first column A219282.
A352522 counts compositions by weak nonexcedances, first column A238874.
A352523 counts compositions by nonfixed points, first column A010054.
A352524 counts compositions by strong excedances, first column A008930.
A352525 counts compositions by weak excedances, first column A177510.
Cf. A088218, A098825, A115994, A238352, A330644, A352516.

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

S(v,u,c=0)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=0); v[1]=1; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1)), O(x^2)); s+=apply(p->polcoef(p,1), v)); s} \\ Andrew Howroyd, Jan 02 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

A352875 Number of integer compositions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 1, 2, 5, 10, 21, 42, 86, 174, 351, 708, 1424, 2861, 5743, 11520, 23092, 46269, 92673, 185562, 371469, 743491, 1487870, 2977164, 5956616, 11916910, 23839736, 47688994, 95393322, 190811346, 381662507, 763389209, 1526881959, 3053930971, 6108131542, 12216698288

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 0 through a(5) = 10 compositions (empty column indicated by dot):
  .  (1)  (11)  (12)   (13)    (14)
                (111)  (22)    (32)
                       (112)   (113)
                       (121)   (122)
                       (1111)  (131)
                               (221)
                               (1112)
                               (1121)
                               (1211)
                               (11111)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The version for partitions is A001522, ranked by A352827 (unproved).
The version for permutations is A002467, complement A000166.
The complement for partitions is A064428, ranked by A352826 (unproved).
This is the sum of latter columns of A238349, nonfixed A352523.
The complement is counted by A238351.
The complement for reversed partitions is A238394, ranked by A352830.
The version for reversed partitions is A238395, ranked by A352872.
The case of just one fixed point is A240736.
A008290 counts permutations by fixed points, nonfixed A098825.
A011782 counts compositions.
A115720 and A115994 count partitions by Durfee square.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352512 counts fixed points in standard compositions, nonfixed A352513.
A352521 = comps by subdiags, first col A219282, rank stat A352514.
A352522 = comps by weak subdiags, first col A238874, rank stat A352515.
A352524 = comps by superdiags, first col A008930, rank stat A352516.
A352525 = comps by weak superdiags, col k=1 A177510, rank stat A352517.
A352833 counts partitions by fixed points.
Cf. A006918, A008292, A088218, A114088, A123125, A188674, A257990.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pq[#]>0&]],{n,0,15}]

S(v,u,c)={vector(#v, k, c + sum(i=1, k-1, v[k-i]*u[i]))}
seq(n)={my(v=vector(1+n), s=vector(#v, i, 2^(i-2))); v[1]=1; s[1]=0; for(i=1, n, v=S(v, vector(n, j, if(j==i,'x,1)), O(x)); s-=apply(p->polcoef(p,0), v)); s} \\ Andrew Howroyd, Jan 02 2023

a(n) = 2^(n-1) - A238351(n) for n >= 1. - Andrew Howroyd, Jan 02 2023

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2023

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]],#

A353315 Triangle read by rows where T(n,k) is the number of integer partitions of n with k parts on or below the diagonal (weak non-excedances).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 1, 2, 2, 3, 2, 1, 0, 1, 2, 3, 3, 3, 2, 1, 0, 1, 3, 4, 4, 4, 3, 2, 1, 0, 1, 3, 6, 5, 5, 4, 3, 2, 1, 0, 1, 4, 7, 8, 6, 6, 4, 3, 2, 1, 0, 1, 4, 9, 10, 9, 7, 6, 4, 3, 2, 1, 0, 1, 6, 10, 14, 12, 10, 8, 6, 4, 3, 2, 1, 0, 1

Offset: 0

Gus Wiseman, May 15 2022

			Triangle begins:
  1
  0  1
  1  0  1
  1  1  0  1
  1  2  1  0  1
  1  2  2  1  0  1
  2  2  3  2  1  0  1
  2  3  3  3  2  1  0  1
  3  4  4  4  3  2  1  0  1
  3  6  5  5  4  3  2  1  0  1
  4  7  8  6  6  4  3  2  1  0  1
  4  9 10  9  7  6  4  3  2  1  0  1
  6 10 14 12 10  8  6  4  3  2  1  0  1
  6 13 16 17 13 11  8  6  4  3  2  1  0  1
  8 15 21 21 19 14 12  8  6  4  3  2  1  0  1
  9 19 24 28 24 20 15 12  8  6  4  3  2  1  0  1
For example, row n = 9 counts the following partitions (empty column indicated by dot):
  9   72   522   3222   22221  222111  2211111  21111111  .  111111111
  54  81   621   4221   32211  321111  3111111
  63  333  711   5211   42111  411111
      432  3321  6111   51111
      441  4311  33111
      531

MathOverflow, Why 'excedances' of permutations? [closed].

Row sums are A000041.
Column k = 0 is A003106.
The strong version is A114088.
The opposite version is A115720/A115994, rank statistic A257990.
The version for permutations is A123125, strong A173018.
The version for compositions is A352522, rank statistic A352515.
The strong opposite version is A353318.
A000700 counts self-conjugate partitions, ranked by A088902.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A008292 is the triangle of Eulerian numbers.
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352490 gives the nonexcedance set of A122111, counted by A000701.
Cf. A002620, A006918, A008290, A008930, A098825, A219282, A238874, A300788, A353319.

pgeq[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pgeq[#]==k&]],{n,0,15},{k,0,n}]

cp's OEIS Frontend

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

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A351983 Number of integer compositions of n with exactly one part above the diagonal.

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A352875 Number of integer compositions y of n with a fixed point y(i) = 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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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.

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A353315 Triangle read by rows where T(n,k) is the number of integer partitions of n with k parts on or below the diagonal (weak non-excedances).

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