cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-35 of 35 results.

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

Views

Author

Gus Wiseman, May 15 2022

Keywords

Examples

			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)

Links

Crossrefs

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.

Programs

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

Formula

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

Extensions

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

Views

Author

Gus Wiseman, May 21 2022

Keywords

Examples

			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)

Crossrefs

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.

Programs

  • Mathematica
    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

Views

Author

Gus Wiseman, May 21 2022

Keywords

Examples

			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)

Crossrefs

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.

Programs

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

A186519 Row sums of A186518.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 3, 3, 4, 5, 4, 4, 8, 5, 7, 8, 7, 10, 10, 7, 12, 9, 12, 6, 12, 12, 10, 16, 14, 13, 12, 15, 13, 18, 17, 13, 19, 14, 16, 16, 16, 19, 19, 20, 19, 17, 23, 15, 21, 28, 20, 18, 22, 24, 22, 21, 26, 27, 23, 22, 29, 33, 21, 32, 31, 33, 32, 27, 39, 27, 31, 29, 33, 33, 32, 41, 35, 27, 45, 28, 37, 40, 28, 36, 35, 41, 30, 33, 40, 45, 42, 32, 41, 42, 39, 45, 38, 44, 48, 44, 46, 49, 48, 56, 45, 51, 50, 49, 44, 51, 50, 46, 45, 58, 53, 42, 45, 52, 50, 62, 51, 55, 50, 56, 52, 62, 52, 60, 53, 68, 53, 57, 63
Offset: 1

Views

Author

Mats Granvik, Feb 23 2011

Keywords

Comments

a(n) mod 2 = A008930 mod 2.

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

Views

Author

Gus Wiseman, May 15 2022

Keywords

Examples

			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

Links

Crossrefs

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.

Programs

  • Mathematica
    pgeq[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pgeq[#]==k&]],{n,0,15},{k,0,n}]
Previous Showing 31-35 of 35 results.

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