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.

Showing 1-5 of 5 results.

A363224 Number of integer compositions of n in which the least part appears more than once.

Original entry on oeis.org

0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025
Offset: 1

Views

Author

Gus Wiseman, Jun 04 2023

Keywords

Comments

Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.

Examples

			The a(1) = 0 through a(6) = 21 compositions:
  .  (11)  (111)  (22)    (113)    (33)
                  (112)   (131)    (114)
                  (121)   (311)    (141)
                  (211)   (1112)   (222)
                  (1111)  (1121)   (411)
                          (1211)   (1113)
                          (2111)   (1122)
                          (11111)  (1131)
                                   (1212)
                                   (1221)
                                   (1311)
                                   (2112)
                                   (2121)
                                   (2211)
                                   (3111)
                                   (11112)
                                   (11121)
                                   (11211)
                                   (12111)
                                   (21111)
                                   (111111)

Links

Crossrefs

The complement is counted by A105039.
For partitions instead of compositions we have A117989.
Row sums of columns k > 1 of A238342.
If all parts appear more than once we have A240085, for partitions A007690.
If the least part appears exactly twice we have A241862.
For greatest instead of least we have A363262, see triangle A238341.
A000041 counts integer partitions, strict A000009.
A032020 counts strict compositions.
A067029 gives last exponent in prime factorization, first A071178.
A261982 counts compositions with some part appearing more than once.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
Cf. A002865, A008284, A097979, A243737.

Programs

  • Mathematica
    Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Min@@#]>1&]],{n,15}]
  • PARI
    C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024

Formula

G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024

A363263 Number of integer partitions of n covering an initial interval of positive integers with a unique co-mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 4, 4, 5, 7, 10, 8, 13, 13, 15, 19, 25, 24, 35, 35, 43, 50, 61, 59, 79, 83, 98, 111, 137, 137, 176, 187, 219, 240, 284, 298, 360, 385, 444, 485, 568, 600, 706, 763, 867, 951, 1088, 1168, 1345, 1453, 1641, 1792, 2023, 2179, 2467, 2673, 2988
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2023

Keywords

Comments

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Examples

			The a(1) = 1 through a(10) = 7 partitions:
  1  11  111  211   221    21111   2221     22211     22221      33211
              1111  2111   111111  22111    221111    32211      222211
                    11111          211111   2111111   2211111    322111
                                   1111111  11111111  21111111   2221111
                                                      111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(9) = 5 through a(12) = 8 partitions:
  (22221)      (33211)       (33221)        (2222211)
  (32211)      (222211)      (222221)       (3222111)
  (2211111)    (322111)      (322211)       (3321111)
  (21111111)   (2221111)     (332111)       (32211111)
  (111111111)  (22111111)    (2222111)      (222111111)
               (211111111)   (3221111)      (2211111111)
               (1111111111)  (22211111)     (21111111111)
                             (221111111)    (111111111111)
                             (2111111111)
                             (11111111111)

Crossrefs

For parts instead of multiplicities we have A087897, complement A000009.
For multisets instead of partitions we have A105039, complement A363224.
The complement is counted by A363264.
For mode we have A363484, complement A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A025147, A096765, A117989, A243737, A362612.

Programs

  • Mathematica
    comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]==1&]]],{n,0,30}]

A363264 Number of integer partitions of n covering an initial interval of positive integers with a more than one co-mode.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 2, 1, 2, 3, 3, 2, 7, 5, 9, 12, 13, 13, 22, 19, 29, 33, 39, 43, 63, 63, 82, 94, 111, 119, 159, 164, 203, 229, 272, 301, 370, 400, 479, 538, 628, 692, 826, 904, 1053, 1181, 1353, 1502, 1742, 1919, 2205, 2456, 2790, 3097, 3539, 3911, 4435, 4929
Offset: 0

Views

Author

Gus Wiseman, Jun 06 2023

Keywords

Comments

We define a co-mode in a multiset to be an element that appears at most as many times as each of the others. For example, the co-modes of {a,a,b,b,b,c,c} are {a,c}.

Crossrefs

For parts instead of multiplicities we have A000009, complement A087897.
For multisets instead of partitions we have A363224, complement A105039.
The complement is counted by A363263.
For mode we have A363485, complement A363484.
A000041 counts integer partitions, A000009 covering an initial interval.
A067029 counts minima in prime factorization, co-modes A362613.
A071178 counts maxima in prime factorization, modes A362611.
A097979 counts normal multisets with a unique mode, complement A363262.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A096765, A117989, A243737, A275870, A362612.

Programs

  • Mathematica
    comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[If[n==0,0,Length[Select[IntegerPartitions[n],Union[#]==Range[Max@@#]&&Length[comsi[#]]>1&]]],{n,0,30}]

A363484 Number of integer partitions of n covering an initial interval of positive integers with a unique mode.

Original entry on oeis.org

0, 1, 1, 1, 2, 3, 2, 5, 6, 6, 8, 11, 12, 17, 20, 21, 27, 35, 38, 50, 56, 65, 76, 95, 105, 125, 146, 167, 198, 233, 252, 305, 351, 394, 457, 522, 585, 681, 778, 878, 994, 1135, 1269, 1446, 1638, 1828, 2067, 2339, 2613, 2940, 3301, 3684, 4143, 4634, 5156, 5771
Offset: 0

Views

Author

Gus Wiseman, Jun 05 2023

Keywords

Comments

A mode in a multiset is an element that appears at least as many times as each of the others. For example, the modes of {a,a,b,b,b,c,d,d,d} are {b,d}.

Examples

			The a(1) = 1 through a(8) = 6 partitions:
  (1)  (11)  (111)  (211)   (221)    (21111)   (2221)     (3221)
                    (1111)  (2111)   (111111)  (3211)     (22211)
                            (11111)            (22111)    (32111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

Crossrefs

For parts instead of multiplicities we have A096765, complement A025147.
For multisets instead of partitions we have A097979, complement A363262.
For co-mode we have A363263, complement A363264.
The complement is counted by A363485.
A000041 counts integer partitions, A000009 covering an initial interval.
A362607 counts partitions with multiple modes, co-modes A362609.
A362608 counts partitions with a unique mode, co-mode A362610.
A362614 counts partitions by number of modes, co-modes A362615.
Cf. A002865, A008284, A105039, A117989, A243737, A362612.

Programs

  • Mathematica
    Table[If[n==0,0,Length[Select[IntegerPartitions[n], Union[#]==Range[Max@@#]&&Length[Commonest[#]]==1&]]],{n,0,30}]

A383101 Number of compositions of n such that any part 1 can be m different colors where m is the largest part of the composition.

Original entry on oeis.org

1, 1, 2, 6, 21, 77, 294, 1178, 4978, 22191, 104146, 513385, 2653003, 14349804, 81125023, 478686413, 2943737942, 18838530436, 125268429098, 864256288435, 6177766228172, 45689641883377, 349173454108407, 2754058599745239, 22393206702946457, 187501022603071090
Offset: 0

Views

Author

John Tyler Rascoe, Apr 16 2025

Keywords

Examples

			a(3) = 6 counts: (3), (2,1_a), (2,1_b), (1_a,2), (1_b,2), (1_a,1_a,1_a).

Links

Crossrefs

Cf. A011782, A088305, A105422, A171632, A363262, A382991, A382992.

Programs

  • Maple
    b:= proc(n, p, m) option remember; binomial(n+p, n)*
      m^n+add(b(n-j, p+1, max(m, j)), j=2..n)
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Apr 23 2025
  • PARI
    A_x(N) = {my(x='x+O('x^N)); Vec(1+sum(m=1,N, x^m/((1-m*x-(x^2-x^m)/(1-x))*(1-m*x-(x^2-x^(m+1))/(1-x)))))}
A_x(30)

Formula

G.f.: 1 + Sum_{m>0} x^m/((1 - m*x - (x^2 - x^m)/(1 - x)) * (1 - m*x - (x^2 - x^(m+1))/(1 - x))).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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