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 1-3 of 3 results.

A362614 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 11, 3, 1, 0, 16, 4, 2, 0, 21, 6, 3, 0, 29, 8, 4, 1, 0, 43, 7, 5, 1, 0, 54, 13, 8, 2, 0, 78, 12, 8, 3, 0, 102, 17, 11, 5, 0, 131, 26, 12, 6, 1, 0, 175, 29, 17, 9, 1, 0, 233, 33, 18, 11, 2, 0, 295, 47, 25
Offset: 0

Views

Author

Gus Wiseman, May 04 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

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  11   3   1
   0  16   4   2
   0  21   6   3
   0  29   8   4   1
   0  43   7   5   1
   0  54  13   8   2
   0  78  12   8   3
   0 102  17  11   5
   0 131  26  12   6   1
   0 175  29  17   9   1
Row n = 8 counts the following partitions:
  (8)         (53)    (431)
  (44)        (62)    (521)
  (332)       (71)
  (422)       (3311)
  (611)
  (2222)
  (3221)
  (4211)
  (5111)
  (22211)
  (32111)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Links

Crossrefs

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362607, ranks A362605.
Column k = 1 is A362608, ranks A356862.
This statistic (mode-count) is ranked by A362611.
For co-modes we have A362615, ranked by A362613.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A002865, A003056, A098859, A240219, A359893, A360071, A362609, A362610, A362612, A372542.

Programs

  • Mathematica
    msi[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[msi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Formula

Sum_{k=0..A003056(n)} k * T(n,k) = A372542. - Alois P. Heinz, May 05 2024

A362615 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k co-modes.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 2, 1, 0, 4, 1, 0, 5, 2, 0, 7, 3, 1, 0, 10, 4, 1, 0, 13, 7, 2, 0, 16, 11, 3, 0, 23, 14, 4, 1, 0, 30, 19, 6, 1, 0, 35, 29, 11, 2, 0, 50, 34, 14, 3, 0, 61, 46, 23, 5, 0, 73, 69, 27, 6, 1, 0, 95, 81, 44, 10, 1, 0, 123, 105, 53, 14, 2
Offset: 0

Views

Author

Gus Wiseman, May 04 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

			Triangle begins:
   1
   0   1
   0   2
   0   2   1
   0   4   1
   0   5   2
   0   7   3   1
   0  10   4   1
   0  13   7   2
   0  16  11   3
   0  23  14   4   1
   0  30  19   6   1
   0  35  29  11   2
   0  50  34  14   3
   0  61  46  23   5
   0  73  69  27   6   1
   0  95  81  44  10   1
Row n = 8 counts the following partitions:
  (8)         (53)     (431)
  (44)        (62)     (521)
  (332)       (71)
  (422)       (3221)
  (611)       (3311)
  (2222)      (4211)
  (5111)      (32111)
  (22211)
  (41111)
  (221111)
  (311111)
  (2111111)
  (11111111)

Links

Crossrefs

Row sums are A000041.
Row lengths are A002024.
Removing columns 0 and 1 and taking sums gives A362609, ranks A362606.
Column k = 1 is A362610, ranks A359178.
This statistic (co-mode count) is ranked by A362613.
For mode instead of co-mode we have A362614, ranked by A362611.
A008284 counts partitions by length.
A096144 counts partitions by number of minima, A026794 by maxima.
A238342 counts compositions by number of minima, A238341 by maxima.
A275870 counts collapsible partitions.
Cf. A003056, A098859, A325347, A359893, A362607, A362608, A362612, A372632.

Programs

  • Mathematica
    comsi[ms_]:=Select[Union[ms],Count[ms,#]<=Min@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],Length[comsi[#]]==k&]],{n,0,15},{k,0,Floor[(Sqrt[1+8n]-1)/2]}]

Formula

Sum_{k=0..A003056(n)} k * T(n,k) = A372632(n). - Alois P. Heinz, May 07 2024

A232697 Number of partitions of 2n into parts such that the largest multiplicity equals n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 5, 8, 9, 13, 15, 22, 25, 35, 42, 56, 67, 89, 106, 138, 166, 211, 254, 321, 384, 479, 575, 709, 848, 1040, 1239, 1508, 1795, 2168, 2574, 3095, 3661, 4379, 5171, 6154, 7246, 8592, 10088, 11915, 13960, 16425, 19197, 22520, 26253, 30702, 35718
Offset: 0

Views

Author

Alois P. Heinz, Nov 27 2013

Keywords

Examples

			a(1) = 1: [2].
a(2) = 2: [2,2], [2,1,1].
a(3) = 2: [2,2,2], [3,1,1,1].
a(4) = 3: [2,2,2,2], [2,2,1,1,1,1], [4,1,1,1,1].
a(5) = 3: [2,2,2,2,2], [3,2,1,1,1,1,1], [5,1,1,1,1,1].
a(6) = 5: [2,2,2,2,2,2], [2,2,2,1,1,1,1,1,1], [3,3,1,1,1,1,1,1], [4,2,1,1,1,1,1,1], [6,1,1,1,1,1,1].

Links

Crossrefs

Cf. A000041, A002865, A091602, A232623, A332051.
Partial sums give A133041.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
a:= n-> b(2*n, 1, n)-`if`(n=0, 0, b(2*n, 1, n-1)):
seq(a(n), n=0..60);
  • Mathematica
    CoefficientList[x/(1-x) + (1-x)/QPochhammer[x] + O[x]^60, x] (* Jean-François Alcover, Dec 18 2016 *)

Formula

G.f.: x/(1-x) + Product_{k>=2} 1/(1-x^k).
a(0) = 1, a(n) = 1 + A002865(n) = 1 + A000041(n)-A000041(n-1) for n>0.
a(n) = A091602(2n,n) = A096144(2n,n).
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(3/2)). - Vaclav Kotesovec, Oct 25 2018
Showing 1-3 of 3 results.

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