A242451 -id:A242451 - cp's OEIS frontend

A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 6, 0, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 13, 1, 0, 0, 0, 0, 1, 0, 16, 4, 0, 1, 0, 0, 0, 1, 0, 25, 2, 2, 0, 0, 0, 0, 0, 1, 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1, 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Jun 28 2014

T(0,0) = 1 by convention.
Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.
Row sums are A000041.

			Triangle starts:
00:  1;
01:  0,   1;
02:  0,   1,  1;
03:  0,   2,  0, 1;
04:  0,   3,  1, 0, 1;
05:  0,   6,  0, 0, 0, 1;
06:  0,   7,  2, 1, 0, 0, 1;
07:  0,  13,  1, 0, 0, 0, 0, 1;
08:  0,  16,  4, 0, 1, 0, 0, 0, 1;
09:  0,  25,  2, 2, 0, 0, 0, 0, 0, 1;
10:  0,  33,  6, 1, 0, 1, 0, 0, 0, 0, 1;
11:  0,  49,  4, 2, 0, 0, 0, 0, 0, 0, 0, 1;
12:  0,  61,  9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
13:  0,  90,  6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
14:  0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
15:  0, 156,  9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
16:  0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
17:  0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18:  0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
19:  0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20:  0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are:
01:  [ 1 1 1 1 1 1 1 1 1 ]   9
02:  [ 1 1 1 1 1 1 1 2 ]   1
03:  [ 1 1 1 1 1 1 3 ]   1
04:  [ 1 1 1 1 1 2 2 ]   2
05:  [ 1 1 1 1 1 4 ]   1
06:  [ 1 1 1 1 2 3 ]   1
07:  [ 1 1 1 1 5 ]   1
08:  [ 1 1 1 2 2 2 ]   3
09:  [ 1 1 1 2 4 ]   1
10:  [ 1 1 1 3 3 ]   2
11:  [ 1 1 1 6 ]   1
12:  [ 1 1 2 2 3 ]   1
13:  [ 1 1 2 5 ]   1
14:  [ 1 1 3 4 ]   1
15:  [ 1 1 7 ]   1
16:  [ 1 2 2 2 2 ]   1
17:  [ 1 2 2 4 ]   1
18:  [ 1 2 3 3 ]   1
19:  [ 1 2 6 ]   1
20:  [ 1 3 5 ]   1
21:  [ 1 4 4 ]   1
22:  [ 1 8 ]   1
23:  [ 2 2 2 3 ]   1
24:  [ 2 2 5 ]   1
25:  [ 2 3 4 ]   1
26:  [ 2 7 ]   1
27:  [ 3 3 3 ]   3
28:  [ 3 6 ]   1
29:  [ 4 5 ]   1
30:  [ 9 ]   1
Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)

Cf. A183568, A242451 (the same for compositions).

Cf. A091602 (partitions by max multiplicity of any part).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
T:= (n, k)-> b(n$2, k) -`if`(n=0 and k=0, 0, b(n$2, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 5, 6, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 13, 21, 18, 5, 6, 0, 1, 0, 19, 40, 34, 21, 6, 7, 0, 1, 0, 27, 87, 59, 40, 27, 7, 8, 0, 1, 0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1, 0, 65, 219, 272, 210, 131, 63, 44, 9, 10, 0, 1

Offset: 0

Alois P. Heinz, May 15 2014

T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity larger than k.

			T(6,1) = 11: [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1], [2,4], [4,2], [1,5], [5,1], [6].
T(6,2) = 10: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3], [1,1,4], [1,4,1], [4,1,1].
T(6,3) = 5: [2,2,2], [1,1,1,3], [1,1,3,1], [1,3,1,1], [3,1,1,1].
T(6,4) = 5: [1,1,1,1,2], [1,1,1,2,1], [1,1,2,1,1], [1,2,1,1,1], [2,1,1,1,1].
T(6,6) = 1: [1,1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  3,   0,   1;
  0,  3,   4,   0,   1;
  0,  5,   6,   4,   0,  1;
  0, 11,  10,   5,   5,  0,  1;
  0, 13,  21,  18,   5,  6,  0, 1;
  0, 19,  40,  34,  21,  6,  7, 0, 1;
  0, 27,  87,  59,  40, 27,  7, 8, 0, 1;
  0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1;

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000007, A032020 (for n>0), A243119, A243120, A243121, A243122, A243123, A243124, A243125, A243126, A243127.
T(2n,n) = A232665(n).
Row sums give A011782.
Cf. A242451 (the same for minimal multiplicity).

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n$2, 0, k) -`if`(k=0, 0, b(n$2, 0, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, i-1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, 0, k] - If[k == 0, 0, b[n, n, 0, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)

A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

Original entry on oeis.org

1, 1, 3, 6, 15, 23, 53, 94, 203, 404, 855, 1648, 3416, 6662, 13400, 26406, 53038, 105306, 212051, 422162, 849267, 1696864, 3406077, 6807024, 13642099, 27268122, 54576003, 109096436, 218250874, 436243705, 872533347, 1744312748, 3488432736, 6974783481

Offset: 1

Alois P. Heinz, Jun 21 2014

nonn

			From _Gus Wiseman_, Nov 25 2019: (Start)
The a(1) = 1 through a(5) = 15 compositions:
  (1)  (2)  (3)    (4)      (5)
            (1,2)  (1,3)    (1,4)
            (2,1)  (3,1)    (2,3)
                   (1,1,2)  (3,2)
                   (1,2,1)  (4,1)
                   (2,1,1)  (1,1,3)
                            (1,2,2)
                            (1,3,1)
                            (2,1,2)
                            (2,2,1)
                            (3,1,1)
                            (1,1,1,2)
                            (1,1,2,1)
                            (1,2,1,1)
                            (2,1,1,1)
(End)

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 1..2000 (first 400 terms from Alois P. Heinz)
Vaclav Kotesovec, Graph a(n)/2^n

Column k=1 of A242451.
The complement is counted by A240085.
Cf. A003242, A098504, A114901, A261983, A329740, A329741.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 1) -b(n$2, 0, 2):
seq(a(n), n=1..50);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Min@@Length/@Split[Sort[#]]==1&]],{n,0,10}] (* Gus Wiseman, Nov 25 2019 *)

a(n) = 2^(n-1) - A240085(n). - Gus Wiseman, Nov 25 2019

A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.

Original entry on oeis.org

1, 3, 7, 21, 71, 253, 925, 3433, 12871, 48621, 184757, 705433, 2704157, 10400601, 40116601, 155117521, 601080391, 2333606221, 9075135301, 35345263801, 137846528821, 538257874441, 2104098963721, 8233430727601, 32247603683101, 126410606437753, 495918532948105

Offset: 0

Alois P. Heinz, Jun 21 2014

nonn

			a(2) = 7: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000984, A007318, A065567, A242451, A245732.

a:= proc(n) option remember;
      `if`(n<3, 2^(n+1)-1, ((15*n^2-31*n+12) *a(n-1)
       -2*(3*n-2)*(2*n-3) *a(n-2)) / ((3*n-5)*n))
    end:
seq(a(n), n=0..30);

a[n_] := a[n] = If[n < 3, 2^(n+1) - 1, ((15*n^2 - 31*n + 12)*a[n-1] - 2*(3*n - 2)*(2*n - 3)*a[n-2])/((3*n - 5)*n)]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 07 2014, after Alois P. Heinz *)

A244174 = lambda m: SetPartitions(2*m,[2*m]).cardinality()+2*SetPartitions(2*m,[m,m]).cardinality()
[1] + [A244174(m) for m in (1..26)] # Peter Luschny, Aug 02 2015

a(n) = A242451(3n,n).
Recurrence: see Maple program.
For n>0, a(n) = 1 + C(2n,n) = 1 + A000984(n). - Vaclav Kotesovec, Jun 21 2014
G.f.: 1/(sqrt(1-4*x)) + x/(1-x). - Alois P. Heinz, Jun 22 2014
a(n) = A245732(2n,n). - Alois P. Heinz, Jul 30 2014
a(n) = A065567(2n,n) for n>=1. - Alois P. Heinz, Sep 05 2023

A244169 Number of compositions of n in which the minimal multiplicity of parts equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 925, 1716, 4719, 5005, 11011, 12376, 24494, 28848, 49771, 60985, 94997, 113323, 176576, 205948, 300841, 362000, 502840, 588343, 17972200, 47500741, 164220317, 452654380, 1198032651, 2936508011, 6683727408, 15107475055

Offset: 6

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 6..400

Column k=6 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 6) -b(n$2, 0, 7):
seq(a(n), n=6..50);

A244165 Number of compositions of n in which the minimal multiplicity of parts equals 2.

Original entry on oeis.org

1, 0, 1, 0, 7, 10, 32, 31, 71, 77, 222, 342, 971, 1936, 4681, 9662, 19440, 38304, 76458, 143542, 281447, 536472, 1051100, 2039870, 4055916, 8030958, 16168611, 32510383, 65705473, 132895297, 269206168, 544002516, 1099360989, 2217243856, 4464684513, 8975720721

Offset: 2

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 2..400

Column k=2 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 2) -b(n$2, 0, 3):
seq(a(n), n=2..50);

A244166 Number of compositions of n in which the minimal multiplicity of parts equals 3.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 21, 35, 91, 105, 211, 221, 464, 441, 740, 2571, 5457, 14740, 37990, 89091, 203487, 416751, 877183, 1722277, 3374384, 6381902, 12054327, 22545335, 42054605, 78409434, 147669414, 280480248, 539039384, 1051964654, 2078682442, 4165775779

Offset: 3

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 3..400

Column k=3 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 3) -b(n$2, 0, 4):
seq(a(n), n=3..50);

A244167 Number of compositions of n in which the minimal multiplicity of parts equals 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 0, 71, 126, 336, 330, 776, 841, 1541, 1821, 2951, 3221, 5322, 5697, 43288, 99626, 313917, 807218, 2049995, 4769054, 10240287, 22002219, 45015647, 90239153, 177239582, 342276724, 650127448, 1223160912, 2275920969, 4197371497, 7754873919

Offset: 4

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 4..400

Column k=4 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 4) -b(n$2, 0, 5):
seq(a(n), n=4..50);

A244168 Number of compositions of n in which the minimal multiplicity of parts equals 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 253, 462, 1254, 1287, 2794, 3256, 6117, 6980, 12319, 13630, 22015, 25971, 38144, 43966, 64863, 828898, 2119622, 7035420, 18918950, 48777982, 117594112, 259516217, 574862404, 1204750520, 2487540831, 5003559312, 9935325156

Offset: 5

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 5..400

Column k=5 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 5) -b(n$2, 0, 6):
seq(a(n), n=5..50);

A244170 Number of compositions of n in which the minimal multiplicity of parts equals 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 3433, 6435, 17875, 19448, 43264, 50388, 96968, 119713, 208803, 256597, 422375, 512524, 785708, 976990, 1423465, 1737759, 2488824, 3001562, 4141412, 5012060, 6722158, 407104665, 1108110431, 3914660472, 10999975393

Offset: 7

Alois P. Heinz, Jun 21 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..400

Column k=7 of A242451.

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=[0, $max(1, k)..n/i])))
    end:
a:= n-> b(n$2, 0, 7) -b(n$2, 0, 8):
seq(a(n), n=7..55);

cp's OEIS Frontend

A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A244164 Number of compositions of n in which the minimal multiplicity of parts equals 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A244174 Number of compositions of 3n in which the minimal multiplicity of parts equals n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A244169 Number of compositions of n in which the minimal multiplicity of parts equals 6.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A244165 Number of compositions of n in which the minimal multiplicity of parts equals 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A244166 Number of compositions of n in which the minimal multiplicity of parts equals 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A244167 Number of compositions of n in which the minimal multiplicity of parts equals 4.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

A244168 Number of compositions of n in which the minimal multiplicity of parts equals 5.

Views