A364909 -id:A364909 - cp's OEIS frontend

A364910 Number of integer partitions of 2n whose distinct parts sum to n.

1, 1, 1, 3, 3, 4, 12, 11, 19, 23, 54, 55, 103, 115, 178, 289, 389, 507, 757, 970, 1343, 2033, 2579, 3481, 4840, 6312, 8317, 10998, 15459, 19334, 26368, 33480, 44709, 56838, 74878, 93369, 128109, 157024, 206471, 258357, 338085, 417530, 544263, 669388, 859570, 1082758, 1367068

Offset: 0

Gus Wiseman, Aug 16 2023

nonn

Also the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of n.

			The a(0) = 1 through a(7) = 11 partitions:
  ()  (11)  (22)  (33)     (44)      (55)       (66)         (77)
                  (2211)   (3311)    (3322)     (4422)       (4433)
                  (21111)  (311111)  (4411)     (5511)       (5522)
                                     (4111111)  (33321)      (6611)
                                                (42222)      (442211)
                                                (322221)     (4222211)
                                                (332211)     (4421111)
                                                (3222111)    (42221111)
                                                (3321111)    (422111111)
                                                (32211111)   (611111111)
                                                (51111111)   (4211111111)
                                                (321111111)
The a(0) = 1 through a(7) = 11 linear combinations:
  0  1*1  1*2  1*3      1*4      1*5      1*6          1*7
               0*2+3*1  0*3+4*1  0*4+5*1  0*4+3*2      0*6+7*1
               1*2+1*1  1*3+1*1  1*3+1*2  0*5+6*1      1*4+1*3
                                 1*4+1*1  1*4+1*2      1*5+1*2
                                          1*5+1*1      1*6+1*1
                                          0*3+0*2+6*1  0*4+0*2+7*1
                                          0*3+1*2+4*1  0*4+1*2+5*1
                                          0*3+2*2+2*1  0*4+2*2+3*1
                                          0*3+3*2+0*1  0*4+3*2+1*1
                                          1*3+0*2+3*1  1*4+0*2+3*1
                                          1*3+1*2+1*1  1*4+1*2+1*1
                                          2*3+0*2+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500 (first 91 terms from David A. Corneth)

The case with no zero coefficients is A000009.
Central diagonal of A116861.
A version based on Heinz numbers is A364906.
Using all partitions (not just strict) we get A364907.
The version for compositions is A364908, strict A364909.
Main diagonal of A364916.
Using strict partitions of any number from 1 to n gives A365002.
These partitions have ranks A365003.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A237113, A364350, A364839, A364911, A364912, A364914.

Table[Length[Select[IntegerPartitions[2n],Total[Union[#]]==n&]],{n,0,15}]

a(n) = {my(res = 0); forpart(p = 2*n,s = Set(p); if(vecsum(s) == n, res++)); res} \\ David A. Corneth, Aug 20 2023

from sympy.utilities.iterables import partitions
def A364910(n): return sum(1 for d in partitions(n<<1,k=n) if sum(set(d))==n) # Chai Wah Wu, Sep 13 2023

a(n) = A116861(2n,n).

a(n) = A364916(n,n).

More terms from David A. Corneth, Aug 20 2023

A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

Original entry on oeis.org

1, 1, 4, 13, 50, 179, 696, 2619, 10119, 38867, 150407, 582065, 2260367, 8786919, 34225256, 133471650, 521216494, 2037608462, 7974105052, 31235316275, 122457794193, 480473181271, 1886555402750, 7412471695859, 29142658077266, 114643347181003, 451237737215201

Offset: 0

Gus Wiseman, Aug 18 2023

nonn

A way of writing n as a (presumed nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(0) = 1 through a(3) = 13 ways:
  0  1*1  1*2      1*3
          0*1+2*1  0*2+3*1
          1*1+1*1  1*2+1*1
          2*1+0*1  0*1+0*1+3*1
                   0*1+1*1+2*1
                   0*1+2*1+1*1
                   0*1+3*1+0*1
                   1*1+0*1+2*1
                   1*1+1*1+1*1
                   1*1+2*1+0*1
                   2*1+0*1+1*1
                   2*1+1*1+0*1
                   3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A000041.
A finer version is A364906.
The version for compositions is A364908, strict A364909.
Using just strict partitions we get A364910, main diagonal of A364916.
Main diagonal of A365004.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A323092 counts double-free partitions, ranks A320340.
Cf. A116861, A364350, A364839, A364911, A364912, A364913, A364914, A364915, A365002, A365003.

b:= proc(n, i, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
     `if`(i<1, 0, b(n, i-1, m)+add(b(n-i, min(i, n-i), m-i*j), j=0..m/i)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..27);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,IntegerPartitions[n]}]],{n,0,5}]

a(n) = Sum_{m:A056239(m)=n} A364906(m).
a(n) = A364912(2n,n).
a(n) = A365004(n,n).

a(9)-a(26) from Alois P. Heinz, Jan 28 2024

A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

Original entry on oeis.org

1, 1, 4, 15, 70, 314, 1542, 7428, 36860, 182911, 917188, 4612480, 23323662, 118273428, 601762636, 3069070533, 15689123386, 80356953555, 412300910566, 2118715503962, 10902791722490, 56175374185014, 289766946825180, 1496239506613985, 7733302967423382

Offset: 0

Gus Wiseman, Aug 22 2023

nonn

A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).

			The a(3) = 15 ways to write 3 as a nonnegative linear combination of an integer composition of 3:
  1*3  0*2+3*1  1*1+1*2  0*1+0*1+3*1
       1*2+1*1  3*1+0*2  0*1+1*1+2*1
                         0*1+2*1+1*1
                         0*1+3*1+0*1
                         1*1+0*1+2*1
                         1*1+1*1+1*1
                         1*1+2*1+0*1
                         2*1+0*1+1*1
                         2*1+1*1+0*1
                         3*1+0*1+0*1

Alois P. Heinz, Table of n, a(n) for n = 0..500

The case with no zero coefficients is A011782.
The version for partitions is A364907, strict A364910.
The strict case is A364909.
A000041 counts integer partitions, strict A000009.
A011782 counts compositions, strict A032020.
A097805 counts compositions by length, strict A072574.
A116861 = positive linear combinations of strict ptns of k, reverse A364916.
A365067 = nonnegative linear combinations of strict partitions of k.
A364912 = positive linear combinations of partitions of k.
A364916 = positive linear combinations of strict partitions of k.
Cf. A364350, A364839, A364906, A364911, A364914, A365002, A365004.

b:= proc(n, m) option remember; `if`(n=0, `if`(m=0, 1, 0),
      add(add(b(n-i, m-i*j), j=0..m/i), i=1..n))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 28 2024

combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]}, Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[n,ptn],{ptn,Join@@Permutations /@ IntegerPartitions[n]}]],{n,0,5}]

a(8)-a(24) from Alois P. Heinz, Jan 28 2024

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A364910 Number of integer partitions of 2n whose distinct parts sum to n.

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A364907 Number of ways to write n as a nonnegative linear combination of an integer partition of n.

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A364908 Number of ways to write n as a nonnegative linear combination of an integer composition of n.

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Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Extensions