A364906 -id:A364906

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A364910

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

+10
18

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

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

LINKS

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

FORMULA

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

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

EXAMPLE

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

MATHEMATICA

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

PROG

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

(Python)

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 16 2023

EXTENSIONS

More terms from David A. Corneth, Aug 20 2023

STATUS

approved

A364907

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

+10
11

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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).

LINKS

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

FORMULA

a(n) = Sum_{m:A056239(m)=n} A364906(m).

a(n) = A364912(2n,n).

a(n) = A365004(n,n).

EXAMPLE

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

MAPLE

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

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2023

EXTENSIONS

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

STATUS

approved

A364908

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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).

LINKS

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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}]

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 22 2023

EXTENSIONS

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

STATUS

approved

A364909

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

+10
4

1, 1, 1, 5, 5, 7, 51, 45, 89, 109, 709, 733, 1495, 1935, 3119, 13785, 16611, 29035, 44611, 68733, 95193, 372897, 435007, 781345, 1177181, 1866659, 2600537, 3906561, 12052631, 14610799, 25407653, 37652265, 59943351, 84060993, 128112805, 172172117, 480353257, 578740011

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

LINKS

Table of n, a(n) for n=0..37.

EXAMPLE

The a(0) = 1 through a(5) = 7 ways:

. 1*1 1*2 1*3 1*4 1*5

0*2+3*1 0*3+4*1 0*4+5*1

1*1+1*2 1*1+1*3 1*1+1*4

1*2+1*1 1*3+1*1 1*2+1*3

3*1+0*2 4*1+0*3 1*3+1*2

1*4+1*1

5*1+0*4

MATHEMATICA

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/@Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 5}]

PROG

(Python)

from math import factorial

from sympy.utilities.iterables import partitions

def A364909(n):

if n == 0: return 1

aset = tuple(set(p) for p in partitions(n) if max(p.values(), default=0)==1)

return sum(factorial(len(t)) for p in partitions(n) for t in aset if set(p).issubset(t)) # Chai Wah Wu, Sep 21 2023

CROSSREFS

The case with no zero coefficients is A032020.

The version for partitions is A364907, strict A364910(n) = A364916(n,n).

The non-strict version is A364908.

A000041 counts integer partitions, strict A000009.

A011782 counts compositions, strict A032020.

A008284 counts partitions by length, strict A008289.

A097805 counts compositions by length, strict A072574.

Cf. A116861, A364350, A364839, A364906, A364912, A364914, A365002, A365004.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 18 2023

EXTENSIONS

a(18)-a(37) from Chai Wah Wu, Sep 21 2023

STATUS

approved

A365003

Heinz numbers of integer partitions where the sum of all parts is twice the sum of distinct parts.

+10
3

1, 4, 9, 25, 36, 48, 49, 100, 121, 160, 169, 196, 225, 289, 361, 441, 448, 484, 529, 567, 676, 750, 810, 841, 900, 961, 1080, 1089, 1156, 1200, 1225, 1369, 1408, 1440, 1444, 1521, 1681, 1764, 1849, 1920, 2116, 2209, 2268, 2352, 2601, 2809, 3024, 3025, 3159

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

LINKS

Table of n, a(n) for n=1..49.

FORMULA

A056239(a(n)) = 2*A066328(a(n)).

EXAMPLE

The prime indices of 750 are {1,2,3,3,3}, with sum 12, while the distinct prime indices {1,2,3} have sum 6, so 750 is in the sequence.

The terms together with their prime indices begin:

1: {}

4: {1,1}

9: {2,2}

25: {3,3}

36: {1,1,2,2}

48: {1,1,1,1,2}

49: {4,4}

100: {1,1,3,3}

121: {5,5}

160: {1,1,1,1,1,3}

169: {6,6}

196: {1,1,4,4}

225: {2,2,3,3}

289: {7,7}

361: {8,8}

441: {2,2,4,4}

448: {1,1,1,1,1,1,4}

MATHEMATICA

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[1000], Total[prix[#]]==2*Total[Union[prix[#]]]&]

CROSSREFS

The LHS is A056239 (sum of prime indices).

The RHS is twice A066328.

Partitions of this type are counted by A364910.

A000041 counts integer partitions, strict A000009.

A001222 counts prime indices, distinct A001221.

A112798 lists prime indices, distinct A304038.

A116861 counts partitions by sum and sum of distinct parts.

A323092 counts double-free partitions, ranks A320340.

Cf. A364350, A364839, A364906, A364907, A364911, A364916.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 23 2023

STATUS

approved

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