Original entry on oeis.org
0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 4, 1, 4, 1, 1, 1, 10, 1, 1, 2, 4, 1, 7, 1, 8, 1, 1, 1, 13, 1, 1, 1, 10, 1, 7, 1, 4, 4, 1, 1, 24, 1, 4, 1, 4, 1, 10, 1, 10, 1, 1, 1, 22, 1, 1, 4, 16, 1, 7, 1, 4, 1, 7, 1, 38, 1, 1, 4, 4, 1
Offset: 1
A251683 counts ordered factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Other cases of odd length:
-
A024429 counts set partitions of odd length.
-
A027193 counts partitions of odd length.
-
A067659 counts strict partitions of odd length.
-
A089677 counts ordered set partitions of odd length.
-
A166444 counts compositions of odd length.
-
A332304 counts strict compositions of odd length.
-
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],OddQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)
A338915
Number of integer partitions of n that have an even number of parts and cannot be partitioned into distinct pairs of not necessarily distinct parts.
Original entry on oeis.org
0, 0, 0, 0, 1, 0, 1, 1, 4, 2, 6, 6, 12, 12, 20, 22, 38, 42, 60, 73, 101, 124, 164, 203, 266, 319, 415, 507, 649, 786, 983, 1198, 1499, 1797, 2234, 2673, 3303, 3952, 4826, 5753, 6999, 8330, 10051, 11943, 14357, 16956, 20322, 23997, 28568, 33657, 39897, 46879
Offset: 0
The a(7) = 1 through a(12) = 12 partitions:
211111 2222 411111 222211 222221 3333
221111 21111111 331111 611111 222222
311111 511111 22211111 441111
11111111 22111111 32111111 711111
31111111 41111111 22221111
1111111111 2111111111 32211111
33111111
42111111
51111111
2211111111
3111111111
111111111111
For example, the partition y = (3,2,2,1,1,1,1,1) can be partitioned into pairs in just three ways:
{{1,1},{1,1},{1,2},{2,3}}
{{1,1},{1,1},{1,3},{2,2}}
{{1,1},{1,2},{1,2},{1,3}}
None of these is strict, so y is counted under a(12).
The Heinz numbers of these partitions are
A320892.
The complement in even-length partitions is
A338916.
A000070 counts non-multigraphical partitions of 2n, ranked by
A339620.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A339655 counts non-loop-graphical partitions of 2n, ranked by
A339657.
The following count partitions of even length and give their Heinz numbers:
-
smcs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[smcs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&smcs[Times@@Prime/@#]=={}&]],{n,0,10}]
A350945
Heinz numbers of integer partitions of which the number of even parts is equal to the number of even conjugate parts.
Original entry on oeis.org
1, 2, 5, 6, 8, 9, 11, 14, 17, 20, 21, 23, 24, 26, 30, 31, 32, 36, 38, 39, 41, 44, 47, 56, 57, 58, 59, 66, 67, 68, 73, 74, 75, 80, 83, 84, 86, 87, 92, 96, 97, 102, 103, 104, 106, 109, 111, 120, 122, 124, 125, 127, 128, 129, 137, 138, 142, 144, 149, 152, 156
Offset: 1
The terms together with their prime indices begin:
1: ()
2: (1)
5: (3)
6: (2,1)
8: (1,1,1)
9: (2,2)
11: (5)
14: (4,1)
17: (7)
20: (3,1,1)
21: (4,2)
23: (9)
24: (2,1,1,1)
These partitions are counted by
A350948.
These are the positions of 0's in
A350950.
A122111 = conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
A349157: # of even parts = # of odd conjugate parts, counted by
A277579.
A350848: # of even conj parts = # of odd conj parts, counted by
A045931.
A350944: # of odd parts = # of odd conjugate parts, counted by
A277103.
A350945: # of even parts = # of even conjugate parts, counted by
A350948.
Cf.
A000070,
A000290,
A027187,
A027193,
A103919,
A236559,
A344607,
A344651,
A345196, `
A350942,
A350950,
A350951.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?EvenQ]==Count[primeMS[#],?EvenQ]&]
A101707
Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).
Original entry on oeis.org
0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750
Offset: 0
a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
From _Gus Wiseman_, Feb 07 2021: (Start)
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
11 . 31 32 33 52 53 54 55
1111 51 3211 71 72 73
3111 3221 3222 91
111111 3311 3321 3322
5111 5211 3331
311111 321111 5221
11111111 5311
7111
322111
331111
511111
31111111
1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
2 . 4 221 6 421 8 432 A
211 222 22111 422 441 433
411 431 621 442
21111 611 22221 622
22211 42111 631
41111 2211111 811
2111111 22222
42211
43111
61111
2221111
4111111
211111111
(End)
- George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.
Note: A-numbers of ranking sequences are in parentheses below.
The even- but not necessarily positive-rank version is
A340601 (
A340602).
The Heinz numbers of these partitions are (
A340604).
- Rank -
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A026804 counts partitions whose least part is odd.
A339890 counts factorizations of odd length.
-
b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
`if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
`if`(r<0, irem(i, 2), r))))
end:
a:= n-> b(n$2, -1)/2:
seq(a(n), n=0..55); # Alois P. Heinz, Jan 29 2021
-
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
a /@ Range[0, 55] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)
A339618
Heinz numbers of non-graphical integer partitions of even numbers.
Original entry on oeis.org
3, 7, 9, 10, 13, 19, 21, 22, 25, 28, 29, 30, 34, 37, 39, 43, 46, 49, 52, 53, 55, 57, 61, 62, 63, 66, 70, 71, 75, 76, 79, 82, 84, 85, 87, 88, 89, 91, 94, 100, 101, 102, 107, 111, 113, 115, 116, 117, 118, 121, 129, 130, 131, 133, 134, 136, 138, 139, 146, 147
Offset: 1
The sequence of terms together with their prime indices begins:
3: {2} 43: {14} 79: {22}
7: {4} 46: {1,9} 82: {1,13}
9: {2,2} 49: {4,4} 84: {1,1,2,4}
10: {1,3} 52: {1,1,6} 85: {3,7}
13: {6} 53: {16} 87: {2,10}
19: {8} 55: {3,5} 88: {1,1,1,5}
21: {2,4} 57: {2,8} 89: {24}
22: {1,5} 61: {18} 91: {4,6}
25: {3,3} 62: {1,11} 94: {1,15}
28: {1,1,4} 63: {2,2,4} 100: {1,1,3,3}
29: {10} 66: {1,2,5} 101: {26}
30: {1,2,3} 70: {1,3,4} 102: {1,2,7}
34: {1,7} 71: {20} 107: {28}
37: {12} 75: {2,3,3} 111: {2,12}
39: {2,6} 76: {1,1,8} 113: {30}
For example, there are three possible multigraphs with degrees (1,1,3,3):
{{1,2},{1,2},{1,2},{3,4}}
{{1,2},{1,2},{1,3},{2,4}}
{{1,2},{1,2},{1,4},{2,3}}.
Since none of these is a graph, the Heinz number 100 belongs to the sequence.
A006881 lists squarefree semiprimes.
A320656 counts factorizations into squarefree semiprimes.
A339659 counts graphical partitions of 2n into k parts.
The following count vertex-degree partitions and give their Heinz numbers:
-
A339617 counts non-graphical partitions of 2n (
A339618 [this sequence]).
The following count partitions of even length and give their Heinz numbers:
-
strs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strs[n/d],Min@@#>d&]],{d,Select[Divisors[n],And[SquareFreeQ[#],PrimeOmega[#]==2]&]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],EvenQ[Length[nrmptn[#]]]&&strs[Times@@Prime/@nrmptn[#]]=={}&]
A344739
Triangle read by rows where T(n,k) is the number of strict integer partitions of n with reverse-alternating sum k, with k ranging from -n to n in steps of 2.
Original entry on oeis.org
1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 2, 1, 0, 1, 0, 1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 0, 1, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1
Offset: 0
Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 1 0 0 1
0 1 1 0 0 1
0 1 1 0 1 0 1
0 1 1 1 0 1 0 1
0 1 1 1 0 1 1 0 1
0 1 1 1 1 0 2 1 0 1
0 1 1 1 2 0 1 2 1 0 1
0 1 1 1 2 1 0 2 2 1 0 1
0 1 1 1 2 2 0 1 3 2 1 0 1
0 1 1 1 2 3 1 0 2 3 2 1 0 1
0 1 1 1 2 3 3 0 1 3 3 2 1 0 1
0 1 1 1 2 3 4 1 0 3 4 3 2 1 0 1
0 1 1 1 2 3 5 3 0 1 4 4 3 2 1 0 1
0 1 1 1 2 3 5 5 1 0 3 5 4 3 2 1 0 1
0 1 1 1 2 3 5 6 4 0 1 5 6 4 3 2 1 0 1
For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
. E1 D2 C3 B4 A5 96 87 . 762 654 843 A32 C21 . F
9321 7431 6432 861 753 942 B31
8421 6531 54321 852 A41
7521 951
The right halves of even-indexed rows are
A344649.
The non-reverse non-strict version is the right half of
A344651, which is
A239830 interleaved with
A239829.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A124754 lists alternating sums of standard compositions (reverse:
A344618).
A316524 is the alternating sum of the prime indices of n (reverse:
A344616).
A344610 counts partitions of n by positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf.
A000070,
A003242,
A006330,
A027187,
A103919,
A114121,
A343941,
A344607,
A344608,
A344650,
A344654.
-
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&sats[#]==k&]],{n,0,12},{k,-n,n,2}]
A344741
Number of integer partitions of 2n with reverse-alternating sum -2.
Original entry on oeis.org
0, 0, 1, 2, 4, 8, 14, 24, 39, 62, 95, 144, 212, 309, 442, 626, 873, 1209, 1653, 2245, 3019, 4035, 5348, 7051, 9229, 12022, 15565, 20063, 25722, 32847, 41746, 52862, 66657, 83768, 104873, 130889, 162797, 201902, 249620, 307789, 378428, 464122, 567721, 692828, 843448
Offset: 0
The a(2) = 1 through a(6) = 14 partitions:
(31) (42) (53) (64) (75)
(3111) (3221) (3331) (4332)
(4211) (4222) (4431)
(311111) (4321) (5322)
(5311) (5421)
(322111) (6411)
(421111) (322221)
(31111111) (333111)
(422211)
(432111)
(531111)
(32211111)
(42111111)
(3111111111)
The version for -1 instead of -2 is
A000070.
The non-reversed negative version is
A000097.
The ordered version appears to be
A001700.
The version for 1 instead of -2 is
A035363.
The whole set of partitions of 2n is counted by
A058696.
The strict case appears to be
A065033.
The version for -1 instead of -2 is
A306145.
The version for 2 instead of -2 is
A344613.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A124754 gives alternating sums of standard compositions (reverse:
A344618).
A316524 is the alternating sum of the prime indices of n (reverse:
A344616).
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
Cf.
A001250,
A003242,
A006330,
A027187,
A028260,
A344604,
A344607,
A344608,
A344650,
A344651,
A344654,
A344739.
-
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]==-2&]],{n,0,30,2}]
- or -
Table[Length[Select[IntegerPartitions[n],EvenQ[Max[#]]&&Count[#,_?OddQ]==2&]],{n,0,30,2}]
A350944
Heinz numbers of integer partitions of which the number of odd parts is equal to the number of odd conjugate parts.
Original entry on oeis.org
1, 2, 6, 9, 10, 12, 15, 18, 20, 30, 35, 49, 54, 55, 56, 70, 75, 77, 81, 84, 88, 90, 98, 108, 110, 112, 125, 132, 135, 143, 154, 162, 168, 169, 176, 180, 187, 210, 221, 260, 264, 270, 286, 294, 315, 323, 330, 338, 340, 350, 361, 363, 364, 374, 391, 416, 420
Offset: 1
The terms together with their prime indices begin:
1: ()
2: (1)
6: (2,1)
9: (2,2)
10: (3,1)
12: (2,1,1)
15: (3,2)
18: (2,2,1)
20: (3,1,1)
30: (3,2,1)
35: (4,3)
49: (4,4)
54: (2,2,2,1)
These partitions are counted by
A277103.
A122111 = conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
A349157: # of even parts = # of odd conjugate parts, counted by
A277579.
A350945: # of even parts = # of even conjugate parts, counted by
A350948.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[conj[primeMS[#]],?OddQ]==Count[primeMS[#],?OddQ]&]
A350948
Number of integer partitions of n with as many even parts as even conjugate parts.
Original entry on oeis.org
1, 1, 0, 3, 1, 5, 3, 7, 6, 10, 10, 18, 19, 27, 31, 40, 47, 65, 75, 98, 115, 142, 170, 217, 257, 316, 376, 458, 544, 671, 792, 952, 1129, 1351, 1598, 1919, 2259, 2681, 3155, 3739, 4384, 5181, 6064, 7129, 8331, 9764, 11380, 13308, 15477, 18047, 20944
Offset: 0
The a(0) = 1 through a(8) = 6 partitions (empty column indicated by dot):
() (1) . (3) (22) (5) (42) (7) (62)
(21) (41) (321) (61) (332)
(111) (311) (2211) (511) (521)
(2111) (4111) (4211)
(11111) (31111) (32111)
(211111) (221111)
(1111111)
For example, both (3,2,1,1,1) and its conjugate (5,2,1) have exactly 1 even part, so are counted under a(8).
Comparing even to odd conjugate parts gives
A277579, ranked by
A349157.
Comparing product of parts to product of conjugate parts gives
A325039.
A103919 counts partitions by sum and alternating sum, reverse
A344612.
A116482 counts partitions by number of even (or even conjugate) parts.
A122111 represents partition conjugation using Heinz numbers.
A351978: # even = # odd = # even conj = # odd conj, ranked by
A350947.
Cf.
A027187,
A130780,
A171966,
A195017,
A239241,
A241638,
A344607,
A344651,
A350848,
A350941,
A350942,
A350943.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Count[#,?EvenQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A350943
Heinz numbers of integer partitions of which the number of even conjugate parts is equal to the number of odd parts.
Original entry on oeis.org
1, 3, 6, 7, 13, 14, 18, 19, 26, 27, 29, 36, 37, 38, 42, 43, 53, 54, 58, 61, 63, 70, 71, 74, 78, 79, 84, 86, 89, 101, 105, 106, 107, 113, 114, 117, 122, 126, 130, 131, 139, 140, 142, 151, 156, 158, 162, 163, 171, 173, 174, 178, 181, 190, 193, 195, 199, 202, 210
Offset: 1
The terms together with their prime indices begin:
1: ()
3: (2)
6: (2,1)
7: (4)
13: (6)
14: (4,1)
18: (2,2,1)
19: (8)
26: (6,1)
27: (2,2,2)
29: (10)
36: (2,2,1,1)
37: (12)
38: (8,1)
42: (4,2,1)
For example, the partition (6,3,2) has conjugate (3,3,2,1,1,1) and 1 = 1 so 195 is in the sequence.
These partitions are counted by
A277579.
These are the positions of 0's in
A350942.
A122111 = conjugation using Heinz numbers.
A316524 = alternating sum of prime indices.
The following rank partitions:
A350848: # of even conj parts = # of odd conj parts, counted by
A045931.
A350944: # of odd parts = # of odd conjugate parts, counted by
A277103.
A350945: # of even parts = # of even conjugate parts, counted by
A350948.
Cf.
A000070,
A000290,
A027187,
A027193,
A103919,
A236559,
A344607,
A344651,
A345196,
A350950,
A350951.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],Count[primeMS[#],?OddQ]==Count[conj[primeMS[#]],?EvenQ]&]
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