A351981
Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 2, 2, 4, 2, 1, 6, 8, 7, 9, 13, 14, 15, 19, 21, 23, 32, 40, 41, 45, 66, 81, 80, 96, 124, 139, 160, 194, 221, 246, 303, 360, 390, 446, 546, 634, 703, 810, 971, 1115, 1250, 1448, 1685, 1910
Offset: 0
The a(n) partitions for selected n:
n = 3 9 15 18 19 20 21
-----------------------------------------------------------
21 4221 622221 633222 633322 644321 643332
4311 632211 643221 643321 653321 654321
642111 643311 644221 654221 665211
651111 644211 644311 654311 82222221
653211 653221 82222211 83222211
663111 653311 84221111 84222111
654211 86111111 85221111
664111 86211111
87111111
For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).
These partitions are ranked by
A351980.
There are four statistics:
There are four other pairings of statistics:
-
A045931: # of even parts = # of odd parts:
-
A277103: # of odd parts = # of odd conjugate parts, ranked by
A350944.
-
A350948: # of even parts = # of even conjugate parts, ranked by
A350945.
There are two other double-pairings of statistics:
-
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[#],?OddQ]&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A345196
Number of integer partitions of n with reverse-alternating sum equal to the reverse-alternating sum of their conjugate.
Original entry on oeis.org
1, 1, 0, 1, 1, 1, 1, 3, 4, 4, 4, 8, 11, 11, 11, 20, 27, 29, 31, 48, 65, 70, 74, 109, 145, 160, 172, 238, 314, 345, 372, 500, 649, 721, 782, 1019, 1307, 1451, 1577, 2015, 2552, 2841, 3098, 3885, 4867, 5418, 5914, 7318, 9071, 10109, 11050
Offset: 0
The a(5) = 1 through a(12) = 11 partitions:
(311) (321) (43) (44) (333) (541) (65) (66)
(2221) (332) (531) (4321) (4322) (552)
(4111) (2222) (32211) (32221) (4331) (4332)
(4211) (51111) (52111) (4421) (4422)
(6311) (4431)
(222221) (6411)
(422111) (33222)
(611111) (53211)
(222222)
(422211)
(621111)
The non-reverse version is
A277103.
Comparing even parts to odd conjugate parts gives
A277579.
Comparing signs only gives
A340601.
A000041 counts partitions of 2n with alternating sum 0, ranked by
A000290.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A120452 counts partitions of 2n with rev-alt sum 2 (negative:
A344741).
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.
A000070,
A000097,
A006330,
A027187,
A027193,
A236559,
A239829,
A257991,
A344607,
A344608,
A344651,
A344654.
-
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],sats[#]==sats[conj[#]]&]],{n,0,15}]
A350841
Heinz numbers of integer partitions with a difference < -1 and a conjugate difference < -1.
Original entry on oeis.org
20, 28, 40, 44, 52, 56, 63, 68, 76, 80, 84, 88, 92, 99, 100, 104, 112, 116, 117, 124, 126, 132, 136, 140, 148, 152, 153, 156, 160, 164, 168, 171, 172, 176, 184, 188, 189, 196, 198, 200, 204, 207, 208, 212, 220, 224, 228, 232, 234, 236, 244, 248, 252, 260, 261
Offset: 1
The terms together with their prime indices begin:
20: (3,1,1)
28: (4,1,1)
40: (3,1,1,1)
44: (5,1,1)
52: (6,1,1)
56: (4,1,1,1)
63: (4,2,2)
68: (7,1,1)
76: (8,1,1)
80: (3,1,1,1,1)
84: (4,2,1,1)
88: (5,1,1,1)
92: (9,1,1)
99: (5,2,2)
Heinz number rankings are in parentheses below.
These partitions are counted by
A350839.
A116932 = partitions with no successions or gaps of size 1, strict
A025157.
-
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],(Min@@Differences[Reverse[primeMS[#]]]<-1)&&(Min@@Differences[conj[primeMS[#]]]<-1)&]
A352129
Number of strict integer partitions of n with as many even conjugate parts as odd conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 5, 5, 6, 6, 9, 8, 10, 12, 13, 15, 17, 20, 20, 26, 26, 32, 35, 39, 44, 50, 55, 61, 71, 76, 87, 96, 108, 117, 135, 145, 164, 181, 200, 222, 246, 272, 298, 334, 363, 404, 443
Offset: 0
The a(n) strict partitions for selected n:
n = 3 13 15 18 20 22
------------------------------------------------------------------
(2,1) (6,5,2) (10,5) (12,6) (12,7,1) (12,8,2)
(6,4,2,1) (6,4,3,2) (8,7,3) (8,5,4,3) (8,6,5,3)
(6,5,3,1) (8,5,3,2) (8,6,4,2) (8,7,5,2)
(8,6,3,1) (8,7,4,1) (12,7,2,1)
(8,6,3,2,1) (8,6,4,3,1)
(8,7,4,2,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are three double-pairings of statistics:
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A100824
Number of partitions of n with at most one odd part.
Original entry on oeis.org
1, 1, 1, 2, 2, 4, 3, 7, 5, 12, 7, 19, 11, 30, 15, 45, 22, 67, 30, 97, 42, 139, 56, 195, 77, 272, 101, 373, 135, 508, 176, 684, 231, 915, 297, 1212, 385, 1597, 490, 2087, 627, 2714, 792, 3506, 1002, 4508, 1255, 5763, 1575, 7338, 1958, 9296, 2436, 11732, 3010, 14742
Offset: 0
From _Gus Wiseman_, Jan 21 2022: (Start)
The a(1) = 1 through a(9) = 12 partitions with at most one odd part:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(21) (22) (32) (42) (43) (44) (54)
(41) (222) (52) (62) (63)
(221) (61) (422) (72)
(322) (2222) (81)
(421) (432)
(2221) (441)
(522)
(621)
(3222)
(4221)
(22221)
(End)
The case of alternating sum 0 (equality) is
A000070.
A multiplicative version is
A339846.
A058695 = partitions of odd numbers.
A277103 = partitions with the same number of odd parts as their conjugate.
Cf.
A000984,
A001791,
A008549,
A097805,
A119620,
A182616,
A236559,
A236913,
A236914,
A304620,
A344607,
A345958,
A347443.
-
seq(coeff(convert(series((1+x/(1-x^2))/mul(1-x^(2*i),i=1..100),x,100),polynom),x,n),n=0..60); (C. Ronaldo)
-
nmax = 50; CoefficientList[Series[(1+x/(1-x^2)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)
Table[Length[Select[IntegerPartitions[n],Count[#,?OddQ]<=1&]],{n,0,30}] (* _Gus Wiseman, Jan 21 2022 *)
-
a(n) = if(n%2==0, numbpart(n/2), sum(i=1, (n+1)\2, numbpart((n-2*i+1)\2))) \\ David A. Corneth, Jan 23 2022
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 19 2005
A346701
Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
Original entry on oeis.org
1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 6, 13, 7, 5, 4, 17, 6, 19, 10, 7, 11, 23, 6, 5, 13, 9, 14, 29, 10, 31, 8, 11, 17, 7, 6, 37, 19, 13, 10, 41, 14, 43, 22, 15, 23, 47, 12, 7, 10, 17, 26, 53, 9, 11, 14, 19, 29, 59, 10, 61, 31, 21, 8, 13, 22, 67, 34, 23, 14, 71
Offset: 1
The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.
The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.
Positions of last appearances are
A000290 without the first term 0.
Positions of first appearances are
A342768.
The non-reverse version is
A346703.
The even non-reverse version is
A346704.
A001221 counts distinct prime factors.
A103919 counts partitions by sum and alternating sum, reverse
A344612.
A209281 (shifted) adds up the odd bisection of standard compositions.
A344606 counts alternating permutations of prime indices.
A344617 gives the sign of the alternating sum of prime indices.
A346700 gives the sum of the even bisection of reversed prime indices.
Cf.
A025047,
A027187,
A027193,
A053738,
A106356,
A277103,
A341446,
A344653,
A345957,
A345958,
A345959,
A346698,
A346702.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]
A352130
Number of strict integer partitions of n with as many odd parts as even conjugate parts.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 5, 6, 7, 7, 8, 9, 11, 12, 13, 14, 16, 18, 21, 23, 25, 28, 31, 34, 37, 41, 45, 50, 55, 60, 65, 72, 79, 86, 93, 102, 111, 121, 132, 143, 155, 169, 183, 197, 213, 231, 251, 271, 292, 315, 340, 367, 396
Offset: 0
The a(n) strict partitions for selected n:
n = 2 7 9 13 14 15 16
--------------------------------------------------------------------
(2) (6,1) (8,1) (12,1) (14) (14,1) (16)
(4,2,1) (4,3,2) (6,4,3) (6,5,3) (6,5,4) (8,5,3)
(6,2,1) (8,3,2) (10,3,1) (8,4,3) (12,3,1)
(10,2,1) (6,4,3,1) (10,3,2) (6,5,4,1)
(8,3,2,1) (12,2,1) (8,4,3,1)
(6,5,3,1) (10,3,2,1)
(6,4,3,2,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are three double-pairings of statistics:
Cf.
A027187,
A027193,
A103919,
A122111,
A236559,
A325039,
A344607,
A344651,
A345196,
A350950,
A350951.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A352131
Number of strict integer partitions of n with same number of even parts as odd conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 2, 3, 2, 2, 3, 4, 3, 4, 5, 5, 5, 6, 7, 7, 8, 10, 10, 10, 12, 14, 15, 14, 17, 21, 20, 20, 25, 28, 28, 29, 34, 39, 39, 40, 47, 52, 53, 56, 64, 70, 71, 77, 86, 92, 97, 104, 114, 122
Offset: 0
The a(n) strict partitions for selected n:
n = 3 10 14 18 21 24
----------------------------------------------------------------------
(2,1) (6,4) (8,6) (10,8) (11,10) (8,7,5,4)
(4,3,2,1) (5,4,3,2) (6,5,4,3) (8,6,4,3) (9,8,4,3)
(6,5,2,1) (7,6,3,2) (8,7,4,2) (10,8,4,2)
(8,7,2,1) (10,8,2,1) (10,9,3,2)
(6,5,4,3,2,1) (11,10,2,1)
(8,6,4,3,2,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are three double-pairings of statistics:
Cf.
A027187,
A027193,
A103919,
A122111,
A236559,
A325039,
A344607,
A344651,
A345196,
A350942,
A350950,
A350951.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?EvenQ]==Count[conj[#],?OddQ]&]],{n,0,30}]
A352128
Number of strict integer partitions of n with (1) as many even parts as odd parts, and (2) as many even conjugate parts as odd conjugate parts.
Original entry on oeis.org
1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 2, 0, 2, 2, 3, 0, 3, 0, 2, 2, 5, 2, 5, 4, 6, 7, 7, 8, 8, 9, 9, 13, 9, 14, 12, 20, 13, 25, 17, 33, 23, 40, 26, 50, 33, 59, 39, 68, 45, 84, 58, 92, 70, 115, 88, 132, 109, 156, 139, 182, 172, 212, 211
Offset: 0
The a(n) strict partitions for selected n:
n = 3 18 22 28 31 32
-----------------------------------------------------------------------
(2,1) (8,5,3,2) (8,6,5,3) (12,7,5,4) (10,7,5,4,3,2) (12,8,7,5)
(8,6,3,1) (8,7,5,2) (12,8,5,3) (10,7,6,5,2,1) (12,9,7,4)
(12,7,2,1) (12,9,5,2) (10,8,5,4,3,1) (16,9,4,3)
(16,9,2,1) (10,9,6,3,2,1) (12,10,7,3)
(12,10,5,1) (12,11,7,2)
(16,11,4,1)
A130780 counts partitions with no more even than odd parts, strict
A239243.
A171966 counts partitions with no more odd than even parts, strict
A239240.
There are four statistics:
There are four other pairings of statistics:
There are two other double-pairings of statistics:
Cf.
A000070,
A014105,
A088218,
A098123,
A195017,
A236559,
A236914,
A241638,
A325700,
A350839,
A350941.
-
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,?OddQ]==Count[#,?EvenQ]&&Count[conj[#],?OddQ]==Count[conj[#],?EvenQ]&]],{n,0,30}]
A349151
Heinz numbers of integer partitions with alternating sum <= 1.
Original entry on oeis.org
1, 2, 4, 6, 8, 9, 15, 16, 18, 24, 25, 32, 35, 36, 49, 50, 54, 60, 64, 72, 77, 81, 96, 98, 100, 121, 128, 135, 140, 143, 144, 150, 162, 169, 196, 200, 216, 221, 225, 240, 242, 256, 288, 289, 294, 308, 315, 323, 324, 338, 361, 375, 384, 392, 400, 437, 441, 450
Offset: 1
The terms and their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
9: {2,2}
15: {2,3}
16: {1,1,1,1}
18: {1,2,2}
24: {1,1,1,2}
25: {3,3}
32: {1,1,1,1,1}
35: {3,4}
36: {1,1,2,2}
49: {4,4}
The case of alternating sum 0 is
A000290.
These partitions are counted by
A100824.
These are the positions of 0's and 1's in
A344616.
The case of alternating sum 1 is
A345958.
The conjugate partitions are ranked by
A349150.
A103919 counts partitions by sum and alternating sum (reverse:
A344612).
A122111 is a representation of partition conjugation.
A316524 gives the alternating sum of prime indices.
A344610 counts partitions by sum and positive reverse-alternating sum.
A349157 ranks partitions with as many even parts as odd conjugate parts.
Cf.
A000070,
A000700,
A001222,
A027187,
A027193,
A215366,
A277103,
A277579,
A326841,
A349149,
A349158.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Select[Range[100],ats[Reverse[primeMS[#]]]<=1&]
Comments