Displaying 1-10 of 15 results found.
Number of even parts in the conjugate of the integer partition with Heinz number n.
+10
23
0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 0, 0, 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 1, 3, 1, 0, 0, 0, 1, 0, 0, 2, 1, 3, 2, 0, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 0, 4, 2, 2, 0, 0, 1, 3, 1, 2, 1, 0, 2, 0, 1, 0, 1, 3, 1, 0, 0, 2, 2, 0, 1, 0, 1, 1, 0, 4, 1, 0, 0, 2, 1, 0, 2, 3, 1, 2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so a(n) counts even prime indices of n.
MATHEMATICA
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]}]];
Table[Count[conj[primeMS[n]], _?EvenQ], {n, 100}]
CROSSREFS
Positions of first appearances are A001248. Subtracting from the number of odd conjugate parts gives A350941. Subtracting from the number of odd parts gives A350942. Subtracting from the number of even parts gives A350950. There are four statistics:
There are six possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350848: # of even conj parts = # of odd conj parts, counted by A045931. - A350943: # of even conjugate parts = # of odd parts, counted by A277579. - A350944: # of odd parts = # of odd conjugate parts, counted by A277103. - A350945: # of even parts = # of even conjugate parts, counted by A350948. There are three possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
Cf. A028260, A130780, A171966, A195017, A236559, A239241, A241638, A316524, A325700, A350849, A350951.
Number of integer partitions of n with as many even parts as even conjugate parts.
+10
22
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
EXAMPLE
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).
MATHEMATICA
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}]
CROSSREFS
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.
Heinz numbers of integer partitions with the same number of even parts, odd parts, even conjugate parts, and odd conjugate parts.
+10
19
1, 6, 84, 210, 490, 525, 2184, 2340, 5460, 9464, 12012, 12740, 12870, 13650, 14625, 19152, 22308, 30030, 34125, 43940, 45144, 55770, 59150, 66066, 70070, 70785, 75075, 79625, 82992, 88920
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.
EXAMPLE
The terms together with their prime indices begin:
1: ()
6: (2,1)
84: (4,2,1,1)
210: (4,3,2,1)
490: (4,4,3,1)
525: (4,3,3,2)
2184: (6,4,2,1,1,1)
2340: (6,3,2,2,1,1)
5460: (6,4,3,2,1,1)
9464: (6,6,4,1,1,1)
12012: (6,5,4,2,1,1)
12740: (6,4,4,3,1,1)
12870: (6,5,3,2,2,1)
13650: (6,4,3,3,2,1)
14625: (6,3,3,3,2,2)
19152: (8,4,2,2,1,1,1,1)
For example, the partition (6,6,4,1,1,1) has conjugate (6,3,3,3,2,2), and all four statistics are equal to 3, so 9464 is in the sequence.
MATHEMATICA
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[1000], Count[primeMS[#], _?EvenQ]==Count[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?EvenQ]==Count[conj[primeMS[#]], _?OddQ]&]
CROSSREFS
These partitions are counted by A351978. There are four individual statistics:
There are six possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350848: # of even conj parts = # of odd conj parts, counted by A045931. - A350943: # of even conjugate parts = # of odd parts, counted by A277579. - A350944: # of odd parts = # of odd conjugate parts, counted by A277103. - A350945: # of even parts = # of even conjugate parts, counted by A350948. There are three possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A239241, A241638, A325700, A350849, A350941, A350942, A350950, A350951.
Heinz numbers of integer partitions with as many even parts as even conjugate parts and as many odd parts as odd conjugate parts.
+10
19
1, 2, 6, 9, 20, 30, 56, 75, 84, 125, 176, 210, 264, 294, 315, 350, 416, 441, 490, 525, 624, 660, 735, 924, 990, 1088, 1100, 1386, 1540, 1560, 1632, 1650, 1715, 2184, 2310, 2340, 2401, 2432, 2600, 3267, 3276, 3388, 3640, 3648, 3900, 4080, 4125, 5082, 5324, 5390
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.
FORMULA
Closed under A122111 (conjugation).
EXAMPLE
The terms together with their prime indices begin:
1: ()
2: (1)
6: (2,1)
9: (2,2)
20: (3,1,1)
30: (3,2,1)
56: (4,1,1,1)
75: (3,3,2)
84: (4,2,1,1)
125: (3,3,3)
176: (5,1,1,1,1)
210: (4,3,2,1)
264: (5,2,1,1,1)
294: (4,4,2,1)
315: (4,3,2,2)
350: (4,3,3,1)
416: (6,1,1,1,1,1)
MATHEMATICA
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[1000], Count[primeMS[#], _?OddQ]==Count[conj[primeMS[#]], _?OddQ]&&Count[primeMS[#], _?EvenQ]==Count[conj[primeMS[#]], _?EvenQ]&]
CROSSREFS
These partitions are counted by A351976. There are four other possible pairings of statistics:
There are two other possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Cf. A026424, A028260, A098123, A130780, A171966, A241638, A325700, A350849, A350941, A350942, A350950, A350951.
Heinz numbers of integer partitions with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.
+10
16
1, 6, 65, 84, 210, 216, 319, 490, 525, 532, 731, 1254, 1403, 1924, 2184, 2340, 2449, 2470, 3024, 3135, 3325, 3774, 4028, 4141, 4522, 5311, 5460, 7030, 7314, 7315, 7560, 7776, 7942, 8201, 8236, 9048, 9435, 9464, 10659, 10921, 11484, 11914, 12012, 12025, 12740
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.
FORMULA
Closed under A122111 (conjugation).
EXAMPLE
The terms together with their prime indices begin:
1: ()
6: (2,1)
65: (6,3)
84: (4,2,1,1)
210: (4,3,2,1)
216: (2,2,2,1,1,1)
319: (10,5)
490: (4,4,3,1)
525: (4,3,3,2)
532: (8,4,1,1)
731: (14,7)
1254: (8,5,2,1)
1403: (18,9)
1924: (12,6,1,1)
2184: (6,4,2,1,1,1)
2340: (6,3,2,2,1,1)
2449: (22,11)
2470: (8,6,3,1)
For example, the prime indices of 532 are (8,4,1,1), even/odd counts 2/2, and the prime indices of the conjugate 3024 are (4,2,2,2,1,1,1,1), with even/odd counts 4/4; so 532 belongs to the sequence.
MATHEMATICA
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[1000], #==1||Mean[Mod[primeMS[#], 2]]== Mean[Mod[conj[primeMS[#]], 2]]==1/2&]
CROSSREFS
For the first condition alone:
- ordered version (compositions) A098123There are four statistics:
There are four other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350943: # of even conj parts = # of odd parts, strict counted by A352130. - A350944: # of odd parts = # of odd conjugate parts, counted by A277103. - A350945: # of even parts = # of even conjugate parts, counted by A350948. There are two other possible double-pairings of statistics:
A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
A316524 = alternating sum of prime indices.
Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.
+10
16
0, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, 0, 1, 0, 0, -3, 0, 3, 1, 1, 0, 0, 0, -1, -1, -2, 0, 1, 0, 0, -1, 0, 1, 1, 0, 2, -1, 0, 1, -2, -2, -1, 1, 1, 2, -3, 0, 0, 0, 0, -1, 1, -1, 3, -1, -2, 0, 0, 0, -1, -1, 1, 1, 0, 0, 0, 1, -3, 1, 1, 0
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.
FORMULA
a( A122111(n)) = -a(n), where A122111 represents partition conjugation.
EXAMPLE
The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 1 = 1.
MATHEMATICA
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]}]];
Table[Count[primeMS[n], _?EvenQ]-Count[conj[primeMS[n]], _?EvenQ], {n, 100}]
CROSSREFS
The version comparing even with odd parts is A195017. The version comparing even with odd conjugate parts is A350849. The version comparing even conjugate with odd conjugate parts is A350941. The version comparing odd with even conjugate parts is A350942. The version comparing odd with odd conjugate parts is A350951. There are four individual statistics:
There are five other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350848: # of even conj parts = # of odd conj parts, counted by A045931. - A350943: # of even conjugate parts = # of odd parts, counted by A277579. - A350944: # of odd parts = # of odd conjugate parts, counted by A277103. There are three possible double-pairings of statistics:
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.
+10
16
0, 0, -2, 2, -2, 0, -4, 2, 0, 0, -4, 0, -6, -2, 0, 4, -6, 0, -8, 0, -2, -2, -8, 2, 2, -4, -2, -2, -10, 0, -10, 4, -2, -4, 0, 2, -12, -6, -4, 2, -12, -2, -14, -2, -2, -6, -14, 2, 0, 2, -4, -4, -16, 0, 0, 0, -6, -8, -16, 2, -18, -8, -4, 6, -2, -2, -18, -4, -6, 0
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.
All terms are even.
FORMULA
a( A122111(n)) = -a(n), where A122111 represents partition conjugation.
EXAMPLE
The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 1 - 5 = -4.
MATHEMATICA
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]}]];
Table[Count[primeMS[n], _?OddQ]-Count[conj[primeMS[n]], _?OddQ], {n, 100}]
CROSSREFS
The version comparing even with odd parts is A195017. The version comparing even with odd conjugate parts is A350849. The version comparing even conjugate with odd conjugate parts is A350941. The version comparing odd with even conjugate parts is A350942. The version comparing even with even conjugate parts is A350950. There are four individual statistics:
There are five other possible pairings of statistics:
- A349157: # of even parts = # of odd conjugate parts, counted by A277579. - A350848: # of even conj parts = # of odd conj parts, counted by A045931. - A350943: # of even conjugate parts = # of odd parts, counted by A277579. - A350945: # of even parts = # of even conjugate parts, counted by A350948. There are three possible double-pairings of statistics:
A103919 counts partitions by number of odd parts.
A116482 counts partitions by number of even parts.
A122111 represents partition conjugation using Heinz numbers.
A316524 gives the alternating sum of prime indices.
Cf. A026424, A028260, A053738, A098123, A130780, A171966, A236559, A236914, A239241, A241638, A325700.
Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.
+10
15
1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 6, 9, 11, 11, 16, 21, 22, 24, 31, 41, 46, 48, 64, 82, 91, 98, 120, 155, 175, 188, 237, 297, 329, 357, 437, 544, 607, 658, 803, 987, 1098, 1196, 1432, 1749, 1955, 2126, 2541, 3071, 3417, 3729, 4406, 5291, 5890, 6426
EXAMPLE
The a(n) partitions for selected n:
n = 3 8 11 12 15 16
----------------------------------------------------------
(21) (332) (4322) (4332) (4443) (4444)
(4211) (4331) (4422) (54321) (53332)
(4421) (4431) (632211) (55222)
(611111) (53211) (633111) (55411)
(621111) (642111) (633211)
(81111111) (642211)
(643111)
(7321111)
(82111111)
MATHEMATICA
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]==Count[conj[#], _?OddQ]&&Count[#, _?EvenQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A350949. A122111 represents partition conjugation using Heinz numbers.
A195017 = # of even parts - # of odd parts.
There are four statistics:
There are four other possible pairings of statistics:
There are two other possible double-pairings of statistics:
- A351977: # even = # odd, # even conj = # odd conj, ranked by A350946. - A351981: # even = # odd conj, # odd = # even conj, ranked by A351980. Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.
Number of integer partitions of n with as many even parts as odd parts and as many even conjugate parts as odd conjugate parts.
+10
15
1, 0, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 2, 0, 2, 4, 2, 1, 6, 6, 7, 9, 11, 10, 13, 17, 17, 21, 28, 36, 35, 41, 58, 71, 72, 90, 106, 121, 142, 178, 191, 216, 269, 320, 344, 400, 486, 564, 633, 734, 867, 991, 1130, 1312, 1509, 1702, 1978, 2288, 2582, 2917, 3404
EXAMPLE
The a(n) partitions for selected n (A..C = 10..12):
n = 3 9 15 18 20
----------------------------------------------------------
(21) (63) (A5) (8433) (8543)
(222111) (632211) (8532) (8741)
(642111) (8631) (C611)
(2222211111) (43322211) (43332221)
(44322111) (44432111)
(44421111) (84221111)
(422222111111)
MATHEMATICA
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Count[#, _?OddQ]==Count[#, _?EvenQ]&&Count[conj[#], _?OddQ]==Count[conj[#], _?EvenQ]&]], {n, 0, 30}]
CROSSREFS
These partitions are ranked by A350946. There are four statistics:
There are four additional pairings of statistics:
There are two additional double-pairings of statistics:
Cf. A000041, A000070, A088218, A098123, A130780, A171966, A195017, A236559, A236914, A241638, A350849.
Number of integer partitions of n for which the number of even parts, the number of odd parts, the number of even conjugate parts, and the number of odd conjugate parts are all equal.
+10
15
1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 2, 0, 1, 0, 6, 1, 3, 1, 8, 5, 3, 5, 7, 14, 2, 13, 9, 28, 5, 22, 26, 44, 17, 30, 60, 59, 42, 41, 120, 84, 84, 66, 204, 143, 144, 131, 325, 268, 226, 261, 486, 498, 344, 488, 739, 874
EXAMPLE
The a(n) partitions for selected n (A = 10):
n = 3 12 19 21 23 24 27
--------------------------------------------------------------
21 4332 633322 643332 644333 84332211 655443
4431 643321 654321 654332 84441111 655542
644311 665211 654431 85322211 665541
653221 655322 86322111 666333
654211 655421 86421111 666531
664111 664331 A522221111
665321 A622211111
666311
MATHEMATICA
conj[y_]:=If[Length[y]==0, y, Table[Length[Select[y, #>=k&]], {k, 1, Max[y]}]];
Table[Length[Select[IntegerPartitions[n], Count[#, _?EvenQ]==Count[#, _?OddQ]==Count[conj[#], _?EvenQ]==Count[conj[#], _?OddQ]&]], {n, 0, 30}]
CROSSREFS
The strict case appears to be the indicator function for A014105. These partitions are ranked by A350947. There are four statistics:
There are six pairings of statistics:
- A045931: # of even parts = # of odd parts: There are three double-pairings of statistics:
A103919 and A116482 count partitions by sum and number of odd/even parts.
A195017 = # of even parts - # of odd parts.
Cf. A000070, A122111, A130780, A171966, A236559, A236914, A350849, A350941, A350942, A350950, A350951.
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