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Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.
+10
27
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 0, 0, 3, 6, 4, 1, 0, 1, 0, 0, 1, 7, 7, 5, 1, 0, 1, 0, 0, 1, 4, 8, 10, 5, 1, 0, 1, 0, 0, 0, 3, 6, 14, 11, 6, 1, 0, 1, 0, 0, 0, 1, 5, 12, 16, 14, 6, 1, 0
COMMENTS
Also the number of integer partitions of n whose right half (inclusive) sums to n-k.
EXAMPLE
Triangle begins:
1
1 0
1 1 0
1 1 1 0
1 0 3 1 0
1 0 2 3 1 0
1 0 1 4 4 1 0
1 0 0 3 6 4 1 0
1 0 0 1 7 7 5 1 0
1 0 0 1 4 8 10 5 1 0
1 0 0 0 3 6 14 11 6 1 0
1 0 0 0 1 5 12 16 14 6 1 0
1 0 0 0 1 2 12 14 23 16 7 1 0
1 0 0 0 0 2 7 13 24 27 19 7 1 0
1 0 0 0 0 1 5 9 24 30 35 21 8 1 0
1 0 0 0 0 1 3 7 17 31 42 40 25 8 1 0
1 0 0 0 0 0 2 4 16 23 46 51 51 27 9 1 0
1 0 0 0 0 0 1 3 10 21 37 57 69 57 31 9 1 0
1 0 0 0 0 0 1 2 7 15 34 47 83 81 69 34 10 1 0
For example, row n = 9 counts the following partitions:
(9) . . (333) (432) (54) (63) (72) (81)
(441) (522) (621) (711)
(22221) (531) (3321) (4311)
(111111111) (3222) (4221) (5211)
(32211) (33111) (6111)
(2211111) (42111)
(3111111) (51111)
(21111111) (222111)
(321111)
(411111)
For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Total[Take[#, Floor[Length[#]/2]]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
The left inclusive version is A360675 with rows reversed. A008284 counts partitions by length.
First for prime indices, second for partitions, third for prime factors:
Number of partitions of n whose median is not a part.
+10
24
0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 13, 18, 23, 30, 40, 50, 64, 83, 104, 131, 166, 206, 256, 320, 394, 485, 598, 730, 891, 1088, 1318, 1596, 1932, 2326, 2797, 3360, 4020, 4804, 5735, 6824, 8108, 9624, 11392, 13468, 15904, 18737, 22048, 25914, 30400, 35619, 41686
COMMENTS
Also, the number of partitions p of n such that (1/2)*max(p) is a part of p.
Also the number of even-length integer partitions of n with distinct middle parts. For example, the partition (4,3,2,1) has middle parts {2,3} so is counted under a(10), but (3,2,2,1) has middle parts {2,2} so is not counted under a(8). - Gus Wiseman, May 13 2023
FORMULA
For all n, A027187(n) >= a(n). [Because when a partition of n has an odd number of parts, then it is not counted by this sequence (cf. A238478) and also some of the partitions with an even number of parts might be excluded here. Cf. Examples.] - Antti Karttunen, Feb 27 2014 G.f.: Sum_{n>=1} q^(3*n)/Product_{k=1..2*n} (1-q^k).
a(n) ~ Pi/(2^(17/4)*3^(3/4)*n^(5/4))*exp(Pi*sqrt(2*n/3)). Proved by Blecher and Knopfmacher. (End)
EXAMPLE
a(6) counts these partitions: 51, 42, 2211 which all have an even number of parts, and their medians 3, 3 and 1.5 are not present. Note that the partitions 33 and 3111, although having an even number of parts, are not included in the count of a(6), but instead in that of A238478(6), as their medians, 3 for the former and 1 for the latter, are present in those partitions.
MATHEMATICA
Table[Count[IntegerPartitions[n], p_ /; !MemberQ[p, Median[p]]], {n, 40}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[p]/2]], {n, 50}]
PROG
(Python)
from sympy.utilities.iterable import partitions
def A238479(n): return sum(1 for p in partitions(n) if (m:=max(p, default=0))&1^1 and m>>1 in p) # Chai Wah Wu, Sep 21 2023
CROSSREFS
These partitions have ranks A362617.
Number of partitions of n whose median is a part.
+10
18
1, 2, 2, 4, 5, 8, 11, 17, 22, 32, 43, 59, 78, 105, 136, 181, 233, 302, 386, 496, 626, 796, 999, 1255, 1564, 1951, 2412, 2988, 3674, 4516, 5524, 6753, 8211, 9984, 12086, 14617, 17617, 21211, 25450, 30514, 36475, 43550, 51869, 61707, 73230, 86821, 102706
COMMENTS
Also the number of integer partitions of n with a unique middle part. This means that either the length is odd or the two middle parts are equal. For example, the partition (4,3,2,1) has middle parts {2,3} so is not counted under a(10), but (3,2,2,1) has middle parts {2,2} so is counted under a(8). - Gus Wiseman, May 13 2023
FORMULA
For all n, a(n) >= A027193(n) (because when a partition of n has an odd number of parts, its median is simply the part at the middle). - Antti Karttunen, Feb 27 2014
EXAMPLE
a(6) counts these partitions: 6, 411, 33, 321, 3111, 222, 21111, 111111.
MATHEMATICA
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Median[p]]], {n, 40}]
CROSSREFS
These partitions have ranks A362618.
Numbers > 1 whose distinct prime indices have integer median.
+10
18
2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 30, 31, 32, 34, 37, 39, 40, 41, 42, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 60, 61, 62, 63, 64, 66, 67, 68, 70, 71, 73, 78, 79, 80, 81, 82, 83, 84, 85, 87, 88, 89, 90, 91, 92, 94, 97, 100
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Distinct prime indices are listed by A304038. The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is in the sequence.
The prime indices of 330 are {1,2,3,5}, with distinct parts {1,2,3,5}, with median 5/2, so 330 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]
CROSSREFS
For mean instead of median we have A326621. Positions of even terms in A360457. The complement (without 1) is A360551. Partitions with these Heinz numbers are counted by A360686.
Numbers > 1 whose distinct prime indices have non-integer median.
+10
13
6, 12, 14, 15, 18, 24, 26, 28, 33, 35, 36, 38, 45, 48, 51, 52, 54, 56, 58, 65, 69, 72, 74, 75, 76, 77, 86, 93, 95, 96, 98, 99, 104, 106, 108, 112, 116, 119, 122, 123, 135, 141, 142, 143, 144, 145, 148, 152, 153, 158, 161, 162, 172, 175, 177, 178, 185, 192, 196
COMMENTS
First differs from A325700 in having 330 and lacking 462. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Distinct prime indices are listed by A304038. The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The prime indices of 900 are {1,1,2,2,3,3}, with distinct parts {1,2,3}, with median 2, so 900 is not in the sequence.
The prime indices of 462 are {1,2,4,5}, with distinct parts {1,2,4,5}, with median 3, so 462 is not in the sequence.
MATHEMATICA
Select[Range[2, 100], !IntegerQ[Median[PrimePi/@First/@FactorInteger[#]]]&]
CROSSREFS
For mean instead of median we have the complement of A326621.
Number of integer partitions of n without a nonempty initial consecutive subsequence summing to n/2.
+10
5
1, 1, 1, 3, 2, 7, 6, 15, 11, 30, 27, 56, 44, 101, 93, 176, 149, 297, 271, 490, 432, 792, 744, 1255, 1109, 1958, 1849, 3010, 2764, 4565, 4287, 6842, 6328, 10143, 9673, 14883, 13853, 21637, 20717, 31185, 29343, 44583, 42609, 63261, 60100, 89134, 85893, 124754
COMMENTS
Also the number of n-multisets of positive integers that (1) have integer median, (2) cover an initial interval, and (3) have weakly decreasing multiplicities.
EXAMPLE
The a(1) = 1 through a(7) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7)
(21) (31) (32) (42) (43)
(111) (41) (51) (52)
(221) (222) (61)
(311) (411) (322)
(2111) (2211) (331)
(11111) (421)
(511)
(2221)
(3211)
(4111)
(22111)
(31111)
(211111)
(1111111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(8).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !MemberQ[Accumulate[#], n/2]&]], {n, 0, 15}]
CROSSREFS
The version for compositions is A213173. The complement is counted by A322439 aerated. For mean instead of median we have A362559. Cf. A058398, A108917, A169942, A325676, A353864, A360254, A360672, A360675, A360686, A360687, A362560.
Numbers whose prime factorization has either (1) odd length, or (2) equal middle parts.
+10
5
2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 88, 89, 90, 92, 96, 97, 98, 99, 101
COMMENTS
Also numbers n whose median prime factor is a prime factor of n.
EXAMPLE
The prime factorization of 90 is 2*3*3*5, with middle parts (3,3), so 90 is in the sequence.
MATHEMATICA
prifacs[n_]:=If[n==1, {}, Flatten[ConstantArray@@@FactorInteger[n]]];
Select[Range[2, 100], MemberQ[prifacs[#], Median[prifacs[#]]]&]
CROSSREFS
Partitions of this type are counted by A238478. A359893 counts partitions by median.
A362621 ranks partitions with median equal to maximum, counted by A053263.
A362622 ranks partitions whose maximum is a middle part, counted by A237824.
Number of integer partitions of 2n without a nonempty initial consecutive subsequence summing to n.
+10
4
1, 1, 2, 6, 11, 27, 44, 93, 149, 271, 432, 744, 1109, 1849, 2764, 4287, 6328, 9673, 13853, 20717, 29343, 42609, 60100, 85893, 118475, 167453, 230080, 318654, 433763, 595921, 800878, 1090189, 1456095, 1957032, 2600199, 3465459, 4558785, 6041381, 7908681
EXAMPLE
The a(1) = 1 through a(4) = 11 partitions:
(2) (4) (6) (8)
(31) (42) (53)
(51) (62)
(222) (71)
(411) (332)
(2211) (521)
(611)
(3221)
(3311)
(5111)
(32111)
The partition y = (3,2,1,1,1) has nonempty initial consecutive subsequences (3,2,1,1,1), (3,2,1,1), (3,2,1), (3,2), (3), with sums 8, 7, 6, 5, 3. Since 4 is missing, y is counted under a(4).
MATHEMATICA
Table[Length[Select[IntegerPartitions[2n], !MemberQ[Accumulate[#], n]&]], {n, 0, 15}]
CROSSREFS
The complement is counted by A322439. A304442 counts partitions with all equal run-sums.
A353836 counts partitions by number of distinct run-sums.
Cf. A108917, A169942, A237363, A325676, A353864, A360254, A360672, A360675, A360686, A360952, A362560.
Number of integer partitions of n whose distinct parts have non-integer median.
+10
1
0, 0, 1, 1, 4, 3, 8, 6, 13, 11, 21, 17, 34, 36, 55, 61, 97, 115, 162, 191, 270, 328, 427, 514, 666, 810, 1027, 1211, 1530, 1832, 2260, 2688, 3342, 3952, 4824, 5746, 7010, 8313, 10116, 11915, 14436, 17074, 20536, 24239, 29053, 34170, 40747, 47865, 56830, 66621
COMMENTS
The median of a multiset is either the middle part (for odd length), or the average of the two middle parts (for even length).
EXAMPLE
The a(1) = 0 through a(9) = 13 partitions:
. . (21) (211) (32) (411) (43) (332) (54)
(41) (2211) (52) (611) (63)
(221) (21111) (61) (22211) (72)
(2111) (322) (41111) (81)
(2221) (221111) (441)
(4111) (2111111) (522)
(22111) (3222)
(211111) (6111)
(22221)
(222111)
(411111)
(2211111)
(21111111)
For example, the partition y = (5,3,3,2,1,1) has distinct parts {1,2,3,5}, with median 5/2, so y is counted under a(15).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], !IntegerQ[Median[Union[#]]]&]], {n, 30}]
CROSSREFS
These partitions have ranks A360551. For multiplicities instead of distinct parts we have A360690, ranks A360554. A116608 counts partitions by number of distinct parts.
A360457 gives median of distinct prime indices (times 2).
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