Displaying 1-10 of 17 results found.
Table read by rows: number of partitions of n with k as low median.
+10
26
1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1, 6, 3, 1, 0, 0, 1, 8, 4, 2, 0, 0, 0, 1, 11, 6, 3, 1, 0, 0, 0, 1, 15, 8, 4, 2, 0, 0, 0, 0, 1, 20, 12, 5, 3, 1, 0, 0, 0, 0, 1, 26, 16, 7, 4, 2, 0, 0, 0, 0, 0, 1, 35, 22, 10, 5, 3, 1, 0, 0, 0, 0, 0, 1, 45, 29, 14, 6, 4, 2, 0, 0, 0, 0, 0, 0, 1, 58, 40, 19, 8, 5, 3, 1
COMMENTS
For a multiset with an odd number of elements, the low median is the same as the median. For a multiset with an even number of elements, the low median is the smaller of the two central elements.
Arrange the parts of a partition nonincreasing order. Remove the first part, then the last, then the first remaining part, then the last remaining part, and continue until only a single number, the low median, remains. - Clark Kimberling, May 16 2019
EXAMPLE
For the partition [2,1^2], the sole middle element is 1, so that is the low median. For [3,2,1^2], the two middle elements are 1 and 2; the low median is the smaller, 1.
First 8 rows:
1
1 1
2 0 1
3 1 0 1
4 2 0 0 1
6 3 1 0 0 1
8 4 2 0 0 0 1
11 6 3 1 0 0 0 1
Row n = 8 counts the following partitions:
(71) (62) (53) (44) . . . (8)
(611) (521) (431)
(5111) (422) (332)
(4211) (3221)
(41111) (2222)
(3311) (22211)
(32111)
(311111)
(221111)
(2111111)
(11111111)
(End)
MATHEMATICA
Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 2)/2]]] &, IntegerPartitions[#]]] &, Range[13]] (* Peter J. C. Moses, May 14 2019 *)
CROSSREFS
The high version of this triangle is A124944.
The rank statistic for this triangle is A363941, high version A363942.
A version for mean instead of median is A363945, rank statistic A363943.
A high version for mean instead of median is A363946, rank stat A363944.
A008284 counts partitions by length (or decreasing mean), strict A008289.
A360005(n)/2 returns median of prime indices.
Table, number of partitions of n with k as high median.
+10
23
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 4, 3, 1, 1, 1, 1, 6, 4, 1, 1, 1, 1, 1, 8, 6, 3, 1, 1, 1, 1, 1, 11, 8, 5, 1, 1, 1, 1, 1, 1, 15, 11, 7, 3, 1, 1, 1, 1, 1, 1, 20, 15, 9, 5, 1, 1, 1, 1, 1, 1, 1, 26, 21, 12, 8, 3, 1, 1, 1, 1, 1, 1, 1, 35, 27, 16, 10, 5, 1, 1, 1, 1, 1, 1, 1, 1, 45, 37, 21, 13, 8, 3
COMMENTS
For a multiset with an odd number of elements, the high median is the same as the median. For a multiset with an even number of elements, the high median is the larger of the two central elements.
This table may be read as an upper right triangle with n >= 1 as column index and k >= 1 as row index. - Peter Munn, Jul 16 2017
Arrange the parts of a partition nonincreasing order. Remove the last part, then the first, then the last remaining part, then the first remaining part, and continue until only a single number, the high median, remains. - Clark Kimberling, May 14 2019
EXAMPLE
For the partition [2,1^2], the sole middle element is 1, so that is the high median. For [3,2,1^2], the two middle elements are 1 and 2; the high median is the larger, 2.
Triangle begins:
1
1 1
1 1 1
2 1 1 1
3 1 1 1 1
4 3 1 1 1 1
6 4 1 1 1 1 1
8 6 3 1 1 1 1 1
11 8 5 1 1 1 1 1 1
15 11 7 3 1 1 1 1 1 1
20 15 9 5 1 1 1 1 1 1 1
26 21 12 8 3 1 1 1 1 1 1 1
35 27 16 10 5 1 1 1 1 1 1 1 1
45 37 21 13 8 3 1 1 1 1 1 1 1 1
58 48 29 16 11 5 1 1 1 1 1 1 1 1 1
Row n = 8 counts the following partitions:
(611) (521) (431) (44) (53) (62) (71) (8)
(5111) (422) (332)
(41111) (4211) (3311)
(32111) (3221)
(311111) (2222)
(221111) (22211)
(2111111)
(11111111)
(End)
MATHEMATICA
Map[BinCounts[#, {1, #[[1]] + 1, 1}] &[Map[#[[Floor[(Length[#] + 1)/2]]] &, IntegerPartitions[#]]] &, Range[13]] (* Peter J. C. Moses, May 14 2019 *)
CROSSREFS
The low version of this triangle is A124943.
A008284 counts partitions by length, maximum, or decreasing mean.
A360005(n)/2 returns median of prime indices.
Mean of the multiset of prime indices of n, rounded down.
+10
23
0, 1, 2, 1, 3, 1, 4, 1, 2, 2, 5, 1, 6, 2, 2, 1, 7, 1, 8, 1, 3, 3, 9, 1, 3, 3, 2, 2, 10, 2, 11, 1, 3, 4, 3, 1, 12, 4, 4, 1, 13, 2, 14, 2, 2, 5, 15, 1, 4, 2, 4, 2, 16, 1, 4, 1, 5, 5, 17, 1, 18, 6, 2, 1, 4, 2, 19, 3, 5, 2, 20, 1, 21, 6, 2, 3, 4, 3, 22, 1, 2, 7
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.
Extending the terminology introduced at A124943, this is the "low mean" of prime indices.
EXAMPLE
The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 1.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meandown[y_]:=If[Length[y]==0, 0, Floor[Mean[y]]];
Table[meandown[prix[n]], {n, 100}]
CROSSREFS
Positions of first appearances are 1 and A000040.
The triangle for this statistic (low mean) is A363945.
A360005 gives twice the median of prime indices.
Mean of the multiset of prime indices of n, rounded up.
+10
21
0, 1, 2, 1, 3, 2, 4, 1, 2, 2, 5, 2, 6, 3, 3, 1, 7, 2, 8, 2, 3, 3, 9, 2, 3, 4, 2, 2, 10, 2, 11, 1, 4, 4, 4, 2, 12, 5, 4, 2, 13, 3, 14, 3, 3, 5, 15, 2, 4, 3, 5, 3, 16, 2, 4, 2, 5, 6, 17, 2, 18, 6, 3, 1, 5, 3, 19, 3, 6, 3, 20, 2, 21, 7, 3, 4, 5, 3, 22, 2, 2, 7
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.
Extending the terminology introduced at A124944, this is the "high mean" of prime indices.
EXAMPLE
The prime indices of 360 are {1,1,1,2,2,3}, with mean 3/2, so a(360) = 2.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
meanup[y_]:=If[Length[y]==0, 0, Ceiling[Mean[y]]];
Table[meanup[prix[n]], {n, 100}]
CROSSREFS
Positions of first appearances are 1 and A000040.
The triangle for this statistic (high mean) is A363946.
A360005 gives twice the median of prime indices.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124943, A215366, A327473, A327476, A327482, A359889, A362611, A363723, A363724, A363727, A363951.
High median in the multiset of prime indices of n.
+10
19
0, 1, 2, 1, 3, 2, 4, 1, 2, 3, 5, 1, 6, 4, 3, 1, 7, 2, 8, 1, 4, 5, 9, 1, 3, 6, 2, 1, 10, 2, 11, 1, 5, 7, 4, 2, 12, 8, 6, 1, 13, 2, 14, 1, 2, 9, 15, 1, 4, 3, 7, 1, 16, 2, 5, 1, 8, 10, 17, 2, 18, 11, 2, 1, 6, 2, 19, 1, 9, 3, 20, 1, 21, 12, 3, 1, 5, 2, 22, 1, 2
COMMENTS
The high median (see A124944) in a multiset is either the middle part (for odd length), or the greatest of the two middle parts (for even length).
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.
EXAMPLE
The prime indices of 90 are {1,2,2,3}, with high median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with high median 3, so a(150) = 3.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
merr[y_]:=If[Length[y]==0, 0, If[OddQ[Length[y]], y[[(Length[y]+1)/2]], y[[1+Length[y]/2]]]];
Table[merr[prix[n]], {n, 100}]
CROSSREFS
Positions of first appearances are 1 and A000040.
The triangle for this statistic (high median) is A124944, low A124943.
Regular median of prime indices is A360005(n)/2.
Low median in the multiset of prime indices of n.
+10
18
0, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 7, 2, 8, 1, 2, 1, 9, 1, 3, 1, 2, 1, 10, 2, 11, 1, 2, 1, 3, 1, 12, 1, 2, 1, 13, 2, 14, 1, 2, 1, 15, 1, 4, 3, 2, 1, 16, 2, 3, 1, 2, 1, 17, 1, 18, 1, 2, 1, 3, 2, 19, 1, 2, 3, 20, 1, 21, 1, 3, 1, 4, 2, 22, 1, 2, 1
COMMENTS
The low median (see A124943) in a multiset is either the middle part (for odd length), or the least of the two middle parts (for even length).
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.
EXAMPLE
The prime indices of 90 are {1,2,2,3}, with low median 2, so a(90) = 2.
The prime indices of 150 are {1,2,3,3}, with low median 2, so a(150) = 2.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mell[y_]:=If[Length[y]==0, 0, If[OddQ[Length[y]], y[[(Length[y]+1)/2]], y[[Length[y]/2]]]];
Table[mell[prix[n]], {n, 30}]
CROSSREFS
Positions of first appearances are 1 and A000040.
The triangle for this statistic (low median) is A124943, high A124944.
Median of prime indices is A360005(n)/2.
Numbers whose prime indices have mean 1 when rounded down.
+10
18
2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 560, 576, 600, 640
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.
EXAMPLE
The terms together with their prime indices begin:
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
32: {1,1,1,1,1}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
64: {1,1,1,1,1,1}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Floor[Mean[prix[#]]]==1&]
CROSSREFS
These partitions are counted by A025065.
For the usual rounding (not low or high) we have A363948, counted by A363947.
A360005 gives twice the median of prime indices.
For mean 2 instead of 1 we have A363950, counted by A026905 redoubled.
Triangle read by rows where T(n,k) is the number of integer partitions of n with low mean k.
+10
16
1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1, 0, 4, 2, 0, 0, 1, 0, 4, 3, 3, 0, 0, 1, 0, 7, 4, 3, 0, 0, 0, 1, 0, 7, 10, 0, 4, 0, 0, 0, 1, 0, 12, 6, 7, 4, 0, 0, 0, 0, 1, 0, 12, 16, 8, 0, 5, 0, 0, 0, 0, 1, 0, 19, 21, 10, 0, 5, 0, 0, 0, 0
COMMENTS
Extending the terminology of A124943, the "low mean" of a multiset is its mean rounded down.
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 2 0 1
0 2 2 0 1
0 4 2 0 0 1
0 4 3 3 0 0 1
0 7 4 3 0 0 0 1
0 7 10 0 4 0 0 0 1
0 12 6 7 4 0 0 0 0 1
0 12 16 8 0 5 0 0 0 0 1
0 19 21 10 0 5 0 0 0 0 0 1
0 19 24 15 12 0 6 0 0 0 0 0 1
0 30 32 18 14 0 6 0 0 0 0 0 0 1
0 30 58 23 16 0 0 7 0 0 0 0 0 0 1
0 45 47 57 0 19 0 7 0 0 0 0 0 0 0 1
Row k = 8 counts the following partitions:
. (41111) (611) . (71) . . . (8)
(32111) (521) (62)
(311111) (5111) (53)
(22211) (431) (44)
(221111) (422)
(2111111) (4211)
(11111111) (332)
(3311)
(3221)
(2222)
MATHEMATICA
meandown[y_]:=If[Length[y]==0, 0, Floor[Mean[y]]];
Table[Length[Select[IntegerPartitions[n], meandown[#]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
For median instead of mean we have rank statistic A363941, high A363942.
The rank statistic for this triangle is A363943.
For high mode instead of mean we have A363953, rank statistic A363487.
Cf. A002865, A026905, A237984, A327472, A327482, A344296, A362612, A363723, A363724, A363731, A363951.
Numbers whose prime indices have rounded-up mean 2.
+10
12
3, 6, 9, 10, 12, 18, 20, 24, 27, 28, 30, 36, 40, 48, 54, 56, 60, 72, 80, 81, 84, 88, 90, 96, 100, 108, 112, 120, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 264, 270, 280, 288, 300, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448
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.
EXAMPLE
The terms together with their prime indices begin:
3: {2}
6: {1,2}
9: {2,2}
10: {1,3}
12: {1,1,2}
18: {1,2,2}
20: {1,1,3}
24: {1,1,1,2}
27: {2,2,2}
28: {1,1,4}
30: {1,2,3}
36: {1,1,2,2}
40: {1,1,1,3}
48: {1,1,1,1,2}
54: {1,2,2,2}
56: {1,1,1,4}
60: {1,1,2,3}
72: {1,1,1,2,2}
80: {1,1,1,1,3}
81: {2,2,2,2}
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[1000], Ceiling[Mean[prix[#]]]==2&]
CROSSREFS
For mean 1 we have A000079 except 1.
Partitions of this type are counted by A026905 redoubled.
Numbers whose prime indices have mean < 3/2.
+10
11
2, 4, 8, 12, 16, 24, 32, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 432, 448, 480, 512, 576, 640, 768, 864, 896, 960, 1024, 1152, 1280, 1536, 1728, 1792, 1920, 2048, 2304, 2560, 2592, 2688, 2816, 2880, 3072, 3200, 3456, 3584, 3840, 4096, 4608
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.
EXAMPLE
The initial terms, prime indices, and means:
2: {1} -> 1
4: {1,1} -> 1
8: {1,1,1} -> 1
12: {1,1,2} -> 4/3
16: {1,1,1,1} -> 1
24: {1,1,1,2} -> 5/4
32: {1,1,1,1,1} -> 1
48: {1,1,1,1,2} -> 6/5
64: {1,1,1,1,1,1} -> 1
72: {1,1,1,2,2} -> 7/5
80: {1,1,1,1,3} -> 7/5
96: {1,1,1,1,1,2} -> 7/6
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], Mean[prix[#]]<3/2&]
CROSSREFS
These partitions are counted by A363947.
A360005 gives twice the median of prime indices.
A363950 ranks partitions with low mean 2, counted by A026905 redoubled.
Cf. A051293, A124944, A327473, A327476, A327482, A359889, A363727, A363942, A363943, A363946, A363951.
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