Displaying 1-10 of 13 results found.
Number of partitions of n whose mean is a part.
+10
90
1, 2, 2, 3, 2, 5, 2, 6, 5, 8, 2, 21, 2, 14, 22, 30, 2, 61, 2, 86, 67, 45, 2, 283, 66, 80, 197, 340, 2, 766, 2, 663, 543, 234, 703, 2532, 2, 388, 1395, 4029, 2, 4688, 2, 4476, 7032, 1005, 2, 17883, 2434, 9713, 7684, 14472, 2, 25348, 17562, 37829, 16786, 3721
COMMENTS
a(n) = 2 if and only if n is a prime.
EXAMPLE
a(6) counts these partitions: 6, 33, 321, 222, 111111.
The a(1) = 1 through a(10) = 8 partitions (A = 10):
1 2 3 4 5 6 7 8 9 A
11 111 22 11111 33 1111111 44 333 55
1111 222 2222 432 22222
321 3221 531 32221
111111 4211 111111111 33211
11111111 42211
52111
1111111111
(End)
MATHEMATICA
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Mean[p]]], {n, 40}]
PROG
(Python)
from sympy.utilities.iterables import partitions
def A237984(n): return sum(1 for s, p in partitions(n, size=True) if not n%s and n//s in p) # Chai Wah Wu, Sep 21 2023
CROSSREFS
The Heinz numbers of these partitions are A327473. A similar sequence for subsets is A065795. Partitions without their mean are A327472.
Number of subsets of {1,2,...,n} that contain the average of their elements.
+10
15
1, 2, 4, 6, 10, 16, 26, 42, 72, 124, 218, 390, 706, 1292, 2388, 4436, 8292, 15578, 29376, 55592, 105532, 200858, 383220, 732756, 1403848, 2694404, 5179938, 9973430, 19229826, 37125562, 71762396, 138871260, 269021848, 521666984, 1012520400, 1966957692, 3824240848
COMMENTS
Also the number of subsets of {1,2,...,n} with sum of entries divisible by the largest element (compare A000016). See the Palmer Melbane link for a bijection. - Joel B. Lewis, Nov 13 2014
FORMULA
a(n) = (1/2)*Sum_{i=1..n} (f(i) - 1) where f(i) = (1/i) * Sum_{d | i and d is odd} 2^(i/d) * phi(d).
EXAMPLE
a(4)=6, since {1}, {2}, {3}, {4}, {1,2,3} and {2,3,4} contain their averages.
The a(1) = 1 through a(6) = 16 subsets:
{1} {1} {1} {1} {1} {1}
{2} {2} {2} {2} {2}
{3} {3} {3} {3}
{1,2,3} {4} {4} {4}
{1,2,3} {5} {5}
{2,3,4} {1,2,3} {6}
{1,3,5} {1,2,3}
{2,3,4} {1,3,5}
{3,4,5} {2,3,4}
{1,2,3,4,5} {2,4,6}
{3,4,5}
{4,5,6}
{1,2,3,6}
{1,4,5,6}
{1,2,3,4,5}
{2,3,4,5,6}
(End)
MATHEMATICA
Table[ Sum[a = Select[Divisors[i], OddQ[ # ] &]; Apply[ Plus, 2^(i/a) * EulerPhi[a]]/i, {i, n}]/2, {n, 34}]
(* second program *)
Table[Length[Select[Subsets[Range[n]], MemberQ[#, Mean[#]]&]], {n, 0, 10}] (* Gus Wiseman, Sep 14 2019 *)
PROG
(PARI) a(n) = (1/2)*sum(i=1, n, (1/i)*sumdiv(i, d, if (d%2, 2^(i/d)*eulerphi(d)))); \\ Michel Marcus, Dec 20 2020 (Python)
from sympy import totient, divisors
def A065795(n): return sum((sum(totient(d)<<k//d-1 for d in divisors(k>>(~k&k-1).bit_length(), generator=True))<<1)//k for k in range(1, n+1))>>1 # Chai Wah Wu, Feb 22 2023
CROSSREFS
Subsets containing n whose mean is an element are A000016. The version for integer partitions is A237984. Subsets not containing their mean are A327471.
Number of compositions of n whose own run-lengths are a subsequence (not necessarily consecutive).
+10
13
1, 1, 0, 0, 1, 2, 3, 2, 2, 8, 17, 26, 43, 77, 129, 210, 351, 569
EXAMPLE
The a(0) = 1 through a(9) = 8 compositions (empty columns indicated by dots):
() (1) . . (22) (122) (1122) (11221) (21122) (333)
(221) (1221) (12211) (22112) (22113)
(2211) (22122)
(31122)
(121122)
(122112)
(211221)
(221121)
For example, the composition y = (2,2,3,3,1) has run-lengths (2,2,1), which form a (non-consecutive) subsequence, so y is counted under a(11).
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], MemberQ[Subsets[#], Length/@Split[#]]&]], {n, 0, 15}]
CROSSREFS
The version for partitions is A325702. These compositions are ranked by A353402. The recursive consecutive version is A353430. A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
A353400 counts compositions with all run-lengths > 2.
Cf. A005811, A103295, A114901, A181591, A238279, A242882, A324572, A333755, A351017, A353401, A353426.
Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.
+10
11
1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93
EXAMPLE
The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
(9) (A) (B) (C) (D) (E)
(333) (2233) (141122) (2244) (161122) (2255)
(121122) (3322) (221123) (4422) (221125) (5522)
(221121) (131122) (221132) (151122) (221134) (171122)
(221131) (221141) (221124) (221143) (221126)
(231122) (221142) (221152) (221135)
(321122) (221151) (221161) (221153)
(241122) (251122) (221162)
(421122) (341122) (221171)
(431122) (261122)
(521122) (351122)
(531122)
(621122)
(122121122)
(221121221)
MATHEMATICA
yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y], Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], yosQ]], {n, 0, 15}]
CROSSREFS
The non-recursive reverse version is A353403. The consecutive version is A353430. These compositions are ranked by A353431. A114901 counts compositions with no runs of length 1.
A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-length.
Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.
Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).
+10
10
0, 1, 10, 21, 26, 43, 53, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 693, 696, 697, 698, 699, 804, 826, 858, 860, 861, 885, 954, 1082, 1141, 1173, 1210, 1338, 1353, 1387, 1392, 1393, 1394, 1396, 1397, 1398, 1466
COMMENTS
First differs from A353432 (the consecutive case) in having 0 and 53. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The initial terms, their binary expansions, and the corresponding standard compositions:
0: 0 ()
1: 1 (1)
10: 1010 (2,2)
21: 10101 (2,2,1)
26: 11010 (1,2,2)
43: 101011 (2,2,1,1)
53: 110101 (1,2,2,1)
58: 111010 (1,1,2,2)
107: 1101011 (1,2,2,1,1)
117: 1110101 (1,1,2,2,1)
174: 10101110 (2,2,1,1,2)
186: 10111010 (2,1,1,2,2)
292: 100100100 (3,3,3)
314: 100111010 (3,1,1,2,2)
346: 101011010 (2,2,1,2,2)
348: 101011100 (2,2,1,1,3)
349: 101011101 (2,2,1,1,2,1)
373: 101110101 (2,1,1,2,2,1)
430: 110101110 (1,2,2,1,1,2)
442: 110111010 (1,2,1,1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[ Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y], Length/@Split[y]];
Select[Range[0, 100], rosQ[stc[#]]&]
CROSSREFS
These compositions are counted by A353390. A005811 counts runs in binary expansion.
A333769 lists run-lengths of compositions in standard order.
Statistics of standard compositions:
Classes of standard compositions:
Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/ A333257, A333755, A353393, A353403, A353430.
Number of compositions of n whose own run-lengths are a consecutive subsequence.
+10
9
1, 1, 0, 0, 1, 2, 2, 2, 2, 8, 12, 16, 20, 35, 46, 59, 81, 109, 144, 202, 282
EXAMPLE
The a(0) = 0 through a(10) = 12 compositions (empty columns indicated by dots, 0 is the empty composition):
0 1 . . 22 122 1122 11221 21122 333 1333
221 2211 12211 22112 22113 2233
22122 3322
31122 3331
121122 22114
122112 41122
211221 122113
221121 131122
221131
311221
1211221
1221121
MATHEMATICA
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], #=={}||MemberQ[Join@@Table[Take[#, {i, j}], {i, Length[#]}, {j, i, Length[#]}], Length/@Split[#]]&]], {n, 0, 15}]
CROSSREFS
The non-consecutive version for partitions is A325702. The non-consecutive reverse version is A353403. These compositions are ranked by A353432. A329739 counts compositions with all distinct run-lengths.
Cf. A008965, A032020, A103295, A103300, A114901, A238279, A324572, A325705, A333224, A333755, A351013, A353401.
Number of compositions of n whose own reversed run-lengths are a subsequence (not necessarily consecutive).
+10
9
1, 1, 0, 0, 3, 2, 5, 12, 16, 30, 45, 94, 159, 285, 477, 864, 1487, 2643
EXAMPLE
The a(0) = 1 through a(7) = 12 compositions:
() (1) . . (22) (1121) (1113) (1123)
(112) (1211) (1122) (1132)
(211) (1221) (2311)
(2211) (3211)
(3111) (11131)
(11212)
(11221)
(12112)
(12211)
(13111)
(21121)
(21211)
MATHEMATICA
Table[Length[Select[Join@@Permutations/@ IntegerPartitions[n], MemberQ[Subsets[#], Reverse[Length/@Split[#]]]&]], {n, 0, 15}]
CROSSREFS
The non-reversed recursive consecutive version is A353430. A325705 counts partitions containing all of their distinct multiplicities.
A329739 counts compositions with all distinct run-lengths, for runs A351013.
Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.
+10
9
0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192
COMMENTS
First differs from A353696 (the consecutive version) in having 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are subsequence but not a consecutive subsequence. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The initial terms, their binary expansions, and the corresponding standard compositions:
0: 0 ()
1: 1 (1)
2: 10 (2)
4: 100 (3)
8: 1000 (4)
10: 1010 (2,2)
16: 10000 (5)
32: 100000 (6)
43: 101011 (2,2,1,1)
58: 111010 (1,1,2,2)
64: 1000000 (7)
128: 10000000 (8)
256: 100000000 (9)
292: 100100100 (3,3,3)
349: 101011101 (2,2,1,1,2,1)
442: 110111010 (1,2,1,1,2,2)
512: 1000000000 (10)
586: 1001001010 (3,3,2,2)
676: 1010100100 (2,2,3,3)
697: 1010111001 (2,2,1,1,3,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y], Length/@Split[y]]&& rorQ[Length/@Split[y]];
Select[Range[0, 100], rorQ[stc[#]]&]
CROSSREFS
The non-recursive version for partitions is A325755, counted by A325702. These compositions are counted by A353391. A005811 counts runs in binary expansion.
Statistics of standard compositions:
Classes of standard compositions:
Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.
Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.
+10
9
0, 1, 10, 21, 26, 43, 58, 107, 117, 174, 186, 292, 314, 346, 348, 349, 373, 430, 442, 570, 585, 586, 629, 676, 696, 697, 804, 826, 860, 861, 885, 1082, 1141, 1173, 1210, 1338, 1387, 1392, 1393, 1394, 1396, 1594, 1653, 1700, 1720, 1721, 1882, 2106, 2165, 2186
COMMENTS
First differs from A353402 (the non-consecutive version) in lacking 53. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The initial terms, their binary expansions, and the corresponding standard compositions:
0: 0 ()
1: 1 (1)
10: 1010 (2,2)
21: 10101 (2,2,1)
26: 11010 (1,2,2)
43: 101011 (2,2,1,1)
58: 111010 (1,1,2,2)
107: 1101011 (1,2,2,1,1)
117: 1110101 (1,1,2,2,1)
174: 10101110 (2,2,1,1,2)
186: 10111010 (2,1,1,2,2)
292: 100100100 (3,3,3)
314: 100111010 (3,1,1,2,2)
346: 101011010 (2,2,1,2,2)
348: 101011100 (2,2,1,1,3)
349: 101011101 (2,2,1,1,2,1)
373: 101110101 (2,1,1,2,2,1)
430: 110101110 (1,2,2,1,1,2)
442: 110111010 (1,2,1,1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y, {i, j}], {i, Length[y]}, {j, i, Length[y]}], Length/@Split[y]];
Select[Range[0, 10000], rorQ[stc[#]]&]
CROSSREFS
These compositions are counted by A353392. A005811 counts runs in binary expansion.
Statistics of standard compositions:
Classes of standard compositions:
Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.
Number of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.
+10
8
1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45
EXAMPLE
The a(n) compositions for selected n (A..E = 10..14):
n=4: n=6: n=9: n=10: n=12: n=14:
-----------------------------------------------------------
(4) (6) (9) (A) (C) (E)
(22) (1122) (333) (2233) (2244) (2255)
(2211) (121122) (3322) (4422) (5522)
(221121) (131122) (151122) (171122)
(221131) (221124) (221126)
(221142) (221135)
(221151) (221153)
(241122) (221162)
(421122) (221171)
(261122)
(351122)
(531122)
(621122)
(122121122)
(221121221)
MATHEMATICA
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y, {i, j}], {i, Length[y]}, {j, i, Length[y]}], Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], yoyQ]], {n, 0, 15}]
CROSSREFS
A114901 counts compositions with no runs of length 1.
A329739 counts compositions with all distinct run-lengths.
Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.
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