A359897
Number of strict integer partitions of n whose parts have the same mean as median.
Original entry on oeis.org
0, 1, 1, 2, 2, 3, 4, 4, 4, 7, 6, 6, 10, 7, 10, 13, 11, 9, 20, 10, 20, 18, 21, 12, 30, 24, 28, 27, 30, 15, 73, 16, 37, 43, 45, 67, 74, 19, 55, 71, 126, 21, 150, 22, 75, 225, 78, 24, 183, 126, 245, 192, 132, 27, 284, 244, 403, 303, 120, 30, 828
Offset: 0
The a(1) = 1 through a(9) = 7 partitions:
(1) (2) (3) (4) (5) (6) (7) (8) (9)
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3) (5,4)
(4,1) (5,1) (5,2) (6,2) (6,3)
(3,2,1) (6,1) (7,1) (7,2)
(8,1)
(4,3,2)
(5,3,1)
The complement is counted by
A359898.
A008289 counts strict partitions by mean.
A240850 counts strict partitions containing their mean, complement
A240851.
Cf.
A065795,
A066571,
A067659,
A082550,
A102627,
A135342,
A316313,
A327473,
A327475,
A328966,
A359909.
-
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Mean[#]==Median[#]&]],{n,0,30}]
A364915
Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.
Original entry on oeis.org
1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 12, 10, 16, 16, 19, 21, 29, 25, 37, 35, 44, 46, 60, 55, 75, 71, 90, 90, 114, 110, 140, 138, 167, 163, 217, 201, 248, 241, 298, 303, 359, 355, 425, 422, 520, 496, 594, 603, 715, 706, 834, 826, 968, 972, 1153, 1147, 1334, 1315, 1530
Offset: 0
The a(1) = 1 through a(10) = 8 partitions (A=10):
1 2 3 4 5 6 7 8 9 A
11 111 22 32 33 43 44 54 55
1111 11111 222 52 53 72 64
111111 322 332 333 73
1111111 2222 522 433
11111111 3222 3322
111111111 22222
1111111111
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is counted under a(12).
The partition (6,4,3,2) has 6=4+2, or 6=3+3, or 6=2+2+2, or 4=2+2, so is not counted under a(15).
For subsets instead of partitions we have
A326083, complement
A364914.
A007865 counts binary sum-free sets w/ re-usable parts, complement
A093971.
A364912 counts linear combinations of partitions of k.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], Function[ptn,!Or@@Table[combs[ptn[[k]],Delete[ptn,k]]!={}, {k,Length[ptn]}]]@*Union]], {n,0,15}]
-
from sympy.utilities.iterables import partitions
def A364915(n):
if n <= 1: return 1
alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
for p in partitions(n,k=n-1):
s = set(p)
if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
c += 1
return c # Chai Wah Wu, Sep 23 2023
A336127
Number of ways to split a composition of n into contiguous subsequences with different sums.
Original entry on oeis.org
1, 1, 2, 8, 16, 48, 144, 352, 896, 2432, 7168, 16896, 46080, 114688, 303104, 843776, 2080768, 5308416, 13762560, 34865152, 87818240, 241172480, 583008256, 1503657984, 3762290688, 9604956160, 23689428992, 60532195328, 156397207552, 385137770496, 967978254336
Offset: 0
The a(0) = 1 through a(4) = 16 splits:
() (1) (2) (3) (4)
(1,1) (1,2) (1,3)
(2,1) (2,2)
(1,1,1) (3,1)
(1),(2) (1,1,2)
(2),(1) (1,2,1)
(1),(1,1) (1),(3)
(1,1),(1) (2,1,1)
(3),(1)
(1,1,1,1)
(1),(1,2)
(1),(2,1)
(1,2),(1)
(2,1),(1)
(1),(1,1,1)
(1,1,1),(1)
The version with equal instead of different sums is
A074854.
Starting with a strict composition gives
A336128.
Starting with a partition gives
A336131.
Starting with a strict partition gives
A336132Partitions of partitions are
A001970.
Partitions of compositions are
A075900.
Compositions of compositions are
A133494.
Compositions of partitions are
A323583.
-
splits[dom_]:=Append[Join@@Table[Prepend[#,Take[dom,i]]&/@splits[Drop[dom,i]],{i,Length[dom]-1}],{dom}];
Table[Sum[Length[Select[splits[ctn],UnsameQ@@Total/@#&]],{ctn,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}]
A363260
Number of integer partitions of n with parts disjoint from first differences of parts, meaning no part is the difference of two consecutive parts.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 17, 21, 28, 35, 46, 57, 70, 87, 110, 130, 165, 198, 238, 285, 349, 410, 498, 583, 702, 819, 983, 1136, 1353, 1570, 1852, 2137, 2520, 2898, 3390, 3891, 4540, 5191, 6028, 6889, 7951, 9082, 10450, 11884, 13650, 15508, 17728, 20113
Offset: 0
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (32) (33) (43) (44)
(31) (41) (51) (52) (53)
(1111) (311) (222) (61) (62)
(11111) (411) (322) (71)
(3111) (331) (332)
(111111) (511) (611)
(4111) (2222)
(31111) (3311)
(1111111) (5111)
(41111)
(311111)
(11111111)
For all differences of pairs parts we have
A364345.
For subsets of {1..n} instead of partitions we have
A364463.
A325325 counts partitions with distinct first-differences.
-
Table[Length[Select[IntegerPartitions[n],Intersection[#,-Differences[#]]=={}&]],{n,0,30}]
-
from collections import Counter
from sympy.utilities.iterables import partitions
def A363260(n): return sum(1 for s,p in map(lambda x: (x[0],tuple(sorted(Counter(x[1]).elements()))), partitions(n,size=True)) if set(p).isdisjoint({p[i+1]-p[i] for i in range(s-1)})) # Chai Wah Wu, Sep 26 2023
A363740
Number of integer partitions of n whose median appears more times than any other part, i.e., partitions containing a unique mode equal to the median.
Original entry on oeis.org
1, 2, 2, 4, 5, 7, 10, 15, 18, 26, 35, 46, 61, 82, 102, 136, 174, 224, 283, 360, 449, 569, 708, 883, 1089, 1352, 1659, 2042, 2492, 3039, 3695, 4492, 5426, 6555, 7889, 9482, 11360, 13602, 16231, 19348, 23005, 27313, 32364, 38303, 45227, 53341, 62800, 73829
Offset: 1
The a(1) = 1 through a(8) = 15 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (221) (33) (322) (44)
(211) (311) (222) (331) (332)
(1111) (2111) (411) (511) (422)
(11111) (3111) (2221) (611)
(21111) (4111) (2222)
(111111) (22111) (3221)
(31111) (5111)
(211111) (22211)
(1111111) (32111)
(41111)
(221111)
(311111)
(2111111)
(11111111)
A008284 counts partitions by length (or decreasing mean), strict
A008289.
-
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n],{Median[#]}==modes[#]&]],{n,30}]
A364911
Triangle read by rows where T(n,k) is the number of integer partitions with sum <= n and with distinct parts summing to k.
Original entry on oeis.org
1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 4, 2, 3, 2, 1, 5, 2, 5, 3, 3, 1, 6, 3, 8, 4, 4, 4, 1, 7, 3, 11, 6, 6, 6, 5, 1, 8, 4, 14, 9, 8, 10, 7, 6, 1, 9, 4, 19, 11, 11, 14, 11, 9, 8, 1, 10, 5, 23, 14, 15, 21, 15, 14, 11, 10, 1, 11, 5, 28, 17, 19, 28, 22, 20, 17, 15, 12
Offset: 0
Triangle begins:
1
1 1
1 2 1
1 3 1 2
1 4 2 3 2
1 5 2 5 3 3
1 6 3 8 4 4 4
1 7 3 11 6 6 6 5
1 8 4 14 9 8 10 7 6
1 9 4 19 11 11 14 11 9 8
1 10 5 23 14 15 21 15 14 11 10
1 11 5 28 17 19 28 22 20 17 15 12
1 12 6 34 21 22 40 28 28 24 24 17 15
1 13 6 40 25 27 50 38 37 34 35 27 22 18
1 14 7 46 29 32 65 49 50 43 51 38 35 26 22
1 15 7 54 33 38 79 62 63 59 68 55 50 41 32 27
Row n = 5 counts the following partitions:
. 1 2 3 4 5
1+1 2+2 1+2 1+3 1+4
1+1+1 1+1+2 1+1+3 2+3
1+1+1+1 1+1+1+2
1+1+1+1+1 1+2+2
Row n = 5 counts the following positive linear combinations:
. 1*1 1*2 1*3 1*4 1*5
2*1 2*2 1*2+1*1 1*3+1*1 1*3+1*2
3*1 1*2+2*1 1*3+2*1 1*4+1*1
4*1 1*2+3*1
5*1 2*2+1*1
Columns are partial sums of columns of
A116861.
Column k = 3 appears to be the partial sums of
A137719.
A114638 counts partitions where (length) = (sum of distinct parts).
A116608 counts partitions by number of distinct parts.
A364350 counts combination-free strict partitions, complement
A364839.
Cf.
A002865,
A066328,
A179009,
A236912,
A237113,
A237667,
A364912,
A364913,
A364915,
A364916,
A365002,
A365004.
-
Table[Length[Select[Array[IntegerPartitions,n+1,0,Join],Total[Union[#]]==k&]],{n,0,9},{k,0,n}]
-
T(n)={[Vecrev(p) | p<-Vec(prod(k=1, n, 1 - y^k + y^k/(1 - x^k), 1/(1 - x) + O(x*x^n)))]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 11 2024
A060016
Triangle T(n,k) = number of partitions of n into k distinct parts, 1 <= k <= n.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 2, 0, 0, 0, 0, 0, 1, 4, 3, 0, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1
Triangle starts
[ 1] 1,
[ 2] 1, 0,
[ 3] 1, 1, 0,
[ 4] 1, 1, 0, 0,
[ 5] 1, 2, 0, 0, 0,
[ 6] 1, 2, 1, 0, 0, 0,
[ 7] 1, 3, 1, 0, 0, 0, 0,
[ 8] 1, 3, 2, 0, 0, 0, 0, 0,
[ 9] 1, 4, 3, 0, 0, 0, 0, 0, 0,
[10] 1, 4, 4, 1, 0, 0, 0, 0, 0, 0,
[11] 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0,
[12] 1, 5, 7, 2, 0, 0, 0, 0, 0, 0, 0, 0,
[13] 1, 6, 8, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0,
[14] 1, 6, 10, 5, 0, 0, 0, 0, 0, 0, 0, 0, ...
T(8,3)=2 since 8 can be written in 2 ways as the sum of 3 distinct positive integers: 5+2+1 and 4+3+1.
- M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
- L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 94, 96 and 307.
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 219.
- D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIV.2, p. 493.
-
b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
-> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
end:
T:= proc(n) local l; l:= subsop(1=NULL, b(n, n));
l[], 0$(n-nops(l))
end:
seq(T(n), n=1..20); # Alois P. Heinz, Dec 12 2012
-
Flatten[Table[Length[Select[IntegerPartitions[n,{k}],Max[Transpose[ Tally[#]][[2]]]==1&]],{n,20},{k,n}]] (* Harvey P. Dale, Feb 27 2012 *)
T[, 1] = 1; T[n, k_] /; 1, ] = 0; Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 26 2015 *)
-
N=16; q='q+O('q^N);
gf=sum(n=0,N, z^n * q^((n^2+n)/2) / prod(k=1,n, 1-q^k ) );
/* print triangle: */
gf -= 1; /* remove row zero */
P=Pol(gf,'q);
{ for (n=1,N-1,
p = Pol(polcoeff(P, n),'z);
p += 'z^(n+1); /* preserve trailing zeros */
v = Vec(polrecip(p));
v = vector(n,k,v[k]); /* trim to size n */
print(v);
); }
/* Joerg Arndt, Oct 20 2012 */
A363719
Number of integer partitions of n satisfying (mean) = (median) = (mode), assuming there is a unique mode.
Original entry on oeis.org
1, 2, 2, 3, 2, 4, 2, 5, 3, 5, 2, 10, 2, 7, 7, 12, 2, 18, 2, 24, 16, 13, 2, 58, 15, 18, 37, 60, 2, 123, 2, 98, 79, 35, 103, 332, 2, 49, 166, 451, 2, 515, 2, 473, 738, 92, 2, 1561, 277, 839, 631, 1234, 2, 2043, 1560, 2867, 1156, 225, 2, 9020
Offset: 1
The a(n) partitions for n = 1, 2, 4, 6, 8, 12, 14, 16 (A..G = 10..16):
1 2 4 6 8 C E G
11 22 33 44 66 77 88
1111 222 2222 444 2222222 4444
111111 3221 3333 3222221 5443
11111111 4332 3322211 6442
5331 4222211 7441
222222 11111111111111 22222222
322221 32222221
422211 33222211
111111111111 42222211
52222111
1^16
Just two statistics:
A008284 counts partitions by length (or negative mean), strict
A008289.
A362608 counts partitions with a unique mode.
-
modes[ms_]:=Select[Union[ms],Count[ms,#]>=Max@@Length/@Split[ms]&];
Table[Length[Select[IntegerPartitions[n], {Mean[#]}=={Median[#]}==modes[#]&]],{n,30}]
A364910
Number of integer partitions of 2n whose distinct parts sum to n.
Original entry on oeis.org
1, 1, 1, 3, 3, 4, 12, 11, 19, 23, 54, 55, 103, 115, 178, 289, 389, 507, 757, 970, 1343, 2033, 2579, 3481, 4840, 6312, 8317, 10998, 15459, 19334, 26368, 33480, 44709, 56838, 74878, 93369, 128109, 157024, 206471, 258357, 338085, 417530, 544263, 669388, 859570, 1082758, 1367068
Offset: 0
The a(0) = 1 through a(7) = 11 partitions:
() (11) (22) (33) (44) (55) (66) (77)
(2211) (3311) (3322) (4422) (4433)
(21111) (311111) (4411) (5511) (5522)
(4111111) (33321) (6611)
(42222) (442211)
(322221) (4222211)
(332211) (4421111)
(3222111) (42221111)
(3321111) (422111111)
(32211111) (611111111)
(51111111) (4211111111)
(321111111)
The a(0) = 1 through a(7) = 11 linear combinations:
0 1*1 1*2 1*3 1*4 1*5 1*6 1*7
0*2+3*1 0*3+4*1 0*4+5*1 0*4+3*2 0*6+7*1
1*2+1*1 1*3+1*1 1*3+1*2 0*5+6*1 1*4+1*3
1*4+1*1 1*4+1*2 1*5+1*2
1*5+1*1 1*6+1*1
0*3+0*2+6*1 0*4+0*2+7*1
0*3+1*2+4*1 0*4+1*2+5*1
0*3+2*2+2*1 0*4+2*2+3*1
0*3+3*2+0*1 0*4+3*2+1*1
1*3+0*2+3*1 1*4+0*2+3*1
1*3+1*2+1*1 1*4+1*2+1*1
2*3+0*2+0*1
The case with no zero coefficients is
A000009.
A version based on Heinz numbers is
A364906.
Using all partitions (not just strict) we get
A364907.
Using strict partitions of any number from 1 to n gives
A365002.
These partitions have ranks
A365003.
-
Table[Length[Select[IntegerPartitions[2n],Total[Union[#]]==n&]],{n,0,15}]
-
a(n) = {my(res = 0); forpart(p = 2*n,s = Set(p); if(vecsum(s) == n, res++)); res} \\ David A. Corneth, Aug 20 2023
-
from sympy.utilities.iterables import partitions
def A364910(n): return sum(1 for d in partitions(n<<1,k=n) if sum(set(d))==n) # Chai Wah Wu, Sep 13 2023
A365312
Number of strict integer partitions with sum <= n that cannot be linearly combined using nonnegative coefficients to obtain n.
Original entry on oeis.org
0, 0, 0, 1, 1, 3, 2, 6, 4, 8, 7, 16, 6, 24, 17, 24, 20, 46, 22, 62, 31, 63, 57, 106, 35, 122, 90, 137, 88, 212, 74, 262, 134, 267, 206, 345, 121, 476, 294, 484, 232, 698, 242, 837, 389, 763, 571, 1185, 318, 1327, 634, 1392, 727, 1927, 640, 2056, 827, 2233, 1328
Offset: 0
The strict partition (7,3,2) has 19 = 1*7 + 2*3 + 3*2 so is not counted under a(19).
The strict partition (9,6,3) cannot be linearly combined to obtain 19, so is counted under a(19).
The a(0) = 0 through a(11) = 16 strict partitions:
. . . (2) (3) (2) (4) (2) (3) (2) (3) (2)
(3) (5) (3) (5) (4) (4) (3)
(4) (4) (6) (5) (6) (4)
(5) (7) (6) (7) (5)
(6) (7) (8) (6)
(4,2) (8) (9) (7)
(4,2) (6,3) (8)
(6,2) (9)
(10)
(4,2)
(5,4)
(6,2)
(6,3)
(6,4)
(7,3)
(8,2)
The complement for positive coefficients is counted by
A088314.
For positive coefficients we have
A088528.
The complement is counted by
A365311.
A364350 counts combination-free strict partitions, non-strict
A364915.
A364839 counts combination-full strict partitions, non-strict
A364913.
Cf.
A093971,
A237113,
A237668,
A326080,
A363225,
A364272,
A364534,
A364914,
A365043,
A365314,
A365320.
-
combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
Table[Length[Select[Select[Join@@Array[IntegerPartitions,n], UnsameQ@@#&],combs[n,#]=={}&]],{n,0,10}]
-
from math import isqrt
from sympy.utilities.iterables import partitions
def A365312(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1,n+1) for b in partitions(m,m=isqrt(1+(n<<3))>>1) if max(b.values()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
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