A098859
Number of partitions of n into parts each of which is used a different number of times.
Original entry on oeis.org
1, 1, 2, 2, 4, 5, 7, 10, 13, 15, 21, 28, 31, 45, 55, 62, 82, 105, 116, 153, 172, 208, 251, 312, 341, 431, 492, 588, 676, 826, 905, 1120, 1249, 1475, 1676, 2003, 2187, 2625, 2922, 3409, 3810, 4481, 4910, 5792, 6382, 7407, 8186, 9527, 10434
Offset: 0
a(6)=7 because 6= 4+1+1= 3+3= 3+1+1+1= 2+2+2= 2+1+1+1+1= 1+1+1+1+1+1. Four unrestricted partitions of 6 are not counted by a(6): 5+1, 4+2, 3+2+1 because at least two different summands are each used once; 2+2+1+1 because each summand is used twice.
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(9) = 15 partitions are the following. The Heinz numbers of these partitions are given by A130091.
1 2 3 4 5 6 7 8 9
11 111 22 221 33 322 44 333
211 311 222 331 332 441
1111 2111 411 511 422 522
11111 3111 2221 611 711
21111 4111 2222 3222
111111 22111 5111 6111
31111 22211 22221
211111 41111 33111
1111111 221111 51111
311111 411111
2111111 2211111
11111111 3111111
21111111
111111111
(End)
- Simon Langowski and Mark Daniel Ward, Table of n, a(n) for n = 0..2000 (terms 0..300 from M. D. Ward, 301..700 from Maciej Ireneusz Wilczynski)
- James Allen Fill, Svante Janson and Mark Daniel Ward, Partitions with Distinct Multiplicities of Parts: On An "Unsolved Problem" Posed By Herbert S. Wilf, The Electronic Journal of Combinatorics, Volume 19, Issue 2 (2012)
- Daniel Kane and Robert C. Rhoades, Asymptotics for Wilf's partitions with distinct multiplicities
- Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
- Simon Langowski, Program to compute Wilf Partitions, 2018
- Stephan Wagner, The Number of Fixed Points of Wilf's Partition Involution, The Electronic Journal of Combinatorics, 20(4) (2013), #P13.
- Doron Zeilberger, Using generatingfunctionology to enumerate distinct-multiplicity partitions; Local copy [Pdf file only, no active links]
-
a098859 = p 0 [] 1 where
p m ms _ 0 = if m `elem` ms then 0 else 1
p m ms k x
| x < k = 0
| m == 0 = p 1 ms k (x - k) + p 0 ms (k + 1) x
| m `elem` ms = p (m + 1) ms k (x - k)
| otherwise = p (m + 1) ms k (x - k) + p 0 (m : ms) (k + 1) x
-- Reinhard Zumkeller, Dec 27 2012
-
a[n_] := Length[sp = Split /@ IntegerPartitions[n]] - Count[sp, {__List, b_List, ___List, c_List, ___List} /; Length[b] == Length[c]]; Table[a[n], {n, 0, 48}] (* _Jean-François Alcover, Jan 17 2013 *)
-
a(n)={((r,k,b,w)->if(!k||!r, if(r,0,1), sum(m=0, r\k, if(!m || !bittest(b,m), self()(r-k*m, k-1, bitor(b, 1<Andrew Howroyd, Aug 31 2019
A047993
Number of balanced partitions of n: the largest part equals the number of parts.
Original entry on oeis.org
1, 0, 1, 1, 1, 1, 3, 2, 4, 4, 6, 7, 11, 11, 16, 19, 25, 29, 40, 45, 60, 70, 89, 105, 134, 156, 196, 232, 285, 336, 414, 485, 591, 696, 839, 987, 1187, 1389, 1661, 1946, 2311, 2702, 3201, 3731, 4400, 5126, 6018, 6997, 8195, 9502, 11093, 12849, 14949, 17281, 20062
Offset: 1
From _Joerg Arndt_, Oct 08 2012: (Start)
a(12) = 7 because the partitions of 12 where the largest part equals the number of parts are
2 + 3 + 3 + 4,
2 + 2 + 4 + 4,
1 + 3 + 4 + 4,
1 + 2 + 2 + 2 + 5,
1 + 1 + 2 + 3 + 5,
1 + 1 + 1 + 4 + 5, and
1 + 1 + 1 + 1 + 2 + 6.
(End)
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(13) = 11 integer partitions:
1 21 22 311 321 322 332 333 4222 4322 4332 4333
331 4211 4221 4321 4331 4422 4432
4111 4311 4411 4421 4431 4441
51111 52111 52211 52221 52222
53111 53211 53221
611111 54111 53311
621111 54211
55111
622111
631111
7111111
(End)
-
a047993 = flip a063995 0 -- Reinhard Zumkeller, Jul 24 2013
-
A047993 := proc(n)
a := 0 ;
for p in combinat[partition](n) do
r := max(op(p))-nops(p) ;
if r = 0 then
a := a+1 ;
end if;
end do:
a ;
end proc:
seq(A047993(n),n=1..20) ; # Emeric Deutsch, Dec 11 2004
-
Table[ Count[Partitions[n], par_List/; First[par]===Length[par]], {n, 12}] or recur: Sum[T[n-(2m-1), m-1, m-1], {m, Ceiling[Sqrt[n]], Floor[(n+1)/2]}] with T[m_, a_, b_]/; b < a := T[m, b, a]; T[m_, a_, b_]/; m > a*b := 0; T[m_, a_, b_]/; (2m > a*b) := T[a*b-m, a, b]; T[m_, 1, b_] := If[b < m, 0, 1]; T[0, , ] := 1; T[m_, a_, b_] := T[m, a, b]=Sum[T[m-a*i, a-1, b-i], {i, 0, Floor[m/a]}];
Table[Sum[ -(-1)^k*(p[n-(3*k^2-k)/2] - p[n-(3*k^2+k)/2]), {k, 1, Floor[(1+Sqrt[1+24*n])/6]}] /. p -> PartitionsP, {n, 1, 64}] (* Wouter Meeussen *)
(* also *)
Table[Count[IntegerPartitions[n], q_ /; Max[q] == Length[q]], {n, 24}]
(* Clark Kimberling, Feb 13 2014 *)
nmax = 100; p = 1; s = 1; Do[p = Normal[Series[p*x^2*(1 - x^(2*k - 1))*(1 + x^k)/(1 - x^k), {x, 0, nmax}]]; s += p;, {k, 1, nmax + 1}]; Take[CoefficientList[s, x], nmax] (* Vaclav Kotesovec, Oct 16 2024 *)
-
N=66; q='q + O('q^N );
S=2+2*ceil(sqrt(N));
gf= sum(k=1, S, (-1)^k * ( q^((3*k^2+k)/2) - q^((3*k^2-k)/2) ) ) / prod(k=1,N, 1-q^k );
/* Joerg Arndt, Oct 08 2012 */
-
my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, x^(2*k-1)*prod(j=1, k-1, (1-x^(k+j-1))/(1-x^j)))) \\ Seiichi Manyama, Jan 24 2022
A106529
Numbers having k prime factors (counted with multiplicity), the largest of which is the k-th prime.
Original entry on oeis.org
2, 6, 9, 20, 30, 45, 50, 56, 75, 84, 125, 126, 140, 176, 189, 196, 210, 264, 294, 315, 350, 396, 416, 440, 441, 490, 525, 594, 616, 624, 660, 686, 735, 875, 891, 924, 936, 968, 990, 1029, 1040, 1088, 1100, 1225, 1386, 1404, 1452, 1456, 1485, 1540, 1560
Offset: 1
Matthew Ryan (mattryan1994(AT)hotmail.com), May 30 2005
a(7)=50 because 50=2*5*5 is, for k=3, the product of k primes, the largest of which is the k-th prime, and 50 is the 7th such number.
A061395 selects maximum prime index.
A112798 lists the prime indices of each positive integer.
Other balance-related sequences:
-
A010054 counts balanced strict partitions.
-
A047993 counts balanced partitions.
-
A090858 counts partitions of rank 1.
-
A098124 counts balanced compositions.
-
A340596 counts co-balanced factorizations.
-
A340598 counts balanced set partitions.
-
A340599 counts alt-balanced factorizations.
-
A340600 counts unlabeled balanced multiset partitions.
-
A340653 counts balanced factorizations.
-
with(numtheory): a := proc (n) options operator, arrow: pi(max(factorset(n)))-bigomega(n) end proc: A := {}: for i from 2 to 1600 do if a(i) = 0 then A := `union`(A, {i}) else end if end do: A; # Emeric Deutsch, May 09 2015
-
Select[Range@ 1560, PrimePi@ FactorInteger[#][[-1, 1]] == PrimeOmega@ # &] (* Michael De Vlieger, May 09 2015 *)
A006141
Number of integer partitions of n whose smallest part is equal to the number of parts.
Original entry on oeis.org
1, 0, 0, 1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 25, 29, 33, 38, 42, 49, 54, 62, 69, 78, 87, 99, 109, 123, 137, 154, 170, 191, 211, 236, 261, 290, 320, 357, 392, 435, 479, 530, 582, 644, 706, 779, 854, 940, 1029, 1133, 1237, 1358, 1485
Offset: 1
G.f. = x + x^4 + x^5 + x^6 + x^7 + x^8 + 2*x^9 + 2*x^10 + 3*x^11 + 3*x^12 + ...
a(15) = 5 because the partitions of 15 where the smallest part equals the number of parts are
3 + 6 + 6,
3 + 5 + 7,
3 + 4 + 8,
3 + 3 + 9, and
2 + 13.
- _Joerg Arndt_, Oct 08 2012
a(15) = 5 because the partitions of 15 with parts differing by at least 2 and part 1 present are: [14,1] obtained from the partition of 11 with one part, [11], added to the first part of the special partition [3,1] of 4 and [11,3,1], [10,4,1], [9,5,1], [8,6,1] from adding all partition of 15 - 9 = 6 with one part, [6], and those with two parts, [5,1], [4,1], [3,3], to the special partition [5,3,1] of 9. - _Wolfdieter Lang_, Oct 31 2016
a(15) = 5 because the partitions of 14 with parts >= 3 and parts differing by at least 2 are [14], [11,3], [10,4], [9,5] and [8,6]. See the second [MacMahon] comment. This follows from the g.f. G[3](q) given in Andrews - Baxter, eq. (5.1) for i=3, (using summation index m) and m*(m+2) = 3 + 5 + ... + (2*m+1). - _Wolfdieter Lang_, Nov 02 2016
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(8) = 1 through a(15) = 5 integer partitions:
(6,2) (7,2) (8,2) (9,2) (10,2) (11,2) (12,2) (13,2)
(3,3,3) (4,3,3) (4,4,3) (5,4,3) (5,5,3) (6,5,3) (6,6,3)
(5,3,3) (6,3,3) (6,4,3) (7,4,3) (7,5,3)
(7,3,3) (8,3,3) (8,4,3)
(9,3,3)
(End)
- G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 92-95.
- G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, pp. 292-294.
- P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 45, Section 293.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
- George E. Andrews and R. J. Baxter, A motivated proof of the Rogers-Ramanujan identities, Amer. Math. Monthly 96 (1989), no. 5, 401-409.
- Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Handout, Math. Dept., Rutgers University, April 2015.
- Shashank Kanade, Some results on the representation theory of vertex operator algebras and integer partition identities, PhD Dissertation, Math. Dept., Rutgers University, April 2015.
- James Lepowsky and Minxian Zhu, A motivated proof of Gordon's identities, arXiv:1205.6570 [math.CO], 2012; The Ramanujan Journal 29.1-3 (2012): 199-211.
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
- Eric Weisstein's World of Mathematics, Rogers-Ramanujan Identities
For the generalized Rogers-Ramanujan series G[1], G[2], G[3], G[4], G[5], G[6], G[7], G[8] see
A003114,
A003106,
A006141,
A264591,
A264592,
A264593,
A264594,
A264595. G[0] = G[1]+G[2] is given by
A003113.
A003106 counts partitions with minimum > length.
A003114 counts partitions with minimum >= length.
A026794 counts partitions by minimum.
A039899 counts partitions with minimum < length.
A039900 counts partitions with minimum <= length.
A239950 counts partitions with minimum equal to number of distinct parts.
Sequences related to balance:
-
A010054 counts balanced strict partitions.
-
A047993 counts balanced partitions.
-
A098124 counts balanced compositions.
-
A106529 ranks balanced partitions.
-
A340596 counts co-balanced factorizations.
-
A340598 counts balanced set partitions.
-
A340599 counts alt-balanced factorizations.
-
A340600 counts unlabeled balanced multiset partitions.
-
A340653 counts balanced factorizations.
-
b:= proc(n, i) option remember; `if`(n<0, 0, `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i,i)))))
end:
a:= n-> add(b(n-j^2, j-1), j=0..isqrt(n)):
seq(a(n), n=1..80); # Alois P. Heinz, Oct 08 2012
-
b[n_, i_] := b[n, i] = If[n<0, 0, If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]]; a[n_] := Sum[b[n-j^2, j-1], {j, 0, Sqrt[n]}]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Mar 17 2014, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Min[#]==Length[#]&]],{n,30}] (* Gus Wiseman, Mar 09 2019 *)
-
{a(n) = if( n<1, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2 / prod(j=1, k-1, 1 - x^j, 1 + O(x ^ (n - k^2 + 1) ))), n))} /* Michael Somos, Jan 22 2008 */
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), Jun 20 2000
A064174
Number of partitions of n with nonnegative rank.
Original entry on oeis.org
1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 42, 56, 73, 96, 125, 161, 207, 265, 336, 426, 536, 672, 840, 1046, 1296, 1603, 1975, 2425, 2970, 3628, 4417, 5367, 6503, 7861, 9482, 11412, 13702, 16423, 19642, 23447, 27938, 33231, 39453, 46767, 55342, 65386, 77135
Offset: 1
a(20) = p(19) - p(15) + p(8) = 490 - 176 + 22 = 336.
- Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
- Cristina Ballantine and Mircea Merca, Bisected theta series, least r-gaps in partitions, and polygonal numbers, arXiv:1710.05960 [math.CO], 2017.
- Rekha Biswal, bijection between number of partitions of 2n satisfying certain conditions with number of partitions of n, Mathoverflow.
- Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.
-
f:= n -> add((-1)^(k+1)*combinat:-numbpart(n-(3*k^2-k)/2),k=1..floor((1+sqrt(24*n+1))/6)):
map(f, [$1..100]); # Robert Israel, Aug 03 2015
-
Table[Count[IntegerPartitions[n], q_ /; First[q] >= Length[q]], {n, 16}]
(* also *)
Table[Count[IntegerPartitions[2n],q_/;Last[q]===1 && Max[q-PadRight[Rest[q],Length[q]]]<=1 && Count[First/@Partition[q,2],?OddQ]==Count[Last/@Partition[q,2],?OddQ]],{n,16}]
(* also *)
Table[Count[IntegerPartitions[n], p_ /; MemberQ[p, Max[Max[p], Length[p]]]], {n, 50}] (* Clark Kimberling, Feb 28 2014 *)
-
{a(n) = my(A=1); A = sum(m=0,n,x^m*prod(k=1,m,(1-x^(m+k-1))/(1-x^k +x*O(x^n)))); polcoeff(A,n)}
for(n=1,60,print1(a(n),", ")) \\ Paul D. Hanna, Aug 03 2015
-
my(N=50, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2))) \\ Seiichi Manyama, May 21 2023
A183558
Number of partitions of n containing a clique of size 1.
Original entry on oeis.org
0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941
Offset: 0
a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].
From _Gus Wiseman_, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2111) (3111) (331) (422)
(21111) (421) (431)
(511) (521)
(2221) (611)
(3211) (3221)
(4111) (4211)
(31111) (5111)
(211111) (32111)
(41111)
(311111)
(2111111)
(End)
Cf.
A000041,
A007690,
A183559,
A183560,
A183561,
A183562,
A183563,
A183564,
A183565,
A183566,
A183567.
-
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);
-
max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* Jean-François Alcover, Oct 01 2014. Edited by Gus Wiseman, Apr 19 2019 *)
A266477
Triangle read by rows in which T(n,k) is the number of partitions of n with product of multiplicities of parts equal to k; n>=0, 1<=k<=A266480(n).
Original entry on oeis.org
1, 1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 2, 2, 2, 0, 1, 5, 4, 2, 1, 1, 1, 1, 6, 6, 2, 3, 1, 2, 0, 2, 8, 7, 4, 4, 1, 2, 1, 0, 2, 1, 10, 8, 6, 6, 3, 2, 1, 3, 0, 1, 0, 2, 12, 13, 6, 6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1, 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0
Offset: 0
Row 4 is [2,2,0,1]. Indeed, the products of the multiplicities of the parts in the partitions [4], [1,3], [2,2], [1,1,2], [1,1,1,1] are 1, 1, 2, 2, 4, respectively.
Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 1, 1;
03 : 2, 0, 1;
04 : 2, 2, 0, 1;
05 : 3, 2, 1, 0, 1;
06 : 4, 2, 2, 2, 0, 1;
07 : 5, 4, 2, 1, 1, 1, 1;
08 : 6, 6, 2, 3, 1, 2, 0, 2;
09 : 8, 7, 4, 4, 1, 2, 1, 0, 2, 1;
10 : 10, 8, 6, 6, 3, 2, 1, 3, 0, 1, 0, 2;
11 : 12, 13, 6, 6, 3, 7, 1, 2, 1, 1, 1, 1, 0, 1, 1;
12 : 15, 15, 9, 11, 3, 6, 2, 5, 3, 3, 0, 2, 0, 0, 0, 2, 0, 1;
-
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
x^max(p, p*n), add(b(n-i*j, i-1, max(p, p*j)), j=0..n/i))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2, 1)):
seq(T(n), n=0..16);
-
Map[Table[Length@ Position[#, k], {k, Max@ #}] &, #] &@ Table[Map[Times @@ Map[Last, Tally@ #] &, IntegerPartitions@ n], {n, 12}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, x^Max[p, p*n], Sum[b[n - i*j, i - 1, Max[p, p*j]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p,x]}]][ b[n, n, 1]]; Table[T[n], {n, 0, 16}] // Flatten (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)
A117144
Partitions of n in which each part k occurs at least k times.
Original entry on oeis.org
1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322
Offset: 0
a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
1 11 111 22 221 222 2221 2222 333
1111 11111 2211 22111 22211 22221
111111 1111111 221111 222111
11111111 2211111
111111111
(End)
Cf.
A001462,
A003114,
A006141,
A033461,
A039900,
A047993,
A052335,
A064174,
A090858,
A114638,
A115584,
A276078,
A280204.
-
g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..80); # Alois P. Heinz, Dec 28 2016
-
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)
A239955
Number of partitions p of n such that (number of distinct parts of p) <= max(p) - min(p).
Original entry on oeis.org
0, 0, 0, 0, 1, 2, 4, 7, 12, 17, 27, 38, 54, 75, 104, 137, 187, 245, 322, 418, 542, 691, 887, 1121, 1417, 1777, 2228, 2767, 3441, 4247, 5235, 6424, 7871, 9594, 11688, 14173, 17168, 20723, 24979, 30008, 36010, 43085, 51479, 61357, 73032, 86718, 102852, 121718
Offset: 0
a(6) counts these 4 partitions: 51, 42, 411, 3111.
Applying the condition to the conjugate gives
A350839, ranked by
A350841.
A116932 counts partitions with differences != -1 or -2, strict
A025157.
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(b(n-i*j, i-1), j=1..n/i)))
end:
a:= n-> combinat[numbpart](n)-add(b(n, k), k=0..n):
seq(a(n), n=0..47); # Alois P. Heinz, Aug 18 2025
-
z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[p_] := f[p] = Max[p] - Min[p]; g[n_] := g[n] = IntegerPartitions[n];
Table[Count[g[n], p_ /; d[p] < f[p]], {n, 0, z}] (*A239954*)
Table[Count[g[n], p_ /; d[p] <= f[p]], {n, 0, z}] (*A239955*)
Table[Count[g[n], p_ /; d[p] == f[p]], {n, 0, z}] (*A239956*)
Table[Count[g[n], p_ /; d[p] > f[p]], {n, 0, z}] (*A034296*)
Table[Count[g[n], p_ /; d[p] >= f[p]], {n, 0, z}] (*A239958*)
(* second program *)
Table[Length[Select[IntegerPartitions[n],Min@@Differences[#]<-1&]],{n,0,30}] (* Gus Wiseman, Jun 26 2022 *)
-
qs(a,q,n) = {prod(k=0,n,1-a*q^k)}
A_q(N) = {if(N<4, vector(N+1,i,0), my(q='q+O('q^(N-2)), g= sum(i=2,N+1, q^i/qs(q,q,i-1)*sum(j=1,i-1, q^(2*j)*qs(q^2,q^2,j-2)))); concat([0,0,0,0], Vec(g)))} \\ John Tyler Rascoe, Aug 16 2025
A117571
Expansion of (1+2*x^2)/((1-x)*(1-x^3)).
Original entry on oeis.org
1, 1, 3, 4, 4, 6, 7, 7, 9, 10, 10, 12, 13, 13, 15, 16, 16, 18, 19, 19, 21, 22, 22, 24, 25, 25, 27, 28, 28, 30, 31, 31, 33, 34, 34, 36, 37, 37, 39, 40, 40, 42, 43, 43, 45, 46, 46, 48, 49, 49, 51, 52, 52, 54, 55, 55, 57, 58, 58, 60, 61, 61, 63, 64, 64, 66, 67, 67, 69, 70, 70, 72
Offset: 0
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