A046682 -id:A046682 - cp's OEIS frontend

A131942 Number of partitions of n in which each odd part has odd multiplicity.

1, 1, 1, 3, 3, 6, 6, 11, 13, 21, 24, 35, 44, 59, 74, 99, 126, 158, 202, 250, 320, 392, 495, 598, 758, 908, 1134, 1358, 1685, 2003, 2466, 2925, 3576, 4234, 5129, 6064, 7308, 8612, 10305, 12135, 14443, 16963, 20085, 23548, 27754, 32482, 38105, 44503, 52042

Offset: 0

Brian Drake, Jul 30 2007

			a(5)=6 because 5, 4+1, 3+2, 2+2+1, 2+1+1+1 and 1+1+1+1+1 have all odd parts with odd multiplicity. The partition 3+1+1 is the partition of 5 which is not counted.

Brian Drake and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Cf. A000041, A015128, A006950, A046682.

A:= series(product( 1/(1-q^(2*n)) *(1+q^(2*n-1)-q^(4*n-2))/(1-q^(4*n-2)), n=1..15),q,25): seq(coeff(A,q,i), i=0..24);

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/ ((1-x^(2*k)) * (1-x^(4*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

G.f.: Product_{n>=1} (1+q^(2n-1)-q^(4n-2))/((1-q^(2n))(1-q^(4n-2))).

a(n) ~ sqrt(Pi^2 + 8*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*n/2)) / (8*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 28, 40, 51

Offset: 0

Gus Wiseman, Nov 18 2018

First differs from A046682 at a(11) = 28, A046682(11) = 29.
A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:
  1 2 3
  1 2
  2 3
Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

			The a(1) = 1 through a(8) = 12 partitions whose Young diagram cannot be partitioned into vertical sections of the same sizes as the parts of the original partition are the same as the half-loop-graphical partitions up to n = 8:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (2221)     (2222)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (22211)
                                               (211111)   (32111)
                                               (1111111)  (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
For example, the half-loop-graphs
  {{1},{1,2},{1,3},{2,3}}
  {{1},{2},{3},{1,2},{1,3}}
both have degrees y = (3,2,2), so y is counted under a(7).

The complement is counted by A321728.
Cf. A000110, A000258, A000700, A000701, A006052, A007016, A008277, A046682, A319056, A319616, A321730, A321737, A321738.
The following pertain to the conjecture.
Half-loop-graphical partitions by length are A029889 or A339843 (covering).
The version for full loops is A339656.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320663/A339888 count unlabeled multiset partitions into singletons/pairs.
A322661 counts labeled covering half-loop-graphs, ranked by A340018/A340019.
A339659 is a triangle counting graphical partitions by length.
Cf. A006129, A025065, A062740, A095268, A096373, A167171, A320461, A338915, A339842, A339844, A339845.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Join@@Table[Subsets[ptnpos[y],{k}],{k,Reverse[Union[y]]}],UnsameQ@@First/@#&];
Table[Length[Select[IntegerPartitions[n],Length[Select[spsu[ptnverts[#],ptnpos[#]],Function[p,Sort[Length/@p]==Sort[#]]]]>0&]],{n,8}]

a(n) is the number of integer partitions y of n such that the coefficient of m(y) in e(y) is nonzero, where m is monomial symmetric functions and e is elementary symmetric functions.

a(n) = A000041(n) - A321728(n).

A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

Original entry on oeis.org

0, 1, 0, 2, 2, 3, 3, 4, 7, 8, 9, 10, 15, 16, 18, 23, 30, 32, 35, 42, 51, 59, 63, 73, 89, 100, 106, 125, 145, 160, 174, 198, 229, 255, 274, 310, 355, 388, 420, 472, 534, 582, 631, 701, 784, 859, 928, 1021, 1144, 1243, 1338, 1475, 1630, 1767, 1909, 2089, 2299

Offset: 0

Naohiro Nomoto, Feb 01 2002

T. D. Noe, Table of n, a(n) for n = 0..1000
Arnold Knopfmacher and Neville Robbins, Identities for the total number of parts in partitions of integers, Util. Math. 67 (2005), 9-18.

Cf. A000700, A000701, A006128, A015723, A046682, A079499, A158691.

CoefficientList[Series[Sum[n*q^(2n-1)*Product[1+q^k, {k, 1, 2n-3, 2}], {n, 1, 30}], {q, 0, 60}], q]

G.f.: A(q) = Sum_{n >= 1} n*q^(2*n-1)*(1+q)*(1+q^3)*...*(1+q^(2*n-3)).
From Peter Bala, Aug 20 2017: (Start)
Let F(q) = Product_{i >= 1} (1 + q^(2*i-1)). Then A(q) = Sum_{n >= 0} ( F(q) - Product_{i = 1..n} (1 + q^(2*i-1)) ).
It follows that the above sum A(q) satisfies -A(q-1) = 1 + q + 3*q^2 + 12*q^3 + 61*q^4 + ..., the g.f. for A158691, row-Fishburn matrices of size n. (End)

Edited by Dean Hickerson, Feb 11 2002

A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

Original entry on oeis.org

1, 1, 3, 9, 37, 152, 780, 3965, 23460, 141471, 944217, 6445643, 48075092, 364921557, 2974423953, 24847873439, 219611194148, 1987556951714, 18930298888792, 184244039718755, 1874490999743203, 19510832177784098, 210941659716920257, 2331530519337226199, 26692555830628617358

Offset: 0

Gus Wiseman, Nov 19 2018

nonn

A vertical section is a partial Young diagram with at most one square in each row. For example, a partition (shown as a coloring by positive integers) into vertical sections of the Young diagram of (322) is:
2 3
2
3

			The a(4) = 37 partitions into vertical sections of integer partitions of 4:
  1 2 3 4
.
  1 2 3   1 2 3   1 2 3   1 2 3
  4       3       2       1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3 4   2 3   3 2   1 3   1 2   3 1   2 1
.
  1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2   1 2
  3     3     2     3     2     1     1     3     2     1
  4     3     3     2     2     3     2     1     1     1
.
  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
  2   2   2   2   2   1   1   2   2   2   2   1   1   2   1
  3   3   2   3   2   2   2   1   1   3   2   1   2   1   1
  4   3   3   2   2   3   2   3   2   1   1   2   1   1   1

Cf. A000110, A000258, A008277, A046682, A122111, A318396, A321728, A321729, A321730, A321731, A321738, A321854.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
ptnpos[y_]:=Position[Table[1,{#}]&/@y,1];
ptnverts[y_]:=Select[Rest[Subsets[ptnpos[y]]],UnsameQ@@First/@#&];
Table[Sum[Length[spsu[ptnverts[y],ptnpos[y]]],{y,IntegerPartitions[n]}],{n,6}]

a(11)-a(24) from Ludovic Schwob, Aug 28 2023

A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2

Offset: 2

Matthew Vandermast, Apr 28 2014

If the prime signature of n (nonincreasing version) is viewed as a partition, row n gives the conjugate partition.

			24 = 2^3*3 is divisible by two prime numbers (2 and 3), one square of a prime (4 = 2^2), and one cube of a prime (8 = 2^3); therefore, row 24 of the table is {2,1,1}.
From _Gus Wiseman_, Mar 31 2022: (Start)
Rows begin:
     1: ()        16: (1,1,1,1)    31: (1)
     2: (1)       17: (1)          32: (1,1,1,1,1)
     3: (1)       18: (2,1)        33: (2)
     4: (1,1)     19: (1)          34: (2)
     5: (1)       20: (2,1)        35: (2)
     6: (2)       21: (2)          36: (2,2)
     7: (1)       22: (2)          37: (1)
     8: (1,1,1)   23: (1)          38: (2)
     9: (1,1)     24: (2,1,1)      39: (2)
    10: (2)       25: (1,1)        40: (2,1,1)
    11: (1)       26: (2)          41: (1)
    12: (2,1)     27: (1,1,1)      42: (3)
    13: (1)       28: (2,1)        43: (1)
    14: (2)       29: (1)          44: (2,1)
    15: (2)       30: (3)          45: (2,1)
(End)

Row lengths are A051903(n); row sums are A001222(n).
Cf. A217171.
These partitions are ranked by A238745.
For prime indices A296150 instead of exponents we get A321649, rev A321650.
A000700 counts self-conjugate partitions, ranked by A088902.
A003963 gives product of prime indices, conjugate A329382.
A008480 gives number of permutations of prime indices, conjugate A321648.
A056239 adds up prime indices, row sums of A112798.
A124010 gives prime signature, sorted A118914, length A001221.
A352486-A352490 are sets related to the fixed points of A122111.
Cf. A000701, A000720, A046682, A238747, A258116, A319005, A330644, A352491.

Table[Length/@Table[Select[Last/@FactorInteger[n],#>=k&],{k,Max@@Last/@FactorInteger[n]}],{n,2,100}] (* Gus Wiseman, Mar 31 2022 *)

Row n is identical to row A124859(n) of table A212171.

A118301 Number of partitions of n into distinct parts with largest part congruent to n modulo 2.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 19, 23, 27, 32, 38, 44, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3958, 4404

Offset: 1

Reinhard Zumkeller, Apr 22 2006

nonn

a(2*n) = A026838(2*n), a(2*n-1) = A026837(2*n-1);
a(n) = A000009(n) - A118302(n);
a(A090864(n)) = A118303(n)/2 = A000009(A090864(n))/2.

			a(11) = #{11,9+2,7+4,7+3+1,5+4+2,5+3+2+1} = 6;
a(12) = #{12,10+2,8+4,8+3+1,6+5+1,6+4+2,6+3+2+1} = 7.

Cf. A046682, A118302.

Conjectural g.f.: A(x) = Limit_{N -> oo} ( Sum_{n = 0..2*N+1} (-1)^(n+1)/Product_{k = 1..n} 1 - x^(2*k-1) ). - Peter Bala, Feb 11 2021

A115198 Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0

Offset: 0

Wolfdieter Lang, Feb 23 2006

The array with 0 and 1 interchanged is A115199.
The partitions appear in the Abramowitz-Stegun (A-St) order (see the reference, pp. 831-2).
A partition of n is (here) called even, resp. odd, if the number of even parts is even, resp. odd. A partition with no (0) even part is therefore even. Because the parity of permutations is linked, via their cycle structure, to the number of even parts of partitions one uses here 1 in order to mark the relevant (even) partitions.
The row length sequence of this array is p(n)=A000041(n) (number of partitions).

			[1];[0,1];[1,0,1];[0,1,1,0,1];[1,0,0,1,1,0,1];...
a(4,4)=0 because it refers to the 4th partition of n=4 of the
mentioned A-St ordering, namely to (1^2,2^1)=(1,1,2) which has an odd number
(1) of even parts.
a(5,4)=1 because (1^1,2^2)=(1,2,2) has an even number of even parts
(the number of even parts is in fact 2).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.
W. Lang: First 10 rows.

The sequence of row lengths is A046682 (number of cycle types for even permutations).

a(n,m)= 1 if sum(e(n,m,2*j),j=1..floor(n/2)) is even, else 0, with the exponents e(n,m,k) of the m-th partition of n in the A-St order; i.e. the sum of the exponents of the even parts of the partition (1^e(n,m,1),2^e(n,m,2),..., n^e(n,m,n)) is even iff a(n,m)=1.

A352143 Numbers whose prime indices and conjugate prime indices are all odd.

Original entry on oeis.org

1, 2, 5, 8, 11, 17, 20, 23, 31, 32, 41, 44, 47, 59, 67, 68, 73, 80, 83, 92, 97, 103, 109, 124, 125, 127, 128, 137, 149, 157, 164, 167, 176, 179, 188, 191, 197, 211, 227, 233, 236, 241, 257, 268, 269, 272, 275, 277, 283, 292, 307, 313, 320, 331, 332, 347, 353

Offset: 1

Gus Wiseman, Mar 18 2022

nonn

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, sum A056239, length A001222.
A number's prime signature is the sequence of positive exponents in its prime factorization, which is row n of A124010, length A001221, sum A001222.
These are the Heinz numbers of integer partitions whose parts and conjugate parts are all odd. They are counted by A053253.

			The terms together with their prime indices begin:
   1: {}
   2: {1}
   5: {3}
   8: {1,1,1}
  11: {5}
  17: {7}
  20: {1,1,3}
  23: {9}
  31: {11}
  32: {1,1,1,1,1}
  41: {13}
  44: {1,1,5}
  47: {15}
  59: {17}
  67: {19}
  68: {1,1,7}
  73: {21}
  80: {1,1,1,1,3}

The restriction to primes is A031368.
These partitions appear to be counted by A053253.
The even version is A066207^2.
For even instead of odd conjugate parts we get A066208^2.
The first condition alone (all odd indices) is A066208, counted by A000009.
The second condition alone is A346635, counted by A000009.
A055922 counts partitions with odd multiplicities, ranked by A268335.
A066207 = indices all even, counted by A035363 (complement A086543).
A109297 = same indices as exponents, counted by A114640.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A124010 gives prime signature, sorted A118914, length A001221, sum A001222.
A162642 counts odd prime exponents, even A162641.
A238745 gives the Heinz number of the conjugate prime signature.
A257991 counts odd indices, even A257992.
A258116 ranks strict partitions with all odd parts, even A258117.
A351979 = odd indices and even multiplicities, counted by A035457.
A352140 = even indices and odd multiplicities, counted by A055922 aerated.
A352141 = even indices and even multiplicities, counted by A035444.
A352142 = odd indices and odd multiplicities, counted by A117958.
Cf. A000290, A000701, A000720, A028260, A045931, A046682, A055396, A061395, A195017, A241638, A325698, A325700.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],And@@OddQ/@primeMS[#]&&And@@OddQ/@conj[primeMS[#]]&]

Intersection of A066208 and A346635.

A058397 Row sums of partition triangle A026820.

Original entry on oeis.org

1, 3, 6, 13, 22, 42, 66, 112, 172, 270, 397, 602, 858, 1245, 1748, 2464, 3381, 4671, 6302, 8537, 11372, 15147, 19914, 26201, 34057, 44250, 56986, 73277, 93497, 119161, 150809, 190590, 239496, 300388, 374912, 467135, 579394, 717384, 884813

Offset: 1

Wolfdieter Lang, Dec 11 2000

nonn

Cf. A026820, A046682.

seq(add(combinat:-numbpart(n,k),k=0..n),n=1..39); # Peter Luschny, Aug 03 2015

T[n_, k_] := T[n, k] = If[n==0 || k==1, 1, T[n, k-1] + If[k>n, 0, T[n-k, k] ]];
a[n_] := Sum[T[n, k], {k, 1, n}];
Array[a,39] (* Jean-François Alcover, Jun 03 2019, after Alois P. Heinz in A026820 *)

a(n)=if(n<1,0,polcoeff(sum(k=1,n,1/prod(i=1,k,(1-x^i)),x*O(x^n)),n))

a(n) = Sum_{k=1..n} A026820(n, k).

a(n) = A278427(2n,n). - John P. McSorley, Nov 28 2016

More terms from Benoit Cloitre, Apr 21 2003

A067627 Triangle T(n,k) = number of conjugacy classes of partitions of n using only k types of piles, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 3, 2, 1, 6, 1, 3, 7, 2, 5, 9, 2, 1, 8, 11, 2, 1, 13, 14, 1, 3, 19, 15, 3, 5, 27, 19, 1, 11, 34, 22, 2, 1, 15, 49, 23, 2, 1, 27, 59, 28, 3, 3, 39, 78, 30, 1, 5, 60, 93, 34, 3, 11, 82, 118, 36, 1, 18, 115, 140, 41, 3, 1, 30, 155, 170, 42, 2, 1, 48

Offset: 1

Naohiro Nomoto, Feb 02 2002

Lengths of rows are 1 1 2 2 2 3 3 3 3 4 4 4 4 4 ... (A003056).

			Triangle turned on its side begins:
1.1.1.2.1.2.1.2.2..2..1..3..1..2..2....etc A038548
....1.1.3.3.6.7.9.11.14.15.19.22.23....etc A270060
..........1.1.3.5..8.13.19.27.34.49....etc
...................1..1..3..5.11.15....etc

Cf. A000700, A000701, A046682, A060177. Diagonals give A038548. row sums give A046682.

compareL := proc(L1,L2)
    if nops(L1) < nops(L2) then
        -1 ;
    elif nops(L1) > nops(L2) then
        1;
    else
        for i from 1 to nops(L1) do
            if op(i,L1) > op(i,L2) then
                return 1 ;
            elif op(i,L1) < op(i,L2) then
                return -1 ;
            end if;
        end do:
        0 ;
    end if;
end proc:
A067627 := proc(n,k)
    local a,p,s,pc ;
    a := 0 ;
    for p in combinat[partition](n) do
        s := convert(p,set) ;
        if nops(s) = k then
            pc := combinat[conjpart](p) ;
            if compareL(p,pc) <= 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc:
for n from 1 to 30 do
for k from A003056(n) to 1 by -1 do
    printf("%4d,",A067627(n,k)) ;
end do:
printf("\n") ;
end do: # R. J. Mathar, May 08 2019

More terms from R. J. Mathar, May 08 2019

cp's OEIS Frontend

A131942 Number of partitions of n in which each odd part has odd multiplicity.

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A321729 Number of integer partitions of n whose Young diagram can be partitioned into vertical sections of the same sizes as the parts of the original partition.

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A067619 Total number of parts in all self-conjugate partitions of n. Also, sum of largest parts of all self-conjugate partitions of n.

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A321737 Number of ways to partition the Young diagram of an integer partition of n into vertical sections.

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A238744 Irregular table read by rows: T (n, k) gives the number of primes p such that p^k divides n; table omits all zero values.

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A118301 Number of partitions of n into distinct parts with largest part congruent to n modulo 2.

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A115198 Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows.

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A352143 Numbers whose prime indices and conjugate prime indices are all odd.

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A058397 Row sums of partition triangle A026820.