A105806 -id:A105806 - cp's OEIS frontend

A072233 Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002

Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters, Dec 18 2006
From Gus Wiseman, Feb 10 2021: (Start)
T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:
  111111111  22221     333      432     54     63    72   81  9
             222111    3222     441     522    621   711
             2211111   3321     4221    531    6111
             21111111  32211    4311    5211
                       33111    42111   51111
                       321111   411111
                       3111111
(End)

			Table begins (upper left corner = T(0,0)):
1 1 1  1  1  1  1  1  1 ...
0 1 1  1  1  1  1  1  1 ...
0 1 2  2  2  2  2  2  2 ...
0 1 2  3  3  3  3  3  3 ...
0 1 3  4  5  5  5  5  5 ...
0 1 3  5  6  7  7  7  7 ...
0 1 4  7  9 10 11 11 11 ...
0 1 4  8 11 13 14 15 15 ...
0 1 5 10 15 18 20 21 22 ...
There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.
From _Wolfdieter Lang_, Dec 03 2012 (Start)
The triangle a(n,k) = T(n-k,k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
00   1
01   0  1
02   0  1  1
03   0  1  1  1
04   0  1  2  1  1
05   0  1  2  2  1  1
06   0  1  3  3  2  1  1
07   0  1  3  4  3  2  1  1
08   0  1  4  5  5  3  2  1  1
09   0  1  4  7  6  5  3  2  1  1
10   0  1  5  8  9  7  5  3  2  1  1
...
Row n=5 is, for k=1..5, [1,2,2,1,1] which gives the number of partitions of n=5 with k parts. See A008284 and the Franklin T. Adams-Watters comment above. (End)
From _Gus Wiseman_, Feb 10 2021: (Start)
Row n = 9 counts the following partitions:
  9  54  333  3222  22221  222111  2211111  21111111  111111111
     63  432  3321  32211  321111  3111111
     72  441  4221  33111  411111
     81  522  4311  42111
         531  5211  51111
         621  6111
         711
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..10010
Combinatorial Object Server, Information on Numerical Partitions
FindStat - Combinatorial Statistic Finder, The length of the partition.

Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m).
Alternating row sums are A081362.
Cf. A008284.
The version for factorizations is A316439.
The version for set partitions is A048993/A080510.
The version for strict partitions is A008289/A059607.
A047993 counts balanced partitions, ranked by A106529.
A063995/A105806 count partitions by Dyson rank.
Cf. A006141, A064174, A096401, A117409, A168659, A215366.

Flatten[Table[Length[IntegerPartitions[n, {k}]], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, Feb 24 2014 *)

from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (0..n)] for n in (0..10)] # Peter Luschny, Aug 01 2015

T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. Product_{j=0..infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1..infinity} 1/(1-xy^j). - Franklin T. Adams-Watters, Dec 18 2006
O.g.f. of column No. k of the triangle a(n,k) is x^k/product(1-x^j,j=1..k), k>=0 (the undefined product for k=0 is put to 1). - Wolfdieter Lang, Dec 03 2012

Corrected by Franklin T. Adams-Watters, Dec 18 2006

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

Vladeta Jovovic, Sep 20 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
This sequence (up to proof) equals "partitions of 2n with even number of parts, ending in 1, with max descent of 1, where the number of odd parts in odd places equals the number of odd parts in even places. (See link and 2nd Mathematica line.) - Wouter Meeussen, Mar 29 2013
Number of partitions p of n such that max(max(p), number of parts of p) is a part of p. - Clark Kimberling, Feb 28 2014
From Gus Wiseman, Mar 09 2019: (Start)
Also the number of integer partitions of n with maximum part greater than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324521. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)     (8)
            (21)  (22)  (32)   (33)   (43)    (44)
                  (31)  (41)   (42)   (52)    (53)
                        (311)  (51)   (61)    (62)
                               (321)  (322)   (71)
                               (411)  (331)   (332)
                                      (421)   (422)
                                      (511)   (431)
                                      (4111)  (521)
                                              (611)
                                              (4211)
                                              (5111)
Also the number of integer partitions of n with maximum part less than or equal to the number of parts. The Heinz numbers of these integer partitions are given by A324562. For example, the a(1) = 1 through a(8) = 12 partitions are:
  (1)  (11)  (21)   (22)    (221)    (222)     (322)      (332)
             (111)  (211)   (311)    (321)     (331)      (2222)
                    (1111)  (2111)   (2211)    (2221)     (3221)
                            (11111)  (3111)    (3211)     (3311)
                                     (21111)   (4111)     (4211)
                                     (111111)  (22111)    (22211)
                                               (31111)    (32111)
                                               (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

			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.

Cf. A063995, A064173 (complement).
Row sums of triangle A105806.
Cf. A003114, A006141, A039900, A047993, A090858, A106529, A133121.
Cf. A324516, A324518, A324520, A324521, A324522, A324560, A324562, A324572.

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

a(n) = (A000041(n) + A047993(n))/2.
a(n) = p(n-1) - p(n-5) + p(n-12) - ... -(-1)^k*p(n-(3*k^2-k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Sum_{n>=1} x^n * Product_{k=1..n} (1 - x^(n+k-1))/(1 - x^k). - Paul D. Hanna, Aug 03 2015
A064173(n) + a(n) = A000041(n). - R. J. Mathar, Feb 22 2023
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k-1)/2). - Seiichi Manyama, May 21 2023

Mathematica programs modified by Clark Kimberling, Feb 12 2014

A064173 Number of partitions of n with positive rank.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 80, 106, 136, 178, 225, 291, 366, 466, 583, 735, 912, 1140, 1407, 1743, 2140, 2634, 3214, 3932, 4776, 5807, 7022, 8495, 10225, 12313, 14762, 17696, 21136, 25236, 30030, 35722, 42367, 50216, 59368, 70138, 82665

Offset: 1

Vladeta Jovovic, Sep 19 2001

nonn

The rank of a partition is the largest summand minus the number of summands.
Also number of partitions of n with negative rank. - Omar E. Pol, Mar 05 2012
Column 1 of A208478. - Omar E. Pol, Mar 11 2012
Number of partitions p of n such that max(max(p), number of parts of p) is not a part of p. - Clark Kimberling, Feb 28 2014
The sequence enumerates the semigroup of partitions of positive rank for each number n. The semigroup is a subsemigroup of the monoid of partitions of nonnegative rank under the binary operation "*": Let A be the positive rank partition (a1,...,ak) where ak > k, and let B=(b1,...bj) with bj > j. Then let A*B be the partition (a1b1,...,a1bj,...,akb1,...,akbj), which has akbj > kj, thus having positive rank. For example, the partition (2,3,4) of 9 has rank 1, and its product with itself is (4,6,6,8,8,9,12,12,16) of 81, which has rank 7. A similar situation holds for partitions of negative rank--they are a subsemigroup of the monoid of nonpositive rank partitions. - Richard Locke Peterson, Jul 15 2018

			a(20) = p(18) - p(13) + p(5) = 385 - 101 + 7 = 291.
From _Gus Wiseman_, Feb 09 2021: (Start)
The a(2) = 1 through a(9) = 13 partitions of positive rank:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (31)  (32)  (33)   (43)   (44)    (54)
                  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (411)  (421)  (71)    (81)
                               (511)  (422)   (432)
                                      (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (5111)  (621)
                                              (711)
                                              (5211)
                                              (6111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
FindStat, St000145: The Dyson rank of a partition
Mircea Merca, Rank partition functions and truncated theta identities, arXiv:2006.07705 [math.CO], 2020.

Note: A-numbers of ranking sequences are in parentheses below.
The negative-rank version is also A064173 (A340788).
The case of odd positive rank is A101707 (A340604).
The case of even positive rank is A101708 (A340605).
These partitions are ranked by (A340787).
A063995/A105806 count partitions by rank.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is a multiple of the greatest part.
A200750 counts partitions whose length and greatest part are coprime.
- Rank -
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A340601 counts partitions of even rank (A340602).
A340692 counts partitions of odd rank (A340603).
- Balance -
A047993 counts balanced partitions (A106529).
A340599 counts alt-balanced factorizations.
A340653 counts balanced factorizations.
Cf. A003114, A006141, A039900, A096401, A117193, A117409, A143773, A324516, A324518, A324520.

A064173 := 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(A064173(n),n=0..40) ;# Emeric Deutsch, Dec 11 2004

Table[Count[IntegerPartitions[n], q_ /; First[q] > Length[q]], {n, 24}] (* Clark Kimberling, Feb 12 2014 *)
Table[Count[IntegerPartitions[n], p_ /; ! MemberQ[p, Max[Max[p], Length[p]]]], {n, 20}] (* Clark Kimberling, Feb 28 2014 *)
P = PartitionsP;
a[n_] := (P[n] - Sum[-(-1)^k (P[n - (3k^2 - k)/2] - P[n - (3k^2 + k)/2]), {k, 1, Floor[(1 + Sqrt[1 + 24n])/6]}])/2;
a /@ Range[48] (* Jean-François Alcover, Jan 11 2020, after Wouter Meeussen in A047993 *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k*prod(j=1, k, (1-x^(k+j-2))/(1-x^j))))) \\ Seiichi Manyama, Jan 25 2022

a(n) = (A000041(n) - A047993(n))/2.
a(n) = p(n-2) - p(n-7) + p(n-15) - ... - (-1)^k*p(n-(3*k^2+k)/2) + ..., where p() is A000041(). - Vladeta Jovovic, Aug 04 2004
G.f.: Product_{k>=1} (1/(1-q^k)) * Sum_{k>=1} ( (-1)^k * (-q^(3*k^2/2+k/2))) (conjectured). - Thomas Baruchel, May 12 2018
G.f.: Sum_{k>=1} x^k * Product_{j=1..k} (1-x^(k+j-2))/(1-x^j). - Seiichi Manyama, Jan 25 2022
a(n)+A064174(n) = A000041(n). - R. J. Mathar, Feb 22 2023

A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 1, 0, 1, 1, 2

Offset: 1

N. J. A. Sloane, Sep 19 2001

The rank of a partition is the largest part minus the number of parts.

The rows are symmetric: for every partition of rank r there is its conjugate with rank -r. [Joerg Arndt, Oct 07 2012]

			The partition 5 = 4+1 has largest summand 4 and 2 summands, hence has rank 4-2 = 2.
Triangle begins:
[ 1]                               1,
[ 2]                            1, 0, 1,
[ 3]                         1, 0, 1, 0, 1,
[ 4]                      1, 0, 1, 1, 1, 0, 1,
[ 5]                   1, 0, 1, 1, 1, 1, 1, 0, 1,
[ 6]                1, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1,
[ 7]             1, 0, 1, 1, 2, 1, 3, 1, 2, 1, 1, 0, 1,
[ 8]          1, 0, 1, 1, 2, 2, 3, 2, 3, 2, 2, 1, 1, 0, 1,
[ 9]       1, 0, 1, 1, 2, 2, 3, 3, 4, 3, 3, 2, 2, 1, 1, 0, 1,
[10]    1, 0, 1, 1, 2, 2, 4, 3, 5, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1,
[11] 1, 0, 1, 1, 2, ...
Row 20 is:
T(20, k) = 1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 20, 22, 30, 33, 40, 42, 48, 45, 48, 42, 40, 33, 30, 22, 20, 14, 12, 8, 7, 4, 4, 2, 2, 1, 1, 0, 1; -19 <= k <= 19.
Another view of the table of p(n,m) = number of partitions of n with rank m, taken from Dyson (1969):
n\m -6 -5  -4  -3  -2  -1   0   1   2   3   4   5   6
-----------------------------------------------------
0   0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,
1   0,  0,  0,  0,  0,  0,  1,  0,  0,  0,  0,  0,  0,
2   0,  0,  0,  0,  0,  1,  0,  1,  0,  0,  0,  0,  0,
3   0,  0,  0,  0,  1,  0,  1,  0,  1,  0,  0,  0,  0,
4   0,  0,  0,  1,  0,  1,  1,  1,  0,  1,  0,  0,  0,
5   0,  0,  1,  0,  1,  1,  1,  1,  1,  0,  1,  0,  0,
6   0,  1,  0,  1,  1,  2,  1,  2,  1,  1,  0,  1,  0,
7   1,  0,  1,  1,  2,  1,  3,  1,  2,  1,  1,  0,  1,
...
The central triangle is the present sequence, the right-hand triangle is A105806. - _N. J. A. Sloane_, Jan 23 2020

Alois P. Heinz, Rows n = 1..145, flattened (first 72 rows from Reinhard Zumkeller)
G. E. Andrews, The number of smallest parts in the partitions of n. [Also Selected Works, p. 603, see N(m,n).] - _N. J. A. Sloane_, Dec 16 2013
A. O. L. Atkin and P. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4, (1954). 84-106. Math. Rev. 15,685d.
Alexander Berkovich and Frank G. Garvan, Some observations on Dyson's new symmetries of partitions, Journal of Combinatorial Theory, Series A 100.1 (2002): 61-93.
Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61. See Table 1.
Freeman J. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169-180.

For the number of partitions of n with rank 0 (balanced partitions) see A047993.
Cf. A105806 (right half of triangle), A005408 (row lengths), A000041 (row sums), A047993 (central terms).
Cf. A000025.

import Data.List (sort, group)
a063995 n k = a063995_tabf !! (n-1) !! (n-1+k)
a063995_row n = a063995_tabf !! (n-1)
a063995_tabf = [[1], [1, 0, 1]] ++ (map
   (\rs -> [1, 0] ++ (init $ tail $ rs) ++ [0, 1]) $ drop 2 $ map
   (map length . group . sort . map rank) $ tail pss) where
      rank ps = maximum ps - length ps
      pss = [] : map (\u -> [u] : [v : ps | v <- [1..u],
                             ps <- pss !! (u - v), v <= head ps]) [1..]
-- Reinhard Zumkeller, Jul 24 2013

Table[ Count[ (First[ # ]-Length[ # ]& /@ IntegerPartitions[ k ]), # ]& /@ Range[ -k+1, k-1 ], {k, 16} ]

Sum_{k=-(n-1)..n-1} (-1)^k * T(n,k) = A000025(n). - Alois P. Heinz, Dec 20 2024

More terms from Vladeta Jovovic and Wouter Meeussen, Sep 19 2001

A340692 Number of integer partitions of n of odd rank.

Original entry on oeis.org

0, 0, 2, 0, 4, 2, 8, 4, 14, 12, 26, 22, 44, 44, 76, 78, 126, 138, 206, 228, 330, 378, 524, 602, 814, 950, 1252, 1466, 1900, 2238, 2854, 3362, 4236, 5006, 6232, 7356, 9078, 10720, 13118, 15470, 18800, 22152, 26744, 31456, 37772, 44368, 53002, 62134, 73894

Offset: 0

Gus Wiseman, Jan 29 2021

nonn

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The a(0) = 0 through a(9) = 12 partitions (empty columns indicated by dots):
  .  .  (2)   .  (4)     (32)   (6)       (52)     (8)         (54)
        (11)     (31)    (221)  (33)      (421)    (53)        (72)
                 (211)          (51)      (3211)   (71)        (432)
                 (1111)         (222)     (22111)  (422)       (441)
                                (411)              (431)       (621)
                                (3111)             (611)       (3222)
                                (21111)            (3221)      (3321)
                                (111111)           (3311)      (5211)
                                                   (5111)      (22221)
                                                   (22211)     (42111)
                                                   (41111)     (321111)
                                                   (311111)    (2211111)
                                                   (2111111)
                                                   (11111111)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of length/maximum instead of rank is A027193 (A026424/A244991).
The case of odd positive rank is A101707 is (A340604).
The strict case is A117193.
The even version is A340601 (A340602).
The Heinz numbers of these partitions are (A340603).
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064173 counts partitions of positive/negative rank (A340787/A340788).
A064174 counts partitions of nonpositive/nonnegative rank (A324521/A324562).
A101198 counts partitions of rank 1 (A325233).
A101708 counts partitions of even positive rank (A340605).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A058695 counts partitions of odd numbers (A300063).
A067659 counts strict partitions of odd length (A030059).
A160786 counts odd-length partitions of odd numbers (A300272).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A003114, A006141, A027187, A039900, A067538, A096401, A117409, A143773, A324518, A325134, A340828, A340854/A340855.

Table[Length[Select[IntegerPartitions[n],OddQ[Max[#]-Length[#]]&]],{n,0,30}]

Having odd rank is preserved under conjugation, and self-conjugate partitions cannot have odd rank, so a(n) = 2*A101707(n) for n > 0.

A340788 Heinz numbers of integer partitions of negative rank.

Original entry on oeis.org

4, 8, 12, 16, 18, 24, 27, 32, 36, 40, 48, 54, 60, 64, 72, 80, 81, 90, 96, 100, 108, 112, 120, 128, 135, 144, 150, 160, 162, 168, 180, 192, 200, 216, 224, 225, 240, 243, 250, 252, 256, 270, 280, 288, 300, 320, 324, 336, 352, 360, 375, 378, 384, 392, 400, 405

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
      4: (1,1)             80: (3,1,1,1,1)
      8: (1,1,1)           81: (2,2,2,2)
     12: (2,1,1)           90: (3,2,2,1)
     16: (1,1,1,1)         96: (2,1,1,1,1,1)
     18: (2,2,1)          100: (3,3,1,1)
     24: (2,1,1,1)        108: (2,2,2,1,1)
     27: (2,2,2)          112: (4,1,1,1,1)
     32: (1,1,1,1,1)      120: (3,2,1,1,1)
     36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
     40: (3,1,1,1)        135: (3,2,2,2)
     48: (2,1,1,1,1)      144: (2,2,1,1,1,1)
     54: (2,2,2,1)        150: (3,3,2,1)
     60: (3,2,1,1)        160: (3,1,1,1,1,1)
     64: (1,1,1,1,1,1)    162: (2,2,2,2,1)
     72: (2,2,1,1,1)      168: (4,2,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 is (A340929).
The even case is A101708 is (A340930).
The positive version is (A340787).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 counts partitions whose length is divisible by maximum.
A200750 counts partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324518 counts partitions with rank equal to greatest part (A324517).
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A325134, A326845, A340604, A340605, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]

For all terms A061395(a(n)) < A001222(a(n)).

A340787 Heinz numbers of integer partitions of positive rank.

Original entry on oeis.org

3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95

Offset: 1

Gus Wiseman, Jan 29 2021

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

The Dyson rank of a nonempty partition is its maximum part minus its length. The rank of an empty partition is undefined.

			The sequence of partitions together with their Heinz numbers begins:
     3: (2)      28: (4,1,1)    49: (4,4)      69: (9,2)
     5: (3)      29: (10)       51: (7,2)      70: (4,3,1)
     7: (4)      31: (11)       52: (6,1,1)    71: (20)
    10: (3,1)    33: (5,2)      53: (16)       73: (21)
    11: (5)      34: (7,1)      55: (5,3)      74: (12,1)
    13: (6)      35: (4,3)      57: (8,2)      76: (8,1,1)
    14: (4,1)    37: (12)       58: (10,1)     77: (5,4)
    15: (3,2)    38: (8,1)      59: (17)       78: (6,2,1)
    17: (7)      39: (6,2)      61: (18)       79: (22)
    19: (8)      41: (13)       62: (11,1)     82: (13,1)
    21: (4,2)    42: (4,2,1)    63: (4,2,2)    83: (23)
    22: (5,1)    43: (14)       65: (6,3)      85: (7,3)
    23: (9)      44: (5,1,1)    66: (5,2,1)    86: (14,1)
    25: (3,3)    46: (9,1)      67: (19)       87: (10,2)
    26: (6,1)    47: (15)       68: (7,1,1)    88: (5,1,1,1)

Freeman J. Dyson, A new symmetry of partitions, Journal of Combinatorial Theory 7.1 (1969): 56-61.
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of Heinz-number sequences are in parentheses below.
These partitions are counted by A064173.
The odd case is A101707 (A340604).
The even case is A101708 (A340605).
The negative version is (A340788).
A001222 counts prime factors.
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A168659 = partitions whose greatest part divides their length (A340609).
A168659 = partitions whose length divides their greatest part (A340610).
A200750 = partitions whose length and maximum are relatively prime.
- Rank -
A047993 counts partitions of rank 0 (A106529).
A063995/A105806 count partitions by Dyson rank.
A064174 counts partitions of nonnegative/nonpositive rank (A324562/A324521).
A101198 counts partitions of rank 1 (A325233).
A257541 gives the rank of the partition with Heinz number n.
A324520 counts partitions with rank equal to least part (A324519).
A340601 counts partitions of even rank (A340602), with strict case A117192.
A340692 counts partitions of odd rank (A340603), with strict case A117193.
Cf. A003114, A006141, A039900, A056239, A096401, A112798, A117409, A316413, A324517, A325134, A326845, A340828.

Select[Range[2,100],PrimePi[FactorInteger[#][[-1,1]]]>PrimeOmega[#]&]

For all terms A061395(a(n)) > A001222(a(n)).

A105805 Irregular triangle read by rows: T(n,k) is the Dyson's rank of the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 1, 0, -1, -3, 4, 2, 1, 0, -1, -2, -4, 5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5, 6, 4, 3, 2, 2, 1, 0, 0, 0, -1, -2, -2, -3, -4, -6, 7, 5, 4, 3, 2, 3, 2, 1, 1, 0, 1, 0, -1, -1, -2, -1, -2, -3, -3, -4, -5, -7, 8, 6, 5, 4, 3, 4, 3, 2, 1, 2, 1, 0, 2, 1, 0, 0, -1, -1, 0, -1, -2, -2, -3, -2, -3, -4, -4, -5, -6, -8, 9, 7, 6, 5, 4, 3, 5, 4, 3

Offset: 1

Wolfdieter Lang, Apr 28 2005

The rank of a partition is the largest part minus the number of parts.
Row lengths give A000041, n >= 1.
Just for n <= 6, row n is antisymmetric due to conjugation of partitions (see links under A105806): a(n, p(n)-(k-1)) = a(n,k), k = 1..floor(p(n)/2). [Comment corrected by Franklin T. Adams-Watters, Jan 17 2006]
First differs from A330368 at a(49) = T(7,5). - Omar E. Pol, Dec 31 2019

			Triangle begins:
  [0];
  [1, -1];
  [2,  0, -2];
  [3,  1,  0, -1, -3];
  [4,  2,  1,  0, -1, -2, -4];
  [5,  3,  2,  1,  1,  0, -1, -1, -2, -3, -5];
  ...
Row 3 for partitions of 3 in the mentioned order: 3,(1,2),1^3 with ranks 2,0,-2.
From _Wolfdieter Lang_, Jul 18 2013: (Start)
Row n = 7 is [6, 4, 3, 2, 2, 1, 0 , 0, 0, -1, -2, -2, -3, -4, -6].
This is also antisymmetric, but by accident, because a(7,7) = 0 for the partition (1,3^2), conjugate to (2^2,3) with a(7,8) = 0, and a(7,9) = 0 for (1^3,4) which is self-conjugate.
Row n=8 (see the link) is no longer antisymmetric. See the _Franklin T. Adams-Watters_ correction above. (End)

A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Freeman J. Dyson, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
Wolfdieter Lang, First 16 rows.

Cf. A000041, A036043, A049085, A209616 (sum of positive ranks), A330368 (another version).

# ASPrts is implemented in A119441
A105805 := proc(n,k)
    local pi;
    pi := ASPrts(n)[k] ;
    max(op(pi))-nops(pi) ;
end proc:
for n from 1 do
    for k from 1 to A000041(n) do
        printf("%d,",A105805(n,k)) ;
    end do:
    printf("\n") ;
end do: # R. J. Mathar, Jul 17 2013

a(n,k) = A049085(n,k) - A036043(n,k). - Alford Arnold, Aug 02 2010

Name clarified by Omar E. Pol, Dec 31 2019

A117195 Triangle read by rows: T(n,k) = number of partitions into distinct parts having rank k, 0<=k

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 2, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1, 0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1

Offset: 1

Reinhard Zumkeller, Mar 03 2006

T(n,0) = A010054(n), T(n,1) = 1-A010054(n) for n>1;

A000009(n) = Sum(T(n,k): 0<=k

A117192(n) = Sum(T(n,k)*(1 - k mod 2): 0<=k

A117193(n) = Sum(T(n,k)*(k mod 2): 0<=k

A117194(n) = Sum(T(n,k)*(1 - k mod 2): 0

			Triangle starts:
[ 1]   1,
[ 2]   0, 1,
[ 3]   1, 0, 1,
[ 4]   0, 1, 0, 1,
[ 5]   0, 1, 1, 0, 1,
[ 6]   1, 0, 1, 1, 0, 1,
[ 7]   0, 1, 1, 1, 1, 0, 1,
[ 8]   0, 1, 1, 1, 1, 1, 0, 1,
[ 9]   0, 1, 1, 2, 1, 1, 1, 0, 1,
[10]   1, 0, 2, 1, 2, 1, 1, 1, 0, 1,
[11]   0, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1,
[12]   0, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1,
[13]   0, 1, 1, 3, 2, 3, 2, 2, 1, 1, 1, 0, 1,
[14]   0, 1, 2, 2, 4, 2, 3, 2, 2, 1, 1, 1, 0, 1, ...
T(12,0) = #{} = 0,
T(12,1) = #{5+4+2+1} = 1,
T(12,2) = #{6+3+2+1, 5+4+3} = 2,
T(12,3) = #{6+5+1, 6+4+2} = 2,
T(12,4) = #{7+4+1, 7+3+2} = 2,
T(12,5) = #{8+3+1, 7+5} = 2,
T(12,6) = #{9+2+1, 8+4} = 2,
T(12,7) = #{9+3} = 1,
T(12,8) = #{10+2} = 1,
T(12,9) = #{11+1} = 1,
T(12,10) = #{} = 0,
T(12,11) = #{12} = 1.

Alois P. Heinz, Rows n = 1..141, flattened
Maria Monks, Number theoretic properties of generating functions related to Dyson's rank for partitions into distinct parts, Proceedings of The American Mathematical Society, vol.138, no.02, pp.481-494, 2009.

Cf. A063995, A105806.

b:= proc(n, i, k) option remember;
      if n<0 or k<0 then []
    elif n=0 then [0$k, 1]
    elif i<1 then []
    else zip ((x, y)-> x+y, b(n, i-1, k), b(n-i, i-1, k-1), 0)
      fi
    end:
T:= proc(n) local j, r; r:= [];
      for j from 0 to n do
        r:= zip ((x, y)-> x+y, r, b(n-j, j-1, j-1), 0)
      od; r[]
    end:
seq (T(n), n=1..20);  # Alois P. Heinz, Aug 29 2011

b[n_, i_, k_] := b[n, i, k] = Which[n<0 || k<0, {}, n == 0, Append[Array[0&, k], 1], i<1, {}, True, Plus @@ PadRight[{b[n, i-1, k], b[n-i, i-1, k-1]}]]; T[n_] := Module[{j, r}, r = {}; For[j = 0, j <= n, j++, r = Plus @@ PadRight[{r, b[n-j, j-1, j-1]}]]; r]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

N=33;  L=1+2*ceil(sqrtint(N));
q='q+O(q^N);
gf=sum(n=1,L, q^(n*(n+1)/2) / prod(k=1,n,1-z*q^k) );
v=Vec(gf);
{ for (n=1,#v,  /* print triangle: */
    p = Pol(v[n], 'z) + 'c0;
    p = polrecip(p);
    rw = Vec(p);  rw[1] -= 'c0;
    print1("[", n, "]   " );
    print( rw );
); }
/* Joerg Arndt, Oct 07 2012 */

G.f.: sum(n>=1, q^(n*(n+1)/2) / prod(k=1..n, 1-z*q^k) ), see Monks reference. [Joerg Arndt, Oct 07 2012]

A330368 Irregular triangle read by rows in which row n lists the ranks of the partitions of n in nonincreasing order.

Original entry on oeis.org

0, 1, -1, 2, 0, -2, 3, 1, 0, -1, -3, 4, 2, 1, 0, -1, -2, -4, 5, 3, 2, 1, 1, 0, -1, -1, -2, -3, -5, 6, 4, 3, 2, 2, 1, 0, 0, 0, -1, -2, -2, -3, -4, -6, 7, 5, 4, 3, 3, 2, 2, 1, 1, 1, 0, 0, -1, -1, -1, -2, -2, -3, -3, -4, -5, -7, 8, 6, 5, 4, 4, 3, 3, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, -1, -1, -1, -2, -2, -2, -3, -3, -4, -4, -5, -6, -8

Offset: 1

Omar E. Pol, Dec 12 2019

The rank of a partition is the largest part minus the number of parts.
For more about this ordering, see A330370.
First differs from A105805 at a(49) = T(7,5).

			Triangle begins:
                              0;
                            1, -1;
                          2,  0, -2;
                      3,  1,  0, -1, -3;
                  4,  2,  1,  0, -1, -2, -4;
          5,  3,  2,  1,  1,  0, -1, -1, -2, -3, -5;
  6,  4,  3,  2,  2,  1,  0,  0,  0, -1, -2, -2, -3, -4, -6;
...

Row n has length A000041(n).
Row sums give A000004.
Cf. A105805, A105806, A194547, A194549, A195822, A208478, A209616, A330370.

Edited by N. J. A. Sloane, Sep 15 2020

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cp's OEIS Frontend

A072233 Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.

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A064174 Number of partitions of n with nonnegative rank.

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A064173 Number of partitions of n with positive rank.

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A063995 Irregular triangle read by rows: T(n,k), n >= 1, -(n-1) <= k <= n-1, = number of partitions of n with rank k.

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A340692 Number of integer partitions of n of odd rank.

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A340788 Heinz numbers of integer partitions of negative rank.

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A340787 Heinz numbers of integer partitions of positive rank.

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