A194549 -id:A194549 - cp's OEIS frontend

A209616 Sum of positive Dyson's ranks of all partitions of n.

0, 1, 2, 4, 7, 12, 18, 29, 42, 63, 89, 128, 176, 246, 333, 453, 603, 807, 1058, 1393, 1807, 2346, 3011, 3867, 4915, 6248, 7879, 9926, 12421, 15529, 19297, 23954, 29585, 36486, 44802, 54937, 67096, 81831, 99459, 120700, 146026, 176410, 212512, 255636, 306734

Offset: 1

Omar E. Pol, Mar 10 2012

nonn

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

			For n = 5 we have:
--------------------------
Partitions        Dyson's
of 5               rank
--------------------------
5               5 - 1 =  4
4+1             4 - 2 =  2
3+2             3 - 2 =  1
3+1+1           3 - 3 =  0
2+2+1           2 - 3 = -1
2+1+1+1         2 - 4 = -2
1+1+1+1+1       1 - 5 = -4
--------------------------
The sum of positive Dyson's ranks of all partitions of 5 is 4+2+1 = 7 so a(5) = 7.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
G. E. Andrews, S. H. G. Chan, and B. Kim, The odd moments of ranks and cranks (See the function R_1), Journal of Combinatorial Theory, Series A, Volume 120, Issue 1, January 2013, Pages 77-91.
F. J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10-15.
Frank Garvan, Dyson's rank function and Andrews's SPT-function

Column 1 of triangle A208482.

Cf. A063995, A064174, A092269, A105805, A194547, A194549, A195822, A208478.

# Maple program based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
R1:=add((-1)^(n+1)*q^(n*(3*n+1)/2)/(1-q^n),n=1..M);
R1:=series(R1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

M = 101;
qinf = Product[1-q^i, {i, 1, M}];
qinf = Series[qinf, {q, 0, M}];
R1 = Sum[(-1)^(n+1) q^(n(3n+1)/2)/(1-q^n), {n, 1, M}];
R1 = Series[R1/qinf, {q, 0, M}];
CoefficientList[R1, q] // Rest (* Jean-François Alcover, Aug 18 2018, translated from Maple *)

my(N=50, x='x+O('x^N)); concat(0, Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k+1)/2)/(1-x^k)))) \\ Seiichi Manyama, May 21 2023

a(n) = A115995(n) - A195012(n). - Omar E. Pol, Apr 06 2012
G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+1)/2) / (1-x^k). - Seiichi Manyama, May 21 2023
a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (Pi*2^(3/2)*sqrt(n)). - Vaclav Kotesovec, Jul 06 2025

More terms from Alois P. Heinz, Mar 10 2012

A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 2, 1, 5, 2, 4, 4, 2, 1, 6, 3, 5, 6, 4, 2, 1, 10, 5, 7, 9, 7, 4, 2, 1, 13, 7, 9, 11, 11, 7, 4, 2, 1, 19, 11, 12, 15, 16, 12, 7, 4, 2, 1, 25, 16, 15, 19, 22, 18, 12, 7, 4, 2, 1, 35, 24, 20, 26, 29, 27, 19, 12, 7, 4, 2, 1

Offset: 1

Omar E. Pol, Mar 07 2012

We define the k-th rank of a partition as the k-th part minus the number of parts >= k. Every partition of n has n ranks. This is a generalization of the Dyson's rank of a partition which is the largest part minus the number of parts. Since the first part of a partition is also the largest part of the same partition so the Dyson's rank of a partition is the case for k = 1.
The sum of the k-th ranks of all partitions of n is equal to zero.
Also T(n,k) = number of partitions of n with negative k-th rank.
It appears that reversed rows converge to A000070, the same as A208482. - Omar E. Pol, Mar 11 2012
From Omar E. Pol, Dec 12 2019: (Start)
1) The k-th part of a partition of n is also the number of parts >= k of its conjugate partition.
2) The k-th rank of a partitions is also the number of parts >= k of its conjugate partition minus the number of parts >= k.
For example: for n = 9 consider the partition [5, 3, 1]. The first part is 5, so the conjugate partition has five parts >= 1. The second part is 3, so the conjugate partition has three parts >= 2. The third part is 1, so the conjugate partition has only one part >= 3. The mentioned conjugate partition is [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The first part is 3, so the conjugate partition has three parts >= 1. The second part is 2, so the conjugate partition has two parts >= 2. the Third part is 2, so the conjugate partition has two parts >= 3, and so on. In this case the conjugate partition is [5, 3, 1].
3) The difference between the k-th part and the (k+1)-st part of the partition of n is also the number of k's in its conjugate partition. For example: consider the partition [5, 3, 1]. The difference between the first and the second part is 5 - 3 = 2, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 3 - 1 = 2, equals the number of 2's in its conjugate partition. The difference between the third and the fourth (virtual) part is 1 - 0 = 1, equals the number of 3's in its conjugate partition [3, 2, 2, 1, 1]. And conversely, consider the partition [3, 2, 2, 1, 1]. The difference between the first and the second part is 3 - 2 = 1, equals the number of 1's in its conjugate partition. The difference between the second and the third part is 2 - 2 = 0, equals the number of 2's in its conjugate partition. The difference between the third and the fourth part is 2 - 1 = 1, equals the number of 3's in its conjugate partition, and so on.
4) The list of n ranks of a partition of n equals the list of n ranks multiplied by -1 of its conjugate partition. For example the nine ranks of the partition [5, 3, 1] of 9 are [2, 1, -1, -1, -1, -1, 0, 0, 0], and the nine ranks of its conjugate partition [3, 2, 2, 1, 1] are [-2, -1, 1, 1, 1, 1, 0, 0, 0].
For a list of partitions of the positive integers ordered by its k-th ranks see A330370. (End)

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The number of partitions of 4 with positive k-th ranks are 2, 1, 2, 1 so row 4 lists 2, 1, 2, 1.
Triangle begins:
   0;
   1,  1;
   1,  1,  1;
   2,  1,  2,  1;
   3,  1,  3,  2,  1;
   5,  2,  4,  4,  2,  1;
   6,  3,  5,  6,  4,  2,  1;
  10,  5,  7,  9,  7,  4,  2,  1;
  13,  7,  9, 11, 11,  7,  4,  2,  1;
  19, 11, 12, 15, 16, 12,  7,  4,  2,  1;
  25, 16, 15, 19, 22, 18, 12,  7,  4,  2,  1;
  35, 24, 20, 26, 29, 27, 19, 12,  7,  4,  2,  1;
  ...

Alois P. Heinz, Rows n = 1..44, flattened

Column 1 is A064173.
Row sums give A208479.
Cf. A063995, A105805, A181187, A194547, A194549, A195822, A208482, A208483, A209616, A330368, A330369, A330370.

More terms from Alois P. Heinz, Mar 11 2012

A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 4, 1, 2, 1, 7, 1, 3, 2, 1, 12, 2, 5, 4, 2, 1, 18, 3, 6, 6, 4, 2, 1, 29, 6, 9, 10, 7, 4, 2, 1, 42, 9, 11, 13, 11, 7, 4, 2, 1, 63, 16, 15, 19, 17, 12, 7, 4, 2, 1, 89, 24, 18, 25, 24, 18, 12, 7, 4, 2, 1, 128, 39, 24, 36, 34, 28, 19, 12, 7, 4, 2, 1

Offset: 1

Omar E. Pol, Mar 07 2012

For the definition of the k-th rank see A208478.
It appears that the sum of the k-th ranks of all partitions of n is equal to zero.
It appears that reversed rows converge to A000070, the same as A208478. - Omar E. Pol, Mar 10 2012

			For n = 4 the partitions of 4 and the four types of ranks of the partitions of 4 are
----------------------------------------------------------
Partitions    First      Second       Third      Fourth
of 4          rank        rank        rank        rank
----------------------------------------------------------
4           4-1 =  3    0-1 = -1    0-1 = -1    0-1 = -1
3+1         3-2 =  1    1-1 =  0    0-1 = -1    0-0 =  0
2+2         2-2 =  0    2-2 =  0    0-0 =  0    0-0 =  0
2+1+1       2-3 = -1    1-1 =  0    1-0 =  1    0-0 =  0
1+1+1+1     1-4 = -3    1-0 =  1    1-0 =  1    1-0 =  1
----------------------------------------------------------
The sums of positive k-th ranks of the partitions of 4 are 4, 1, 2, 1 so row 4 lists 4, 1, 2, 1.
Triangle begins:
0;
1,    1;
2,    1,  1;
4,    1,  2,  1;
7,    1,  3,  2,  1;
12,   2,  5,  4,  2,  1;
18,   3,  6,  6,  4,  2,  1;
29,   6,  9, 10,  7,  4,  2,  1;
42,   9, 11, 13, 11,  7,  4,  2,  1;
63,  16, 15, 19, 17, 12,  7,  4,  2,  1;
89,  24, 18, 25, 24, 18, 12,  7,  4,  2,  1;
128, 39, 24, 36, 34, 28, 19, 12,  7,  4,  2,  1;

Alois P. Heinz, Rows n = 1..44, flattened

Column 1 is A209616. Row sums give A208483.

Cf. A006128, A063995, A064173, A105805, A181187, A194547, A194549, A195822, A208478.

Terms a(1)-a(22) confirmed and additional terms added by John W. Layman, Mar 10 2012

A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n has length A000041(n). The sum of row n is equal to zero.

			Written as a triangle:
  0;
  -1,1;
  -2,0,2;
  -3,-1,1,0,3;
  -4,-2,0,-1,2,1,4;
  -5,-3,-1,-2,1,0,3,-1,2,1,5;
  -6,-4,-2,-3,0,-1,2,-2,1,0,4,0,3,2,6;
  -7,-5,-3,-4,-1,-2,1,-3,0,-1,3,-1,2,1,5,-2,1,0,4,3,2,7;

Alois P. Heinz, Rows n = 1..26, flattened

Cf. A105805, A135010, A138121, A194546, A194548, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, i-t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      l:= NULL; b(n, 1, 0); l
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Dec 22 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] := If [n0==0, l = Append[l, i-t], If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]]; l = {}; b[n, 1, 0]; l];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

a(n) = A194546(n) - A193173(n).

More terms from Alois P. Heinz, Dec 22 2011

A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 2, 2, 1, 4, 3, 3, 2, 2, 2, 1, 4, 3, 3, 2, 3, 2, 2, 1, 5, 4, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1, 5, 4, 4, 3, 4, 3, 3, 2, 3, 3, 2, 2, 2, 1, 6, 5, 5, 4, 4, 4, 3, 4, 3, 3, 3, 2, 4, 3, 3, 2, 3, 2, 2, 2, 1, 6, 5, 5, 4, 5, 4, 4, 3, 4, 4, 3, 3, 3, 2, 4, 3, 3, 3, 2, 3, 2, 2, 2, 1

Offset: 1

Omar E. Pol, Dec 11 2011

			Written as a triangle:
  0;
  1;
  1;
  2,1;
  2,1;
  3,2,2,1;
  3,2,2,1;
  4,3,3,2,2,2,1;
  4,3,3,2,3,2,2,1;
  5,4,4,3,3,3,2,3,2,2,2,1;
  5,4,4,3,4,3,3,2,3,3,2,2,2,1;
  6,5,5,4,4,4,3,4,3,3,3,2,4,3,3,2,3,2,2,2,1;
  6,5,5,4,5,4,4,3,4,4,3,3,3,2,4,3,3,3,2,3,2,2,2,1;

Alois P. Heinz, Rows n = 1..33, flattened
Tilman Piesk, Table for A194602, showing the non-one addends.

Row sums give A138135. Row n has length A187219(n).

Cf. A002865, A135010, A138121, A193173, A194546, A194547, A194549.

T:= proc(n) local b, l;
      b:= proc(n, i, t)
            if n=0 then l:=l, t
          elif i>n then
          else b(n-i, i, t+1); b(n, i+1, t)
            fi
          end;
      if n<2 then 0 else l:= NULL; b(n, 2, 0); l fi
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Dec 19 2011

T[n_] := Module[{b, l}, b[n0_, i_, t_] :=
     If[n0==0, l = Append[l, t],
     If[i>n0, , b[n0-i, i, t+1]; b[n0, i+1, t]]];
     If[n<2, {0}, l = {}; b[n, 2, 0]; l]];
Table[T[n], {n, 1, 15}]  // Flatten (* Jean-François Alcover, Mar 05 2021, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 19 2011

A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Dec 10 2011

Row n lists the first A000041(n) terms of A141285.

The representation of the partitions (for fixed n) is as (weakly) decreasing lists of parts, the order between individual partitions (for the same n) is co-lexicographic, see example. - Joerg Arndt, Sep 13 2013

			For n = 5 the partitions of 5 in colexicographic order are:
  1+1+1+1+1
  2+1+1+1
  3+1+1
  2+2+1
  4+1
  3+2
  5
so the fifth row is the largest in each partition: 1,2,3,2,4,3,5
Triangle begins:
  1;
  1,2;
  1,2,3;
  1,2,3,2,4;
  1,2,3,2,4,3,5;
  1,2,3,2,4,3,5,2,4,3,6;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7;
  1,2,3,2,4,3,5,2,4,3,6,3,5,4,7,2,4,3,6,5,4,8;
...

Wikiversity, Lexicographic and colexicographic order

The sum of row n is A006128(n).
Cf. A135010, A138121, A141285, A194547, A194548, A194549.
Row lengths are A000041.
Let y be the n-th integer partition in colexicographic order (A211992):
- The maximum of y is a(n).
- The length of y is A193173(n).
- The minimum of y is A196931(n).
- The Heinz number of y is A334437(n).
Lexicographically ordered reversed partitions are A026791.
Reverse-colexicographically ordered partitions are A026792.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Reverse-lexicographically ordered partitions are A080577.
Lexicographically ordered partitions are A193073.
Cf. A036037, A063008, A115623, A228100, A228531, A238966, A334301.

colex[f_,c_]:=OrderedQ[PadRight[{Reverse[f],Reverse[c]}]];
Max/@Join@@Table[Sort[IntegerPartitions[n],colex],{n,8}] (* Gus Wiseman, May 31 2020 *)

a(n) = A061395(A334437(n)). - Gus Wiseman, May 31 2020

Definition corrected by Omar E. Pol, Sep 12 2013

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

A330374 Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n.

Original entry on oeis.org

1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2, 1, 4, 2, 2, 0, 2, 3, 2, 4, 2, 2, 0, 2, 2, 6, 4, 4, 2, 2, 0, 2, 4, 6, 6, 4, 4, 2, 2, 0, 2, 4, 10, 6, 8, 4, 4, 2, 2, 0, 2, 6, 10, 12, 6, 8, 4, 4, 2, 2, 0, 2, 7, 16, 12, 12, 8, 8, 4, 4, 2, 2, 0, 2, 11, 16, 18, 14, 12, 8, 8, 4, 4, 2, 2, 0, 2, 11, 26, 20, 20, 14, 14

Offset: 1

Omar E. Pol, Dec 18 2019

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

Since the largest part of a partition equals the number of parts of its conjugate partition, so the rank of a partition also is equal to the difference between the number of parts of its conjugate partition and the number of parts of the partition.

			Triangle begins:
--------------------------------------------------------------------
  n \ k   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14
--------------------------------------------------------------------
[ 1]      1;
[ 2]      0,  2;
[ 3]      1,  0,  2;
[ 4]      1,  2,  0,  2;
[ 5]      1,  2,  2,  0,  2;
[ 6]      1,  4,  2,  2,  0,  2;
[ 7]      3,  2,  4,  2,  2,  0,  2;
[ 8]      2,  6,  4,  4,  2,  2,  0,  2;
[ 9]      4,  6,  6,  4,  4,  2,  2,  0,  2;
[10]      4, 10,  6,  8,  4,  4,  2,  2,  0,  2;
[11]      6, 10, 12,  6,  8,  4,  4,  2,  2,  0,  2;
[12]      7, 16, 12, 12,  8,  8,  4,  4,  2,  2,  0,  2;
[13]     11, 16, 18, 14, 12,  8,  8,  4,  4,  2,  2,  0,  2;
[14]     11, 26, 20, 20, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
[15]     16, 28, 30, 22, 22, 14, 14,  8,  8,  4,  4,  2,  2,  0,  2;
...

Row sums give A000041, n >= 1.
Leading diagonal gives A040000.
Second diagonal gives A000004.
Column k=0 is A047993.
Cf. A063995, A105805, A194547, A194549, A195822, A208478, A209616, A330368.

T(n,k) = A063995(n,k)*A040000(k), 0 <= k < n.

cp's OEIS Frontend

A209616 Sum of positive Dyson's ranks of all partitions of n.

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A208478 Triangle read by rows: T(n,k) = number of partitions of n with positive k-th rank.

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A208482 Triangle read by rows: T(n,k) = sum of positive k-th ranks of all partitions of n.

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A194547 Triangle read by rows: T(n,k) = Dyson's rank of the k-th partition of n, with partitions in lexicographic order.

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A194548 Triangle read by rows: T(n,k) = number of parts in the k-th partition of n that does not contain 1 as a part, with partitions in lexicographic order.

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A194546 Triangle read by rows: T(n,k) is the largest part of the k-th partition of n, with partitions in colexicographic order.

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

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A330374 Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n.

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