A101200 -id:A101200 - cp's OEIS frontend

A101198 Number of partitions of n with rank 1 (the rank of a partition is the largest part minus the number of parts).

0, 1, 0, 1, 1, 2, 1, 3, 3, 5, 5, 8, 8, 13, 14, 20, 23, 31, 35, 48, 55, 72, 84, 108, 126, 160, 187, 233, 275, 340, 398, 489, 574, 697, 819, 988, 1158, 1390, 1627, 1941, 2271, 2696, 3145, 3721, 4335, 5104, 5938, 6967, 8088, 9462, 10964, 12783

Offset: 1

Emeric Deutsch, Dec 12 2004

nonn

Column k=1 in the triangle A063995.

			a(6)=2 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000041, A063995.

Cf. A101198-A101200, A101707-A101709.

with(combinat): for n from 1 to 35 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=1 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..35);

Table[Count[IntegerPartitions[n],?(Max[#]-Length[#]==1&)],{n,60}] (* _Harvey P. Dale, Nov 29 2014 *)

G.f. for the number of partitions of n with rank r is Sum((-1)^k*x^(r*k)*(x^((3*k^2+k)/2)-x^((3*k^2-k)/2)), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
Also Sum(x^(2*n+r+1)*Product((1-x^(2*n+r+1-k))/(1-x^k),k=1..n),n=0..infinity). - Vladeta Jovovic, May 05 2008
a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023

A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 4, 2, 7, 6, 13, 11, 22, 22, 38, 39, 63, 69, 103, 114, 165, 189, 262, 301, 407, 475, 626, 733, 950, 1119, 1427, 1681, 2118, 2503, 3116, 3678, 4539, 5360, 6559, 7735, 9400, 11076, 13372, 15728, 18886, 22184, 26501, 31067, 36947, 43242, 51210, 59818, 70576, 82291, 96750

Offset: 0

Emeric Deutsch, Dec 12 2004

nonn

a(n) + A101708(n) = A064173(n).

			a(7)=2 because the only partitions of 7 with positive odd rank are 421 (rank=1) and 52 (rank=3).
From _Gus Wiseman_, Feb 07 2021: (Start)
Also the number of integer partitions of n into an even number of parts, the greatest of which is odd. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot) are:
  11   .  31     32   33       52     53         54       55
          1111        51       3211   71         72       73
                      3111            3221       3222     91
                      111111          3311       3321     3322
                                      5111       5211     3331
                                      311111     321111   5221
                                      11111111            5311
                                                          7111
                                                          322111
                                                          331111
                                                          511111
                                                          31111111
                                                          1111111111
Also the number of integer partitions of n into an odd number of parts, the greatest of which is even. For example, the a(2) = 1 through a(10) = 13 partitions (empty column indicated by dot, A = 10) are:
  2   .  4     221   6       421     8         432       A
         211         222     22111   422       441       433
                     411             431       621       442
                     21111           611       22221     622
                                     22211     42111     631
                                     41111     2211111   811
                                     2111111             22222
                                                         42211
                                                         43111
                                                         61111
                                                         2221111
                                                         4111111
                                                         211111111
(End)

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
FindStat, St000145: The Dyson rank of a partition

Note: A-numbers of ranking sequences are in parentheses below.
The even-rank version is A101708 (A340605).
The even- but not necessarily positive-rank version is A340601 (A340602).
The Heinz numbers of these partitions are (A340604).
Allowing negative odd ranks gives A340692 (A340603).
- Rank -
A047993 counts balanced (rank zero) partitions (A106529).
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).
A257541 gives the rank of the partition with Heinz number n.
- Odd -
A000009 counts partitions into odd parts (A066208).
A026804 counts partitions whose least part is odd.
A027193 counts partitions of odd length/maximum (A026424/A244991).
A058695 counts partitions of odd numbers (A300063).
A339890 counts factorizations of odd length.
A340385 counts partitions of odd length and maximum (A340386).
Cf. A000041, A027187, A101709, A101199, A101200, A117409, A200750.

b:= proc(n, i, r) option remember; `if`(n=0, max(0, r),
      `if`(i<1, 0, b(n, i-1, r) +b(n-i, min(n-i, i), 1-
      `if`(r<0, irem(i, 2), r))))
    end:
a:= n-> b(n$2, -1)/2:
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 29 2021

Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&OddQ[Max[#]]&]],{n,0,30}] (* Gus Wiseman, Feb 10 2021 *)
b[n_, i_, r_] := b[n, i, r] = If[n == 0, Max[0, r],
     If[i < 1, 0, b[n, i - 1, r] + b[n - i, Min[n - i, i], 1 -
     If[r < 0, Mod[i, 2], r]]]];
a[n_] := b[n, n, -1]/2;
a /@ Range[0, 55] (* Jean-François Alcover, May 23 2021, after Alois P. Heinz *)

a(n) = (A000041(n) - A000025(n))/4. - Vladeta Jovovic, Dec 14 2004
G.f.: Sum((-1)^(k+1)*x^((3*k^2+k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004
a(n) = A340692(n)/2. - Gus Wiseman, Feb 07 2021

More terms from Joerg Arndt, Oct 07 2012

a(0)=0 prepended by Alois P. Heinz, Jan 29 2021

A101708 Number of partitions of n having positive even rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 4, 3, 7, 6, 14, 13, 23, 24, 41, 43, 67, 75, 111, 126, 177, 204, 282, 328, 437, 514, 674, 793, 1021, 1207, 1533, 1814, 2273, 2691, 3344, 3956, 4865, 5754, 7027, 8296, 10060, 11864, 14302, 16836, 20183, 23715, 28301, 33191, 39423, 46152, 54607, 63794, 75200, 87687, 103018

Offset: 0

Emeric Deutsch, Dec 12 2004

nonn

			a(7)=4 because the only partitions of 7 with positive even rank are 7 (rank=6), 61 (rank=4), 511 (rank=2) and 43 (rank=2).

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Cf. A000041, A101707, A064173.

Cf. A101198-A101200, A101709.

Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],?(EvenQ[#] && Positive[#]&)], {n,50}] (* _Harvey P. Dale, Feb 26 2012 *)

G.f.: Sum((-1)^(k+1)*x^((3*k^2+3*k)/2)/(1+x^k), k>=1)/Product(1-x^k, k>=1). - Vladeta Jovovic, Dec 20 2004

a(n) = A064173(n) - A101707(n) for n >= 1.

More terms from Joerg Arndt, Oct 07 2012

Offset changed to 0 by Georg Fischer, Dec 23 2023

A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 3, 1, 2, 1, 1, 0, 1, 2, 3, 2, 2, 1, 1, 0, 1, 4, 3, 3, 2, 2, 1, 1, 0, 1, 4, 5, 3, 4, 2, 2, 1, 1, 0, 1, 6, 5, 6, 3, 4, 2, 2, 1, 1, 0, 1, 7, 8, 6, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 8, 9, 7, 6, 4, 4, 2, 2, 1, 1, 0, 1, 11, 13, 10, 10, 7, 7, 4, 4, 2, 2, 1, 1, 0, 1

Offset: 1

Wolfdieter Lang, Mar 11 2005

The array with all ranks (including negative ones) is A063995.
a(n,-r)=a(n,r) for negative rank -r with r from 1,2,...,n-1 (due to conjugation of partitions of n; see the link).
Dyson's rank of a partition of n is the maximal part minus the number of parts, i.e. the number of columns minus the number of rows of the Ferrers diagram (see the link) of the partition.

			Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  1, 1, 1, 0, 1;
  1, 2, 1, 1, 0, 1; ...
Row 6, second entry is 2 because there are 2 partitions of n=6 with rank r=2-1=1, namely (3^2) and (1^2,4).
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 A063995, the right-hand triangle is the present sequence. - _N. J. A. Sloane_, Jan 23 2020

Lars Blomberg, Table of n, a(n) for n = 1..5050
Wolfdieter Lang, First 16 rows.
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, Problems for solution nr. 4261, Am. Math. Month. 54 (1947) 418.
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.
Eric Weisstein's World of Mathematics, Conjugation of partitions of n.
Eric Weisstein's World of Mathematics, Ferrers diagram.

For the full triangle see A063995.
Columns for r=0..5 are given in A047993, A101198, A101199, A101200, A363213, A363214.
Row sums = A064174.

a(n, r)= number of partitions of n with rank r, with r from 0, 1, ..., n-1.

G.f. of column r: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(r*k) * ( x^(k*(3*k-1)/2) - x^(k*(3*k+1)/2) ). - Seiichi Manyama, May 21 2023

A101709 Number of partitions of n having nonnegative even rank (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 7, 5, 11, 10, 20, 20, 34, 35, 57, 62, 92, 104, 151, 171, 237, 274, 371, 433, 571, 670, 870, 1025, 1306, 1543, 1947, 2299, 2864, 3387, 4183, 4943, 6052, 7143, 8688, 10242, 12371, 14566, 17503, 20567, 24583, 28841, 34319, 40188, 47618, 55654, 65700, 76643, 90149, 104968

Offset: 1

Emeric Deutsch, Dec 12 2004

nonn

A101709(n)=A101707(n)+A047993(n). A000041(n)=2*A101707(n)+A101708(n)+A101709(n).

			a(5)=3 because the partitions of 5 with nonnegative even ranks are 5 (rank=4), 41 (rank=2) and 311 (rank=0).

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Cf. A000041, A101707, A101708, A047993.

Cf. A101198-A101200.

G.f.: Sum((-1)^(k+1)*x^((3*k^2-k)/2)/(1+x^k), k=1..infinity)/Product(1-x^k, k=1..infinity). - Vladeta Jovovic, Dec 20 2004

More terms, Joerg Arndt, Oct 07 2012

A101199 Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 2, 2, 3, 3, 6, 6, 9, 10, 15, 16, 23, 27, 36, 42, 55, 64, 84, 98, 124, 147, 185, 217, 270, 318, 391, 461, 562, 661, 802, 942, 1132, 1331, 1592, 1864, 2220, 2597, 3077, 3593, 4240, 4940, 5811, 6758, 7916, 9192, 10737, 12438, 14488, 16755, 19459, 22465, 26024, 29987

Offset: 1

Emeric Deutsch, Dec 12 2004

nonn

Column k=2 in the triangle A063995.

			a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.

George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000041, A063995, A101198, A101200.

with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=2 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);

Table[Count[Max[#]-Length[#]&/@IntegerPartitions[n],2],{n,60}] (* Harvey P. Dale, Dec 22 2018 *)

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3 * 2^(9/2) * n^(3/2)). - Vaclav Kotesovec, May 26 2023

More terms from Joerg Arndt, Oct 07 2012

A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 120, 154, 201, 256, 330, 415, 529, 662, 833, 1035, 1293, 1595, 1976, 2425, 2982, 3640, 4449, 5401, 6565, 7935, 9592, 11543, 13891, 16645, 19943, 23808, 28408, 33792, 40172, 47619, 56413, 66661, 78708, 92724, 109149, 128213, 150486, 176293

Offset: 1

Seiichi Manyama, May 22 2023

nonn

			a(6) = 2 counts these partitions: 6, 5+1.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

With rank r or higher: A064174 (r=0), A064173 (r=1), A123975 (r=2), this sequence (r=3), A363231 (r=4).

Cf. A000041, A101200, A115067.

a(n) = sum(k=1, sqrtint(n), (-1)^(k-1)*numbpart(n-k*(3*k+5)/2));

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^(k-1) * x^(k*(3*k+5)/2).
a(n) = p(n-4) - p(n-11) + p(n-21) - ... + (-1)^(k-1) * p(n-k*(3*k+5)/2) + ..., where p() is A000041().
a(n) ~ exp(Pi*sqrt(2*n/3)) / (8*n*sqrt(3)) * (1 - (1/(2*Pi) + 31*Pi/144) / sqrt(n/6)). - Vaclav Kotesovec, May 26 2023

cp's OEIS Frontend

A101198 Number of partitions of n with rank 1 (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A101707 Number of partitions of n having positive odd rank (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A101708 Number of partitions of n having positive even rank (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

Extensions

A105806 Triangle of number of partitions of n with nonnegative Dyson rank r=0,1,...,n-1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A101709 Number of partitions of n having nonnegative even rank (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

Extensions

A101199 Number of partitions of n with rank 2 (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A363230 Number of partitions of n with rank 3 or higher (the rank of a partition is the largest part minus the number of parts).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs