A097364 -id:A097364 - cp's OEIS frontend

A218572 Number of partitions p of n such that max(p)-min(p) = 9.

1, 1, 3, 3, 7, 8, 14, 18, 28, 35, 53, 66, 92, 117, 157, 196, 259, 319, 411, 507, 638, 777, 970, 1171, 1438, 1728, 2098, 2501, 3012, 3563, 4251, 5008, 5923, 6931, 8152, 9486, 11078, 12835, 14900, 17177, 19844, 22768, 26169, 29916, 34219, 38954, 44387, 50338

Offset: 11

Alois P. Heinz, Nov 02 2012

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..1000
G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014.

terms = 48; offset = 11; max = terms + offset; s[k0_ /; k0 > 0] := Sum[x^(2*k + k0)/Product[ (1 - x^(k + j)), {j, 0, k0}], {k, 1, Ceiling[max/2]}] + O[x]^max // CoefficientList[#, x] &; Drop[s[9], offset] (* Jean-François Alcover, Sep 11 2017, after Alois P. Heinz *)

G.f.: Sum_{k>0} x^(2*k+9)/Product_{j=0..9} (1-x^(k+j)).

a(n) = A097364(n,9) = A116685(n,9) = A194621(n,9) - A194621(n,8) = A218511(n) - A218510(n).

A244966 Triangle read by rows: T(n,k) is the difference between the largest and the smallest part of the k-th partition in the list of colexicographically ordered partitions of n, with n>=1 and 1<=k<=p(n), where p(n) is the number of partitions of n.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 18 2014

The number of t's in row n gives A097364(n,t), with n>=1 and 0<=t

Rows converge to A244967, which is A141285 - 1.

Row n has length A000041(n).

Row sums give A116686.

			Triangle begins:
0;
0, 0;
0, 1, 0;
0, 1, 2, 0, 0;
0, 1, 2, 1, 3, 1, 0;
0, 1, 2, 1, 3, 2, 4, 0, 2, 0, 0;
0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 1, 3, 1, 0;
0, 1, 2, 1, 3, 2, 4, 1, 3, 2, 5, 2, 4, 3, 6, 0, 2, 1, 4, 2, 0, 0;
...
For n = 6 we have:
--------------------------------------------------------
.                        Largest  Smallest   Difference
k    Partition of 6        part     part       T(6,k)
--------------------------------------------------------
1:  [1, 1, 1, 1, 1, 1]      1    -    1     =     0
2:  [2, 1, 1, 1, 1]         2    -    1     =     1
3:  [3, 1, 1, 1]            3    -    1     =     2
4:  [2, 2, 1, 1]            2    -    1     =     1
5:  [4, 1, 1]               4    -    1     =     3
6:  [3, 2, 1]               3    -    1     =     2
7:  [5, 1]                  5    -    1     =     4
8:  [2, 2, 2]               2    -    2     =     0
9:  [4, 2]                  4    -    2     =     2
10: [3, 3]                  3    -    3     =     0
11: [6]                     6    -    6     =     0
--------------------------------------------------------
So the 6th row of triangle is [0,1,2,1,3,2,4,0,2,0,0] and the row sum is A116686(6) = 15.
Note that in the 6th row there are four 0's so A097364(6,0) = 4, there are two 1's so A097364(6,1) = 2, there are three 2's so A097364(6,2) = 3, there is only one 3 so A097364(6,3) = 1, there is only one 4 so A097364(6,4) = 1 and there are no 5's so A097364(6,5) = 0.

G. E. Andrews, M. Beck and N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv:1406.3374 [math.NT], 2014

Cf. A000005, A000041, A008805, A049820, A097364, A116686, A128508, A135010, A141285, A196931, A218567-A218573, A244967.

T(n,k) = A141285(k) - A196931(n,k), n>=1, 1<=k<=A000041(n).

A244967 A141285(n) - 1.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jul 18 2014

nonn

The rows of the triangle in A244966 converge to this sequence.

Cf. A000041, A097364, A135010, A141285, A244966.

A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 3, 2, 2, 0, 7, 0, 3, 6, 10, 0, 13, 0, 17, 10, 5, 0, 40, 12, 6, 18, 34, 0, 62, 0, 50, 24, 8, 60, 125, 0, 9, 32, 169, 0, 165, 0, 95, 176, 11, 0, 373, 114, 198, 54, 143, 0, 384, 254, 574, 66, 14, 0, 1090, 0, 15, 748, 633, 448, 782, 0, 286

Offset: 1

Gus Wiseman, Apr 10 2023

nonn

In terms of partition diagrams, these are partitions whose rectangle from the left (length times minimum) has the same size as the complement.

			The a(4) = 1 through a(12) = 7 partitions:
  (31)  .  (321)  .  (62)    (441)  (32221)  .  (93)
                     (3221)  (522)  (33211)     (642)
                     (3311)                     (4431)
                                                (5322)
                                                (322221)
                                                (332211)
                                                (333111)
The partition y = (4,4,3,1) has maximum 4 and minimum 1 and mean 3, and 4 - 1 = 3, so y is counted under a(12). The diagram of y is:
  o o o o
  o o o o
  o o o .
  o . . .
Both the rectangle from the left and the complement have size 4.

Positions of zeros are 1 and A000040.
For length instead of mean we have A237832.
For minimum instead of mean we have A118096.
These partitions have ranks A362047.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, A058398 by mean.
A067538 counts partitions with integer mean.
A097364 counts partitions by (maximum) - (minimum).
A243055 subtracts the least prime index from the greatest.
A326844 gives the diagram complement size of Heinz partition.
Cf. A237984, A240219, A326836, A326837, A327482, A237755, A237824, A349156, A359360, A360068, A360241, A361853.

Table[Length[Select[IntegerPartitions[n],Max@@#-Min@@#==Mean[#]&]],{n,30}]

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A218572 Number of partitions p of n such that max(p)-min(p) = 9.

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A244966 Triangle read by rows: T(n,k) is the difference between the largest and the smallest part of the k-th partition in the list of colexicographically ordered partitions of n, with n>=1 and 1<=k<=p(n), where p(n) is the number of partitions of n.

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A244967 A141285(n) - 1.

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A361862 Number of integer partitions of n such that (maximum) - (minimum) = (mean).

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