A091605 -id:A091605 - cp's OEIS frontend

A091602 Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times.

1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 4, 3, 2, 1, 0, 1, 5, 4, 3, 1, 1, 0, 1, 6, 7, 3, 3, 1, 1, 0, 1, 8, 8, 6, 3, 2, 1, 1, 0, 1, 10, 12, 7, 5, 3, 2, 1, 1, 0, 1, 12, 15, 11, 6, 5, 2, 2, 1, 1, 0, 1, 15, 21, 14, 10, 5, 5, 2, 2, 1, 1, 0, 1, 18, 26, 20, 12, 9, 5, 4, 2, 2, 1, 1, 0, 1, 22, 35, 25, 18, 11, 8, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 1

Christian G. Bower, Jan 23 2004

From Gary W. Adamson, Mar 13 2010: (Start)
The triangle by rows = finite differences starting from the top, of an array in which row 1 = p(x)/p(x^2), row 2 = p(x)/p(x^3), ... row k = p(x)/p(x^k); such that p(x) = polcoeff A000041: (1 + x + 2x^2 + 3x^3 + 5x^4 + 7x^5 + ...)
Note that p(x)/p(x^2) = polcoeff A000009: (1 + x + x^2 + 2x^3 + 2x^4 + ...).
Refer to the example. (End)

			Triangle starts:
   1:  1;
   2:  1,  1;
   3:  2,  0,  1;
   4:  2,  2,  0,  1;
   5:  3,  2,  1,  0,  1;
   6:  4,  3,  2,  1,  0,  1;
   7:  5,  4,  3,  1,  1,  0,  1;
   8:  6,  7,  3,  3,  1,  1,  0,  1;
   9:  8,  8,  6,  3,  2,  1,  1,  0,  1;
  10: 10, 12,  7,  5,  3,  2,  1,  1,  0,  1;
  11: 12, 15, 11,  6,  5,  2,  2,  1,  1,  0,  1;
  12: 15, 21, 14, 10,  5,  5,  2,  2,  1,  1,  0,  1;
  13: 18, 26, 20, 12,  9,  5,  4,  2,  2,  1,  1,  0,  1;
  14: 22, 35, 25, 18, 11,  8,  5,  4,  2,  2,  1,  1,  0,  1;
  ...
In the partition 5+2+2+2+1+1, 2 is repeated 3 times, no part is repeated more than 3 times.
From _Gary W. Adamson_, Mar 13 2010: (Start)
First few rows of the array =
  ...
  1, 1, 1, 2, 2, 3,  4,  5,  6,  8, 10, ... = p(x)/p(x^2) = A000009
  1, 1, 2, 2, 4, 5,  7,  9, 13, 16, 22, ... = p(x)/p(x^3)
  1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, ... = p(x)/p(x^4)
  1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ... = p(x)/p(x^5)
  1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ... = p(x)/p(x^6)
  ...
Finally, taking finite differences from the top and deleting the first "1", we obtain triangle A091602 with row sums = A000041 starting with offset 1:
  1;
  1, 1;
  2, 0, 1;
  2, 2, 0, 1;
  3, 2, 1, 0, 1;
  4, 3, 2, 1, 0, 1;
  ...
(End)

Alois P. Heinz, Rows n = 1..141, flattened

Row sums: A000041. Inverse: A091603. Square: A091604.
Columns 1-6: A000009, A091605-A091609. Convergent of columns: A002865.
Cf. A000009. - Gary W. Adamson, Mar 13 2010
T(2n,n) gives: A232697.
Cf. A213177, A264397.

g:=sum(t^k*(product((1-x^((k+1)*j))/(1-x^j),j=1..50)-product((1-x^(k*j))/(1-x^j),j=1..50)),k=1..50): gser:=simplify(series(g,x=0,20)): for n from 1 to 13 do P[n]:=coeff(gser,x^n) od: for n from 1 to 13 do seq(coeff(P[n],t^j),j=1..n) od;
# yields sequence in triangular form - Emeric Deutsch, Mar 30 2006
b:= proc(n, i, k) option remember; `if`(n=0, 1,
      `if`(i>n, 0, add(b(n-i*j, i+1, min(k,
       iquo(n-i*j, i+1))), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, 1, k) -`if`(k=0, 0, b(n, 1, k-1)):
seq(seq(T(n, k), k=1..n), n=1..20);
# Alois P. Heinz, Nov 27 2013

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i>n, 0, Sum[b[n-i*j, i+1, Min[k, Quotient[n-i*j, i+1]]], {j, 0, Min[n/i, k]}]]]; t[n_, k_] := b[n, 1, k] - If[k == 0, 0, b[n, 1, k-1]]; Table[t[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 17 2014, after Alois P. Heinz's second Maple program *)

G.f.: G = G(t,x) = sum(k>=1, t^k*(prod(j>=1, (1-x^((k+1)*j))/(1-x^j) ) -prod(j>=1, (1-x^(k*j))/(1-x^j) ) ) ). - Emeric Deutsch, Mar 30 2006

Sum_{k=1..n} k * T(n,k) = A264397(n). - Alois P. Heinz, Nov 20 2015

A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

Original entry on oeis.org

0, 1, 0, 0, 1, 1, 0, 3, 1, 3, 1, 2, 0, 7, 2, 6, 7, 11, 3, 19, 8, 22, 16, 32, 17, 48, 21, 50, 39, 71, 35, 101, 58, 120, 89, 156, 97, 228, 133, 267, 203, 352, 228, 483, 322, 571, 444, 734, 524, 989, 683, 1160, 942, 1490, 1103, 1919, 1438, 2302, 1890, 2881, 2243, 3683, 2842, 4384, 3703, 5461

Offset: 0

Gus Wiseman, Mar 03 2025

nonn

Are there only 4 zeros?

			The partition (3,2,2,1,1,1,1,1,1) has greatest multiplicity 6 and distinct parts (3,2,1) with sum 6, so is counted under a(13).
The a(1) = 1 through a(13) = 7 partitions:
  1  .  .  22  2111  .  2221   22211  333     331111  5111111   .  33331
                        22111         222111          32111111     322222
                        31111         411111                       3331111
                                                                   4411111
                                                                   61111111
                                                                   322111111
                                                                   421111111

For greatest part instead of multiplicity we have A000005.
Counting partitions by the LHS gives A091602, rank statistic A051903.
Counting partitions by the RHS gives A116861, rank statistic A066328.
These partitions are ranked by A381632, for part instead of multiplicity A246655.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts balanced partitions, ranks A106529.
A091605 counts partitions with greatest multiplicity 2.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
Cf. A027193, A047966, A048767, A051904, A212166, A237984, A239455, A241131, A362608, A363724.

Table[Length[Select[IntegerPartitions[n],Max@@Length/@Split[#]==Total[Union[#]]&]],{n,0,30}]

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 2, 4, 5, 8, 6, 15, 13, 19, 25, 33, 36, 54, 58, 80, 96, 122, 141, 188, 217, 274, 326, 408, 474, 600, 695, 859, 1012, 1233, 1440, 1763, 2050, 2475, 2899, 3476, 4045, 4850, 5630, 6695, 7797, 9216, 10689, 12628, 14611, 17162, 19875, 23253

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(3) = 1 through a(10) = 8 partitions:
  (21)  (31)  (41)  (51)    (61)   (71)    (81)      (91)
                    (321)   (421)  (431)   (531)     (541)
                    (2211)         (521)   (621)     (631)
                                   (3311)  (32211)   (721)
                                           (222111)  (4321)
                                                     (4411)
                                                     (33211)
                                                     (42211)

First differences of A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A360014.
Equal case of A381544 (ranks A381439).
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A051904, A091605, A116861, A237984, A239455, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]==Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 9, 12, 16, 24, 30, 41, 56, 72, 94, 124, 158, 205, 262, 331, 419, 531, 663, 829, 1033, 1281, 1581, 1954, 2393, 2936, 3584, 4366, 5300, 6433, 7764, 9374, 11277, 13548, 16225, 19425, 23166, 27623, 32842, 39004, 46212, 54719, 64610, 76251

Offset: 0

Gus Wiseman, Apr 05 2025

nonn

			The a(2) = 1 through a(9) = 12 partitions:
  (2)  (3)  (4)   (5)    (6)    (7)     (8)      (9)
            (22)  (32)   (33)   (43)    (44)     (54)
                  (221)  (42)   (52)    (53)     (63)
                         (222)  (322)   (62)     (72)
                                (331)   (332)    (333)
                                (2221)  (422)    (432)
                                        (2222)   (441)
                                        (3221)   (522)
                                        (22211)  (3222)
                                                 (3321)
                                                 (4221)
                                                 (22221)

The complement (greater than or equal to) is A241131 except first, ranks A360015.
The opposite version (greater than) is A241131 shifted except first, ranks A360013.
These partitions have ranks A382856, complement A360015.
The weak version (less than or equal to) is A381544, ranks A381439.
For equality we have A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]

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A091602 Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times.

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A381079 Number of integer partitions of n whose greatest multiplicity is equal to their sum of distinct parts.

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A381544 Number of integer partitions of n not containing more ones than any other part.

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A382303 Number of integer partitions of n with exactly as many ones as the next greatest multiplicity.

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A382526 Number of integer partitions of n with fewer ones than greatest multiplicity.

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