A007690 -id:A007690 - cp's OEIS frontend

A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

1, 1, 2, 3, 5, 6, 9, 10, 14, 18, 21, 26, 35, 39, 46, 58, 68, 79, 97, 111, 131, 155, 177, 206, 246, 278, 318, 373, 423, 483, 563, 632, 722, 827, 931, 1058, 1209, 1354, 1528, 1736, 1951, 2188, 2475, 2762, 3097, 3488, 3886, 4342, 4876, 5414, 6038, 6741, 7482

Offset: 0

Gus Wiseman, Jun 15 2025

nonn

			The partition y = (6,5,5,5,3,3,2,1) has maximal gapless runs ((6,5,5,5),(3,3,2,1)), with lengths (4,4), so y is counted under a(30).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (2111)   (222)     (322)      (71)
                            (11111)  (321)     (2221)     (332)
                                     (2211)    (3211)     (2222)
                                     (21111)   (22111)    (3221)
                                     (111111)  (211111)   (3311)
                                               (1111111)  (22211)
                                                          (32111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)

The strict case is A384886, distinct A384178.
For distinct instead of equal lengths we have A384884.
For anti-runs instead of runs we have A384888, distinct A384885.
For subsets instead of strict partitions we have A243815.
Without counting decreases by 0 we get A384904.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A044813, A047993, A325324, A325325, A356226, A356230, A356233, A356234, A384175, A384177, A384880.

Table[Length[Select[IntegerPartitions[n],SameQ@@Length/@Split[#,#2>=#1-1&]&]],{n,0,15}]

A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 8, 9, 13, 15, 18, 22, 28, 31, 38, 45, 53, 62, 74, 86, 105, 123, 146, 171, 208, 242, 290, 340, 399, 469, 552, 639, 747, 862, 999, 1150, 1326, 1514, 1736, 1979, 2256, 2560, 2909, 3283, 3721, 4191, 4726, 5311, 5973, 6691, 7510, 8396, 9395

Offset: 0

Gus Wiseman, Jun 13 2025

nonn

			The partition y = (8,6,3,3,3,1) has maximal anti-runs ((8,6,3),(3),(3,1)), with lengths (3,1,2), so y is counted under a(24).
The partition z = (8,6,5,3,3,1) has maximal anti-runs ((8,6),(5,3),(3,1)), with lengths (2,2,2), so z is not counted under a(26).
The a(1) = 1 through a(9) = 9 partitions:
  (1)  (2)  (3)  (4)    (5)      (6)      (7)      (8)      (9)
                 (3,1)  (4,1)    (4,2)    (5,2)    (5,3)    (6,3)
                        (3,1,1)  (5,1)    (6,1)    (6,2)    (7,2)
                                 (4,1,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (4,2,2)  (4,4,1)
                                          (5,1,1)  (4,3,1)  (5,2,2)
                                                   (5,2,1)  (5,3,1)
                                                   (6,1,1)  (6,2,1)
                                                            (7,1,1)

For subsets instead of strict partitions we have A384177, for runs A384175.
The strict case is A384880.
For runs instead of anti-runs we have A384884, strict A384178.
For equal instead of distinct lengths we have A384888, for runs A384887.
A000041 counts integer partitions, strict A000009.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
A098859 counts Wilf partitions (distinct multiplicities), complement A336866.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A355394 counts partitions without a neighborless part, singleton case A355393.
A356236 counts partitions with a neighborless part, singleton case A356235.
A356606 counts strict partitions without a neighborless part, complement A356607.
Cf. A008284, A047966, A242882, A287170, A325324, A325325, A329739, A356226, A356230, A356234, A384886.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[#,#2<#1-1&]&]],{n,0,15}]

A184639 Number of partitions of n having no parts with multiplicity 4.

Original entry on oeis.org

1, 1, 2, 3, 4, 7, 10, 14, 19, 27, 37, 50, 67, 88, 115, 153, 196, 253, 324, 412, 524, 661, 828, 1036, 1290, 1603, 1980, 2443, 2997, 3671, 4487, 5460, 6631, 8034, 9703, 11703, 14075, 16890, 20226, 24175, 28838, 34332, 40801, 48394, 57307, 67765, 79974

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(4) = 4, because 4 partitions of 4 have no parts with multiplicity 4: [1,1,2], [2,2], [1,3], [4].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183561, A183568, A007690, A116645, A118807, A184640, A184641, A184642, A184643, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[l, If[j == 4, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183561(n).
a(n) = A183568(n,0) - A183568(n,4).
G.f.: Product_{j>0} (1-x^(4*j)+x^(5*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-4*x) + exp(-6*x)) dx = 0.77101366877372033648945034346499691865027592089088481444183... - Vaclav Kotesovec, Jun 12 2025

A184640 Number of partitions of n having no parts with multiplicity 5.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 14, 21, 28, 39, 51, 72, 92, 124, 160, 210, 266, 349, 438, 562, 704, 892, 1107, 1395, 1720, 2141, 2631, 3249, 3965, 4873, 5916, 7216, 8730, 10585, 12742, 15387, 18443, 22151, 26466, 31646, 37659, 44873, 53212, 63149, 74666, 88295

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(5) = 6, because 6 partitions of 5 have no parts with multiplicity 5: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183562, A183568, A007690, A116645, A118807, A184639, A184641, A184642, A184643, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 5, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183562(n).
a(n) = A183568(n,0) - A183568(n,5).
G.f.: Product_{j>0} (1-x^(5*j)+x^(6*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-5*x) + exp(-7*x)) dx = 0.78834765570757713777493985857868631321765157344539753651545... - Vaclav Kotesovec, Jun 12 2025

A184641 Number of partitions of n having no parts with multiplicity 6.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 15, 21, 29, 40, 54, 72, 96, 127, 166, 216, 279, 358, 457, 580, 735, 924, 1159, 1446, 1799, 2228, 2752, 3388, 4158, 5087, 6207, 7551, 9165, 11093, 13401, 16144, 19412, 23286, 27882, 33310, 39727, 47289, 56191, 66647, 78923, 93299

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(6) = 10, because 10 partitions of 6 have no parts with multiplicity 6: [1,1,1,1,2], [1,1,2,2], [2,2,2], [1,1,1,3], [1,2,3], [3,3], [1,1,4], [2,4], [1,5], [6].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183563, A183568, A007690, A116645, A118807, A184639, A184640, A184642, A184643, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 6, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183563(n).
a(n) = A183568(n,0) - A183568(n,6).
G.f.: Product_{j>0} (1-x^(6*j)+x^(7*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-6*x) + exp(-8*x)) dx = 0.79818518024793359047735154473665146019665210453617381247423... - Vaclav Kotesovec, Jun 12 2025

A184642 Number of partitions of n having no parts with multiplicity 7.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 22, 29, 41, 54, 75, 97, 130, 168, 222, 283, 368, 465, 597, 750, 949, 1183, 1488, 1841, 2292, 2822, 3487, 4267, 5239, 6376, 7782, 9429, 11439, 13798, 16661, 20007, 24043, 28763, 34420, 41021, 48894, 58066, 68956, 81627, 96592

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

			a(7) = 14, because 14 partitions of 7 have no parts with multiplicity 7: [1,1,1,1,1,2], [1,1,1,2,2], [1,2,2,2], [1,1,1,1,3], [1,1,2,3], [2,2,3], [1,3,3], [1,1,1,4], [1,2,4], [3,4], [1,1,5], [2,5], [1,6], [7].

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183564, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184643, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=7, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 7, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)
Table[Count[IntegerPartitions[n],?(FreeQ[Length/@Split[#],7]&)],{n,0,50}] (* _Harvey P. Dale, Sep 21 2024 *)

a(n) = A000041(n) - A183564(n).
a(n) = A183568(n,0) - A183568(n,7).
G.f.: Product_{j>0} (1-x^(7*j)+x^(8*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-7*x) + exp(-9*x)) dx = 0.80430417180776436899064351977235191494130305607975798117531... - Vaclav Kotesovec, Jun 12 2025

A184643 Number of partitions of n having no parts with multiplicity 8.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 30, 41, 55, 75, 99, 131, 172, 223, 288, 372, 474, 603, 764, 962, 1206, 1509, 1876, 2326, 2878, 3543, 4351, 5330, 6506, 7921, 9623, 11655, 14085, 16987, 20434, 24529, 29392, 35138, 41930, 49947, 59381, 70474, 83512, 98779

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183565, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184644, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=8, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 8, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183565(n).
a(n) = A183568(n,0) - A183568(n,8).
G.f.: Product_{j>0} (1-x^(8*j)+x^(9*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-8*x) + exp(-10*x)) dx = 0.80836901097063952622501649557292291036896118821761722817375... - Vaclav Kotesovec, Jun 12 2025

A184644 Number of partitions of n having no parts with multiplicity 9.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 42, 55, 76, 99, 133, 172, 227, 290, 376, 477, 612, 769, 975, 1217, 1528, 1895, 2359, 2907, 3592, 4400, 5403, 6584, 8034, 9742, 11823, 14272, 17234, 20713, 24897, 29803, 35674, 42542, 50719, 60272, 71592, 84794

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183566, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184645.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=9, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 9, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183566(n).
a(n) = A183568(n,0) - A183568(n,9).
G.f.: Product_{j>0} (1-x^(9*j)+x^(10*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-9*x) + exp(-11*x)) dx = 0.81120660452677002313966407107688916817839171627473737415672... - Vaclav Kotesovec, Jun 12 2025

A184645 Number of partitions of n having no parts with multiplicity 10.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 56, 76, 100, 133, 174, 227, 293, 378, 482, 614, 777, 980, 1229, 1538, 1913, 2375, 2936, 3619, 4445, 5447, 6650, 8102, 9844, 11929, 14421, 17397, 20934, 25141, 30130, 36035, 43014, 51253, 60952, 72367, 85771, 101488

Offset: 0

Alois P. Heinz, Jan 18 2011

nonn

In general, if k>=1 and g.f. = Product_{j>0} (1 - x^(k*j) + x^((k+1)*j)) / (1-x^j), then a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-k*x) + exp(-(k+2)*x)) dx. - Vaclav Kotesovec, Jun 12 2025

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A183567, A183568, A007690, A116645, A118807, A184639, A184640, A184641, A184642, A184643, A184644.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[1]-l[2])(b(n, n)):
seq(a(n), n=0..50);

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[Function[l, If[j == 10, {l[[1]], l[[1]]}, l]][b[n - i*j, i - 1]], {j, 0, n/i}]]];
a[n_] := b[n, n][[1]] - b[n, n][[2]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

a(n) = A000041(n) - A183567(n).
a(n) = A183568(n,0) - A183568(n,10).
G.f.: Product_{j>0} (1-x^(10*j)+x^(11*j))/(1-x^j).
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-10*x) + exp(-12*x)) dx = 0.81326581550954971049947225462608121545474493920551191360132... - Vaclav Kotesovec, Jun 12 2025

A384881 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal runs of consecutive parts decreasing by 1.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Jun 25 2025

			The partition (5,4,2,1,1) has maximal runs ((5,4),(2,1),(1)) so is counted under T(13,3) = 23.
Row n = 9 counts the following partitions:
  9    63    333    6111    33111   411111   3111111   111111111
  54   72    441    22221   51111   2211111  21111111
  432  81    522    42111   222111
       621   531    321111
       3321  711
             3222
             4221
             4311
             5211
             32211
Triangle begins:
  1
  0  1
  0  1  1
  0  2  0  1
  0  1  3  0  1
  0  2  2  2  0  1
  0  2  3  3  2  0  1
  0  2  5  3  2  2  0  1
  0  1  8  4  4  2  2  0  1
  0  3  5 10  4  3  2  2  0  1
  0  2  9  9  9  5  3  2  2  0  1
  0  2 11 13  9  9  4  3  2  2  0  1
  0  2 13 15 17  8 10  4  3  2  2  0  1
  0  2 14 23 16 17  8  9  4  3  2  2  0  1
  0  2 16 26 26 19 16  9  9  4  3  2  2  0  1
  0  4 13 37 32 26 19 16  8  9  4  3  2  2  0  1

John Tyler Rascoe, Rows n = 0..100, flattened

Row sums are A000041.
Column k = 1 is A001227.
For distinct parts instead of maximal runs we have A116608.
The strict case appears to be A116674.
For anti-runs instead of runs we have A268193.
Partitions with distinct runs of this type are counted by A384882, gapless A384884.
For prime indices see A385213, A287170, A066205, A356229.
A007690 counts partitions with no singletons, complement A183558.
A034296 counts flat or gapless partitions, ranks A066311 or A073491.
Cf. A000009, A000217, A008284, A047966, A047993, A098859, A325325, A375136, A384887.

Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1==#2+1&]]==k&]],{n,0,10},{k,0,n}]

tri(n) = {(n*(n+1)/2)}
B_list(N) = {my(v = vector(N, i, 0)); v[1] = q*t; for(m=2,N, v[m] = t * (q^tri(m) + sum(i=1,m-1, q^tri(i) * v[m-i] * (q^((m-i)*(i-1))/(1 - q^(m-i)) - q^((m-i)*i) + O('q^(N-tri(i)+1)))))); v}
A_qt(max_row) = {my(N = max_row+1, B = B_list(N), g = 1 + sum(m=1,N, B[m]/(1 - q^m)) + O('q^(N+1))); vector(N, n, Vecrev(polcoeff(g, n-1)))} \\ John Tyler Rascoe, Aug 18 2025

G.f.: 1 + Sum_{m>0} B(m,q,t)/(1 - q^m) where B(m,q,t) = t * (q^tri(m) + Sum_{i=1..m-1} q^tri(i) * B(m-i,q,t) * ((q^((m-i)*(i-1))/(1 - q^(m-i))) - q^((m-i)*i))) and tri(n) = A000217(n). - John Tyler Rascoe, Aug 18 2025

cp's OEIS Frontend

A384887 Number of integer partitions of n with all equal lengths of maximal gapless runs (decreasing by 0 or 1).

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A384885 Number of integer partitions of n with all distinct lengths of maximal anti-runs (decreasing by more than 1).

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A184639 Number of partitions of n having no parts with multiplicity 4.

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A184640 Number of partitions of n having no parts with multiplicity 5.

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A184641 Number of partitions of n having no parts with multiplicity 6.

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A184642 Number of partitions of n having no parts with multiplicity 7.

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A184643 Number of partitions of n having no parts with multiplicity 8.

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A184644 Number of partitions of n having no parts with multiplicity 9.

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