A100882 -id:A100882 - cp's OEIS frontend

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A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

1, 1, 2, 3, 5, 6, 11, 13, 19, 26, 36, 43, 64, 77, 102, 129, 169, 205, 268, 323, 413, 504, 629, 751, 947, 1131, 1384, 1661, 2024, 2393, 2919, 3442, 4136, 4884, 5834, 6836, 8162, 9531, 11262, 13155, 15493, 17981, 21138, 24472, 28571, 33066, 38475, 44305

Offset: 0

David S. Newman, Nov 21 2004

nonn

From Gus Wiseman, Jan 21 2019: (Start)
Also the number of semistandard Young tableaux where the rows are constant and the entries sum to n. For example, the a(8) = 19 tableaux are:
  8   44   2222   11111111
.
  1   2   11   3   111   22   1111   11   11111   1111   111111
  7   6   6    5   5     4    4      33   3       22     2
.
  1   1   11   111
  2   3   2    2
  5   4   4    3
(End)

			a(5) = 6 because, of the 7 unrestricted partitions of 5, only one, 2 + 2 + 1, has a decreasing sequence of frequencies. Two is used twice, but 1 is used only once.

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

Cf. A100881, A100882, A100884.

Cf. A000085, A000219, A003293, A006951, A100471, A323582.

b:= proc(n, i, t) option remember; `if`(n<0, 0, `if`(n=0, 1,
      `if`(i=1, `if`(n>=t, 1, 0), `if`(i=0, 0, b(n, i-1, t)+
       add(b(n-i*j, i-1, j), j=t..floor(n/i))))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Jul 03 2014

b[n_, i_, t_] := b[n, i, t] = If[n<0, 0, If[n == 0, 1, If[i == 1, If[n >= t, 1, 0], If[i == 0, 0, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t, Floor[n/i]}]]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ[Length/@Split[#]]&]],{n,20}] (* Gus Wiseman, Jan 21 2019 *)

More terms from Vladeta Jovovic, Nov 23 2004

A100471 Number of integer partitions of n whose sequence of frequencies is strictly increasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 18, 20, 27, 32, 40, 44, 60, 67, 82, 93, 114, 129, 161, 175, 209, 239, 285, 315, 372, 416, 484, 545, 631, 698, 811, 890, 1027, 1146, 1304, 1437, 1631, 1805, 2042, 2252, 2539, 2785, 3143, 3439, 3846, 4226, 4722, 5159

Offset: 0

David S. Newman, Nov 21 2004

nonn

			a(4) = 4 because of the 5 unrestricted partitions of 4, only one, 3+1 uses each of its summands just once and 1,1 is not an increasing sequence.
From _Gus Wiseman_, Jan 23 2019: (Start)
The a(1) = 1 through a(8) = 11 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (211)   (2111)   (222)     (511)      (422)
                    (1111)  (11111)  (411)     (4111)     (611)
                                     (3111)    (22111)    (2222)
                                     (21111)   (31111)    (5111)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Cf. A000219, A000837 (frequencies are relatively prime), A047966 (frequencies are equal), A098859 (frequencies are distinct), A100881, A100882, A100883, A304686 (Heinz numbers of these partitions).

a100471 n = p 0 (n + 1) 1 n where
   p m m' k x | x == 0    = if m < m' || m == 0 then 1 else 0
              | x < k     = 0
              | m == 0    = p 1 m' k (x - k) + p 0 m' (k + 1) x
              | otherwise = p (m + 1) m' k (x - k) +
                            if m < m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012

b:= proc(n,i,t) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n>t, 1, 0)
    elif i=0 then 0
    else      b(n, i-1, t)
         +add(b(n-i*j, i-1, j), j=t+1..floor(n/i))
      fi
    end:
a:= n-> b(n, n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==1, If[n>t, 1, 0], i == 0, 0 , True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, t+1, Floor[n/i]}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],OrderedQ@*Split]],{n,20}] (* Gus Wiseman, Jan 23 2019 *)

Corrected and extended by Vladeta Jovovic, Nov 24 2004

Name edited by Gus Wiseman, Jan 23 2019

A100881 Number of partitions of n in which the sequence of frequencies of the summands is decreasing.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 4, 6, 5, 7, 8, 8, 9, 13, 10, 13, 15, 16, 18, 21, 17, 24, 28, 26, 26, 36, 32, 38, 42, 40, 46, 52, 48, 63, 63, 59, 63, 85, 77, 81, 92, 89, 102, 116, 98, 122, 134, 130, 140, 157, 145, 165, 182, 190, 191, 207, 195, 235, 259, 232, 252, 293, 279

Offset: 0

David S. Newman, Nov 21 2004

nonn

			a(7) = 4 because in each of the four partitions [7], [3,3,1], [2,2,2,1], [1,1,1,1,1,1,1] the frequency with which a summand is used decreases as the summand decreases.

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

Cf. A100882, A100883, A100884.

Cf. A100471, A098859.

a100881 = p 0 0 1 where
   p m m' k x | x == 0    = if m > m' || m == 0 then 1 else 0
              | x < k     = 0
              | m == 0    = p 1 m' k (x - k) + p 0 m' (k + 1) x
              | otherwise = p (m + 1) m' k (x - k) +
                            if m > m' then p 0 m (k + 1) x else 0
-- Reinhard Zumkeller, Dec 27 2012

b:= proc(n, i, t) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else      b(n, i-1, t)
         +add(b(n-i*j, i-1, j), j=1..min(t-1, floor(n/i)))
      fi
    end:
a:= n-> b(n, n, n+1):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 21 2011

b[n_, i_, t_] := b[n, i, t] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1, t] + Sum[b[n-i*j, i-1, j], {j, 1, Min[t-1, Floor[n/i]]}]]; a[n_] := b[n, n, n+1]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Feb 21 2011

A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 29, 56, 101, 181, 327, 583, 1023, 1820, 3207, 5631, 9905, 17394, 30489, 53481, 93725, 164169, 287606, 503672, 881834, 1544018, 2703161, 4731860, 8283291, 14499392, 25379278, 44422866, 77754798, 136093756, 238204369, 416923752, 729728031

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(6) = 29 compositions:
  (6)    (141)  (213)   (1113)  (21111)
  (51)   (114)  (132)   (222)   (12111)
  (15)   (33)   (123)   (2211)  (11121)
  (42)   (321)  (3111)  (2121)  (11112)
  (24)   (312)  (1311)  (1212)  (111111)
  (411)  (231)  (1131)  (1122)
Missing are: (2112), (1221), (11211).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Unimodal Sequence

The version for the compositions themselves (not run-lengths) is A329398.
Compositions with equal run-lengths are A329738.
The case of partitions is A332745.
The version for unsorted prime signature is the complement of A332831.
The complement is counted by A332833.
Unimodal compositions are A001523.
Partitions with weakly decreasing run-lengths are A100882.
Partitions with weakly increasing run-lengths are A100883.
Compositions that are not unimodal are A115981.
Compositions whose negation is unimodal are A332578.
Compositions whose run-lengths are unimodal are A332726.
Neither weakly increasing nor weakly decreasing compositions are A332834.
Compositions with weakly increasing run-lengths are A332836.
Compositions that are neither unimodal nor is their negation are A332870.
Cf. A001462, A181819, A329744, A329766, A332273, A332640, A332641, A332746.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,20}]

a(n) = 2 * A332836(n) - A329738(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

A316496 Number of totally strong integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Gus Wiseman, Jul 29 2018

nonn

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

			The a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.

Cf. A181819, A182850, A182857, A304660, A305563, A316597.
The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The dual version is A332275.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
The alternating version is A332339 (see also A317256).
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 07 2020

A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 51, 68, 87, 113, 143, 183, 228, 289, 354, 443, 544, 672, 812, 1001, 1202, 1466, 1758, 2123, 2525, 3046, 3606, 4308, 5089, 6054, 7102, 8430, 9855, 11621, 13571, 15915, 18500, 21673, 25103, 29245, 33835, 39296, 45277, 52470

Offset: 0

Gus Wiseman, Feb 29 2020

nonn

Also partitions whose run-lengths and negated run-lengths are both unimodal.

			The a(8) = 21 partitions are:
  (8)     (44)     (2222)
  (53)    (332)    (22211)
  (62)    (422)    (32111)
  (71)    (431)    (221111)
  (521)   (3311)   (311111)
  (611)   (4211)   (2111111)
  (5111)  (41111)  (11111111)
Missing from this list is only (3221).

The complement is counted by A332641.
The Heinz numbers of partitions not in this class are A332831.
The case of run-lengths of compositions is A332835.
Only weakly decreasing is A100882.
Only weakly increasing is A100883.
Unimodal compositions are A001523.
Non-unimodal compositions are A115981.
Partitions with unimodal run-lengths are A332280.
Partitions whose negated run-lengths are unimodal are A332638.
Compositions with unimodal run-lengths are A332726.
Compositions that are neither weakly increasing nor decreasing are A332834.
Cf. A025065, A059204, A181819, A328509, A332281, A332283, A332577, A332578, A332640, A332727, A332742, A332833.

Table[Length[Select[IntegerPartitions[n],Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]

A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 19, 21, 30, 31, 42, 50, 62, 69, 91, 99, 126, 144, 175, 198, 246, 275, 331, 379, 452, 509, 612, 686, 811, 922, 1076, 1219, 1428, 1604, 1863, 2108, 2434, 2739, 3162, 3551, 4075, 4593, 5240, 5885, 6721, 7527, 8556, 9597, 10870

Offset: 0

Reinhard Zumkeller, Jan 20 2010

nonn

A000009(n) <= a(n) <= A000041(n).
Equivalently, the number of partitions of n such that (maximal multiplicity of parts) = (multiplicity of the maximal part), as in the Mathematica program. - Clark Kimberling, Apr 04 2014
Also the number of integer partitions of n whose greatest part is a mode, meaning it appears at least as many times as each of the others. The name "Number of partitions of n such that smaller parts do not occur more frequently than greater parts" seems to describe A100882 = "Number of partitions of n in which the sequence of frequencies of the summands is nonincreasing," which first differs from this at n = 10 due to the partition (3,3,2,1,1). - Gus Wiseman, May 07 2023

			a(5) = #{5, 4+1, 3+2, 2+2+1, 5x1} = 5;
a(6) = #{6, 5+1, 4+2, 3+3, 3+2+1, 2+2+2, 2+2+1+1, 6x1} = 8;
a(7) = #{7, 6+1, 5+2, 4+3, 4+2+1, 3+3+1, 2+2+2+1, 7x1} = 8;
a(8) = #{8, 7+1, 6+2, 5+3, 5+2+1, 4+4, 4+3+1, 3+3+2, 3+3+1+1, 2+2+2+2, 2+2+2+1+1, 8x1} = 12.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to partitions

For median instead of mode we have A053263.
The complement is counted by A240302.
The case where the maximum is the only mode is A362612.
A000041 counts integer partitions, strict A000009.
A362608 counts partitions with a unique mode, complement A362607.
A362611 counts modes in prime factorization.
A362614 counts partitions by number of modes.
Cf. A002865, A008284, A098859, A237984, A275870, A325347, A362610.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; m[p_] := Max[Map[Length, Split[p]]]  (* maximal multiplicity *)
Table[Count[f[n], p_ /; m[p] == Count[p, Max[p]]], {n, 0, z}] (* this sequence *)
Table[Count[f[n], p_ /; m[p] > Count[p, Max[p]]], {n, 0, z}]  (* A240302 *)
(* Clark Kimberling, Apr 04 2014 *)
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0, If[k == 0, 1, 0],
     If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1,
     If[k == -1, j, If[k == 0, 0, If[j > k, 0, k]]]], {j, 1, n/i}]]];
a[n_] := PartitionsP[n] - b[n, n, -1];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz in A240302 *)
Table[Length[Select[IntegerPartitions[n],MemberQ[Commonest[#],Max[#]]&]],{n,0,30}] (* Gus Wiseman, May 07 2023 *)

{ my(N=53, x='x+O('x^N));
my(gf=1+sum(i=1,N,sum(j=1,floor(N/i),x^(i*j)*prod(k=1,i-1,(1-x^(k*(j+1)))/(1-x^k)))));
Vec(gf) } \\ John Tyler Rascoe, Mar 09 2024

a(n) = p(n,0,1,1) with p(n,i,j,k) = if k<=n then p(n-k,i,j+1,k) +p(n,max(i,j),1,k+1) else (if j0 then 0 else 1).

a(n) + A240302(n) = A000041(n). - Clark Kimberling, Apr 04 2014.

G.f.: 1 + Sum_{i, j>0} x^(i*j) * Product_{k=1..i-1} ((1 - x^(k*(j+1)))/(1 - x^k)). - John Tyler Rascoe, Mar 09 2024

A332831 Numbers whose unsorted prime signature is neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

90, 126, 198, 234, 270, 300, 306, 342, 350, 378, 414, 522, 525, 540, 550, 558, 588, 594, 600, 630, 650, 666, 702, 738, 756, 774, 810, 825, 846, 850, 918, 950, 954, 975, 980, 990, 1026, 1050, 1062, 1078, 1098, 1134, 1150, 1170, 1176, 1188, 1200, 1206, 1242

Offset: 1

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Author

Gus Wiseman, Mar 02 2020

Keywords

nonn

Comments

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

Examples

The sequence of terms together with their prime indices begins: 90: {1,2,2,3} 126: {1,2,2,4} 198: {1,2,2,5} 234: {1,2,2,6} 270: {1,2,2,2,3} 300: {1,1,2,3,3} 306: {1,2,2,7} 342: {1,2,2,8} 350: {1,3,3,4} 378: {1,2,2,2,4} 414: {1,2,2,9} 522: {1,2,2,10} 525: {2,3,3,4} 540: {1,1,2,2,2,3} 550: {1,3,3,5} 558: {1,2,2,11} 588: {1,1,2,4,4} 594: {1,2,2,2,5} 600: {1,1,1,2,3,3} 630: {1,2,2,3,4} For example, the prime signature of 540 is (2,3,1), so 540 is in the sequence.

Crossrefs

The version for run-lengths of partitions is A332641.

The version for run-lengths of compositions is A332833.

The version for compositions is A332834.

Prime signature is A124010.

Unimodal compositions are A001523.

Partitions with weakly increasing run-lengths are A100883.

Partitions with weakly increasing or decreasing run-lengths are A332745.

Compositions with weakly increasing or decreasing run-lengths are A332835.

Compositions with weakly increasing run-lengths are A332836.

Cf. A100882, A115981, A328509, A329398, A332281, A332640, A332643, A332746, A332836, A332870.

Programs

Mathematica

Select[Range[1000],!Or[LessEqual@@Last/@FactorInteger[#],GreaterEqual@@Last/@FactorInteger[#]]&]

Formula

Intersection of A071365 and A112769.

A304406 Number of partitions of n in which the sequence of the sum of the same summands is nonincreasing.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 6, 5, 9, 8, 11, 11, 20, 16, 20, 21, 32, 30, 41, 38, 50, 48, 62, 64, 89, 81, 97, 100, 123, 123, 151, 154, 187, 183, 221, 221, 279, 272, 312, 316, 377, 376, 446, 460, 531, 547, 628, 641, 754, 746, 841, 856, 990, 1007, 1145, 1167, 1325, 1346, 1519, 1567, 1776

Offset: 0

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Author

Seiichi Manyama, May 12 2018

Keywords

nonn

Comments

Number of integer partitions of n with weakly increasing run-sums. - Gus Wiseman, Oct 21 2022

Examples

n | | Sequence of the sum of the same summands --+----------------------+----------------------------------------- 1 | 1 | 1 2 | 2 | 2 | 1+1 | 2 3 | 3 | 3 | 1+1+1 | 3 4 | 4 | 4 | 2+2 | 4 | 2+1+1 | 2, 2 | 1+1+1+1 | 4 5 | 5 | 5 | 2+1+1+1 | 3, 2 | 1+1+1+1+1 | 5 6 | 6 | 6 | 3+3 | 6 | 3+1+1+1 | 3, 3 | 2+2+2 | 6 | 2+1+1+1+1 | 4, 2 | 1+1+1+1+1+1 | 6

Crossrefs

Cf. A100882.

These partitions are ranked by A357861.

The complement is A357865, ranked by A357850.

The opposite version is A304405, ranked by A357875.

The strict version is A304430, ranked by A357864.

The strict opposite version is A304428, ranked by A357862.

Number of rows in A354584 summing to n that are weakly decreasing.

A000041 counts integer partitions, strict A000009.

A304442 counts partitions with equal run-sums, distinct A353837.

Cf. A047966, A087980, A098859, A239312, A275870, A353832, A353833, A353864.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],LessEqual@@Total/@Split[#]&]],{n,0,30}] (* Gus Wiseman, Oct 21 2022 *)

A332641 Number of integer partitions of n whose run-lengths are neither weakly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 14, 22, 33, 48, 69, 96, 136, 184, 248, 330, 443, 574, 756, 970, 1252, 1595, 2040, 2558, 3236, 4041, 5054, 6256, 7781, 9547, 11782, 14394, 17614, 21423, 26083, 31501, 38158, 45930, 55299, 66262, 79477, 94803, 113214

Offset: 0

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Author

Gus Wiseman, Feb 26 2020

Keywords

nonn

Comments

Also partitions whose run-lengths and negated run-lengths are not both unimodal. A sequence of positive integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Examples

The a(8) = 1 through a(13) = 14 partitions: (3221) (4221) (5221) (4331) (4332) (5332) (32221) (6221) (5331) (6331) (33211) (42221) (7221) (8221) (322211) (43221) (43321) (332111) (44211) (44311) (52221) (53221) (322221) (62221) (422211) (332221) (3321111) (333211) (422221) (442111) (522211) (3222211) (33211111)

Links

MathWorld, Unimodal Sequence

Crossrefs

The complement is counted by A332745.

The Heinz numbers of these partitions are A332831.

The case of run-lengths of compositions is A332833.

Partitions whose run-lengths are weakly increasing are A100883.

Partitions whose run-lengths are weakly decreasing are A100882.

Partitions whose run-lengths are not unimodal are A332281.

Partitions whose negated run-lengths are not unimodal are A332639.

Unimodal compositions are A001523.

Non-unimodal permutations are A059204.

Non-unimodal compositions are A115981.

Partitions with unimodal run-lengths are A332280.

Partitions whose negated run-lengths are unimodal are A332638.

Compositions whose negation is not unimodal are A332669.

The case of run-lengths of compositions is A332833.

Compositions that are neither increasing nor decreasing are A332834.

Cf. A025065, A181819, A328509, A332282, A332284, A332577, A332578, A332579, A332640, A332642, A332726, A332727, A332742, A332835.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!Or[LessEqual@@Length/@Split[#],GreaterEqual@@Length/@Split[#]]&]],{n,0,30}]

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A100883 Number of partitions of n in which the sequence of frequencies of the summands is nondecreasing.

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A100471 Number of integer partitions of n whose sequence of frequencies is strictly increasing.

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A100881 Number of partitions of n in which the sequence of frequencies of the summands is decreasing.

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A332835 Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing.

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A316496 Number of totally strong integer partitions of n.

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A332745 Number of integer partitions of n whose run-lengths are either weakly increasing or weakly decreasing.

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A171979 Number of partitions of n such that smaller parts do not occur more frequently than greater parts.

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