A325325 -id:A325325 - cp's OEIS frontend

A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

0, 0, 0, 0, 1, 2, 1, 3, 4, 4, 7, 9, 11, 18, 25, 30, 41, 55, 63, 87, 98, 125, 147, 192, 213, 271, 313, 389, 444, 551, 621, 767, 874, 1055, 1209, 1444, 1646, 1965, 2244, 2644, 2991

Offset: 0

Gus Wiseman, May 18 2025

A partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

A partition is section-sum iff its conjugate is Look-and-Say, meaning it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The a(4) = 1 through a(11) = 9 partitions:
  211  221   21111  2221    422      22221     442        222221
       2111         22111   22211    222111    4222       322211
                    211111  221111   2211111   222211     332111
                            2111111  21111111  322111     422111
                                               2221111    2222111
                                               22111111   3221111
                                               211111111  22211111
                                                          221111111
                                                          2111111111
Conjugates of the a(4) = 1 through a(11) = 9 partitions:
  (3,1)  (3,2)  (5,1)  (4,3)  (5,3)      (5,4)  (6,4)      (6,5)
         (4,1)         (5,2)  (6,2)      (6,3)  (7,3)      (7,4)
                       (6,1)  (7,1)      (7,2)  (8,2)      (8,3)
                              (3,3,1,1)  (8,1)  (9,1)      (9,2)
                                                (6,3,1)    (10,1)
                                                (3,3,2,2)  (6,3,2)
                                                (4,4,1,1)  (6,4,1)
                                                           (7,3,1)
                                                           (6,3,1,1)

Ranking sequences are shown in parentheses below.
These partitions are ranked by (A383516).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A351592 counts non Wilf Look-and-Say partitions (A384006).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383519 counts section-sum Wilf partitions (A383520).
Cf. A047966, A048767, A320348, A325324, A325325, A353837, A381431, A383511, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]=={}&]], {n,0,30}]

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}]

A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 6, 7, 9, 10, 11, 13, 14, 15, 18, 20, 21, 24, 26, 28, 33, 36, 38, 43, 46, 49, 56, 60, 63, 71, 76, 80, 90, 96, 100, 112, 120, 125, 139, 149, 155, 171, 183, 190, 208, 223, 232, 252, 269, 280, 304, 325, 338, 364, 387, 403

Offset: 0

Gus Wiseman, May 01 2019

nonn

The augmented differences aug(y) of an integer partition y of length k are given by aug(y)i = y_i - y{i + 1} + 1 if i < k and aug(y)_k = y_k. For example, aug(6,5,5,3,3,3) = (2,1,3,1,1,3).

The Heinz numbers of these partitions are given by A325396.

			The a(1) = 1 through a(11) = 6 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (10)   (11)
            (21)  (31)  (41)  (42)  (52)   (62)   (63)   (73)   (83)
                              (51)  (61)   (71)   (72)   (82)   (92)
                                    (421)  (521)  (81)   (91)   (101)
                                                  (621)  (631)  (731)
                                                         (721)  (821)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049988, A240026, A320466, A325349, A325350, A325351, A325356, A325357, A325359, A325393.

Mathematica
```
aug[y_]:=Table[If[i
				
```

A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 5, 6, 8, 9, 9, 13, 13, 15, 19, 20, 20, 28, 28, 30, 36, 40, 40, 50, 50, 56, 64, 68, 68, 86, 86, 92, 102, 112, 114, 133, 133, 146, 158, 173, 173, 202, 202, 215, 237, 256, 256, 287, 287, 324, 340, 359, 359, 403, 423, 446, 464, 495, 495

Offset: 0

Gus Wiseman, May 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325398.

			The a(1) = 1 through a(9) = 6 reversed partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)
            (12)  (13)  (14)  (15)  (16)   (17)   (18)
                        (23)  (24)  (25)   (26)   (27)
                                    (34)   (35)   (36)
                                    (124)  (125)  (45)
                                                  (126)
The smallest reversed strict partition with strictly increasing differences not counted by this sequence is (1,2,4,7), whose first and second differences are (1,2,3) and (1,1) respectively.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A179254, A240026, A325353, A325354, A325357, A325393, A325395, A325398, A325404, A325406, A325456, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],And@@Table[Less@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 8, 7, 9, 11, 10, 12, 15, 13, 16, 19, 18, 20, 24, 22, 26, 29, 28, 31, 37, 33, 38, 43, 42, 44, 52, 48, 55, 59, 58, 62, 72, 65, 74, 80, 80, 82, 94, 88, 99, 103, 104, 108, 123, 114, 126, 133, 135, 137, 155, 145, 161, 166, 169, 174

Offset: 0

Gus Wiseman, May 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The Heinz numbers of these partitions are given by A325399.

			The a(1) = 1 through a(9) = 5 partitions:
  (1)  (2)  (3)   (4)   (5)   (6)   (7)   (8)    (9)
            (21)  (31)  (32)  (42)  (43)  (53)   (54)
                        (41)  (51)  (52)  (62)   (63)
                                    (61)  (71)   (72)
                                          (431)  (81)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A049988, A320466, A325353, A325354, A325358, A325391, A325396, A325399, A325404, A325406, A325457, A325468.

Table[Length[Select[IntegerPartitions[n],And@@Table[Greater@@Differences[#,k],{k,0,Length[#]}]&]],{n,0,30}]

A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 71, 73, 74, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 106, 107, 109

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A301899 in lacking 105. First differs from A325399 in having 42.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325391.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    6: {1,2}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

A subsequence of A005117.

Cf. A056239, A112798, A325357, A325391, A325395, A325397, A325399, A325400, A325405, A325406, A325456, A325467.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@Table[Less@@Differences[primeMS[#],k],{k,0,PrimeOmega[#]}]&]

A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A109427 in lacking 54.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences.
The enumeration of these partitions by sum is given by A325354.

			Most small numbers are in the sequence. However, the sequence of non-terms together with their prime indices begins:
   18: {1,2,2}
   36: {1,1,2,2}
   50: {1,3,3}
   54: {1,2,2,2}
   60: {1,1,2,3}
   70: {1,3,4}
   72: {1,1,1,2,2}
   75: {2,3,3}
   90: {1,2,2,3}
   98: {1,4,4}
  100: {1,1,3,3}
  108: {1,1,2,2,2}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  140: {1,1,3,4}
  144: {1,1,1,1,2,2}
  147: {2,4,4}
  150: {1,2,3,3}
  154: {1,4,5}
  162: {1,2,2,2,2}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A056239, A112798, A240026, A325354, A325360, A325362, A325394, A325397, A325398, A325399, A325405, A325467.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],And@@Table[Greater@@Differences[primeptn[#],k],{k,0,PrimeOmega[#]}]&]

A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 1, 3, 2, 0, 0, 1, 4, 2, 3, 1, 0, 0, 1, 1, 5, 5, 2, 1, 0, 0, 1, 3, 5, 6, 3, 3, 1, 0, 0, 1, 3, 4, 8, 7, 1, 4, 2, 0, 0, 1, 3, 6, 11, 7, 5, 2, 4, 2, 1, 0, 1, 1, 6, 13, 8, 9, 9, 0, 4, 3, 1, 0, 1, 6, 7, 11, 12, 9

Offset: 0

Gus Wiseman, May 03 2019

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The zeroth differences of a sequence are the sequence itself, while the k-th differences for k > 0 are the differences of the (k-1)-th differences. The distinct differences of any degree are the union of the k-th differences for all k >= 0. For example, the k-th differences of (1,1,2,4) for k = 0...3 are:
  (1,1,2,4)
  (0,1,2)
  (1,1)
  (0)
so there are a total of 4 distinct differences of any degree, namely {0,1,2,4}.

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  2  0
  0  1  2  2  0
  0  1  1  3  2  0
  0  1  4  2  3  1  0
  0  1  1  5  5  2  1  0
  0  1  3  5  6  3  3  1  0
  0  1  3  4  8  7  1  4  2  0
  0  1  3  6 11  7  5  2  4  2  1
  0  1  1  6 13  8  9  9  0  4  3  1
  0  1  6  7 11 12  9 10  8  4  3  2  2
  0  1  1  7 18  9 14 19  5 10  3  5  4  1
  0  1  3  9 17  9 22 20 15  9  7  6  5  4  1
  0  1  4  8 22 11 16 24 22 19 10 11  2  8  7  2
  0  1  4 10 23 15 24 23 27 27 12 14 11  8  8  5  5
Row n = 8 counts the following partitions:
  (8)  (44)        (17)       (116)     (134)   (1133)   (111122)
       (2222)      (26)       (125)     (233)   (11123)
       (11111111)  (35)       (1115)    (1223)  (11222)
                   (224)      (1124)
                   (1111112)  (11114)
                              (111113)

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Row sums are A000041.

Cf. A049597, A049988, A279945, A320348, A325324, A325325, A325349, A325404, A325466.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,0,Length[#]}]]==k&]],{n,0,16},{k,0,n}]

A325457 Heinz numbers of integer partitions with strictly decreasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320470.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A320470, A320510, A325328, A325352, A325360, A325361, A325368, A325399, A325456, A325461, A320470, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Differences[primeptn[#]]&]

A325874 Number of integer partitions of n whose differences of all degrees > 1 are nonzero.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 6, 8, 12, 13, 19, 24, 26, 33, 45, 52, 66, 78, 92, 113, 129, 160, 192, 231, 268, 305, 361, 436, 501, 591, 665, 783, 897, 1071, 1228, 1361, 1593, 1834, 2101, 2452, 2685, 3129, 3526, 4067, 4568, 5189, 5868, 6655, 7565, 8468, 9400

Offset: 0

Gus Wiseman, Jun 02 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2). The zeroth differences are the sequence itself, while k-th differences for k > 0 are the differences of the (k-1)-th differences. If m is the length of the sequence, its differences of all degrees are the union of the zeroth through m-th differences.

The case for all degrees including 1 is A325852.

			The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)   (3)   (4)    (5)    (6)     (7)    (8)     (9)
       (11)  (21)  (22)   (32)   (33)    (43)   (44)    (54)
                   (31)   (41)   (42)    (52)   (53)    (63)
                   (211)  (221)  (51)    (61)   (62)    (72)
                          (311)  (411)   (322)  (71)    (81)
                                 (2211)  (331)  (332)   (441)
                                         (421)  (422)   (522)
                                         (511)  (431)   (621)
                                                (521)   (711)
                                                (611)   (4221)
                                                (3221)  (4311)
                                                (3311)  (5211)
                                                        (32211)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts

Cf. A049988, A238423, A325325, A325468, A325545, A325849, A325850, A325851, A325852, A325875, A325876.

Table[Length[Select[IntegerPartitions[n],!MemberQ[Union@@Table[Differences[#,i],{i,2,Length[#]}],0]&]],{n,0,30}]

cp's OEIS Frontend

A383509 Number of Look-and-Say partitions of n that are not section-sum partitions.

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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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A325358 Number of integer partitions of n whose augmented differences are strictly decreasing.

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A325391 Number of reversed integer partitions of n whose k-th differences are strictly increasing for all k >= 0.

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A325393 Number of integer partitions of n whose k-th differences are strictly decreasing for all k >= 0.

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A325398 Heinz numbers of reversed integer partitions whose k-th differences are strictly increasing for all k >= 0.

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A325400 Heinz numbers of reversed integer partitions whose k-th differences are weakly increasing for all k >= 0.

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Mathematica

A325406 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree.

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A325457 Heinz numbers of integer partitions with strictly decreasing differences.

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