A325404 -id:A325404 - cp's OEIS frontend

A325325 Number of integer partitions of n with distinct differences between successive parts.

1, 1, 2, 2, 4, 5, 5, 8, 11, 12, 16, 22, 21, 30, 34, 42, 49, 64, 67, 87, 95, 117, 132, 160, 169, 207, 230, 274, 301, 360, 395, 463, 506, 602, 656, 762, 834, 960, 1042, 1220, 1311, 1505, 1643, 1859, 2000, 2341, 2491, 2827, 3083, 3464, 3747, 4302, 4561, 5154

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The Heinz numbers of these partitions are given by A325368.

			The a(0) = 1 through a(9) = 12 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)
                                            (331)  (332)   (441)
                                            (421)  (422)   (522)
                                            (511)  (431)   (621)
                                                   (521)   (711)
                                                   (611)   (4221)
                                                   (4211)  (4311)
                                                           (5211)
For example, (5,2,1,1) has differences (-3,-1,0), which are distinct, so (5,2,1,1) is counted under a(9).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400 (terms 0..123 from Alois P. Heinz)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007294, A007862, A049988, A098859, A240026, A240027, A320348, A320466, A320470, A325324, A325349, A325352, A325368, A325404, A325468.

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

A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 9, 7, 13, 12, 13, 16, 22, 17, 28, 28, 31, 36, 50, 45, 63, 62, 74, 78, 102, 92, 123, 123, 146, 148, 191, 181, 228, 233, 280, 283, 348, 350, 420, 437, 518, 523, 616, 641, 727, 774, 884, 911, 1038, 1102, 1240, 1292, 1463, 1530, 1715, 1861, 2002

Offset: 1

Seiichi Manyama, Oct 11 2018

nonn

Also the number of integer partitions of n whose parts cover an initial interval of positive integers with distinct multiplicities. Also the number of integer partitions of n whose multiplicities cover an initial interval of positive integers and are distinct (see A048767 for a bijection). - Gus Wiseman, May 04 2019

			n = 9
[9]        *********  a_1 = 9.
           ooooooooo
------------------------------------
[8, 1]             *        a_2 = 1.
            *******o  a_1 - a_2 = 7.
            oooooooo
------------------------------------
[7, 2]            **        a_2 = 2.
             *****oo  a_1 - a_2 = 5.
             ooooooo
------------------------------------
[5, 4]          ****        a_2 = 4.
               *oooo  a_1 - a_2 = 1.
               ooooo
------------------------------------
a(9) = 4.
From _Gus Wiseman_, May 04 2019: (Start)
The a(1) = 1 through a(11) = 9 strict partitions with distinct differences (where the last part is taken to be 0) are the following (A = 10, B = 11). The Heinz numbers of these partitions are given by A325388.
  (1)  (2)  (3)  (4)   (5)   (6)   (7)   (8)   (9)   (A)    (B)
                 (31)  (32)  (51)  (43)  (53)  (54)  (64)   (65)
                       (41)        (52)  (62)  (72)  (73)   (74)
                                   (61)  (71)  (81)  (82)   (83)
                                                     (91)   (92)
                                                     (631)  (A1)
                                                            (632)
                                                            (641)
                                                            (731)
The a(1) = 1 through a(10) = 6 partitions covering an initial interval of positive integers with distinct multiplicities are the following. The Heinz numbers of these partitions are given by A325326.
  1  11  111  211   221    21111   2221     22211     22221      222211
              1111  2111   111111  22111    221111    2211111    322111
                    11111          211111   2111111   21111111   2221111
                                   1111111  11111111  111111111  22111111
                                                                 211111111
                                                                 1111111111
The a(1) = 1 through a(10) = 6 partitions whose multiplicities cover an initial interval of positive integers and are distinct are the following (A = 10). The Heinz numbers of these partitions are given by A325337.
  (1)  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (A)
                 (211)  (221)  (411)  (322)  (332)  (441)  (433)
                        (311)         (331)  (422)  (522)  (442)
                                      (511)  (611)  (711)  (622)
                                                           (811)
                                                           (322111)
(End)

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..500 (terms 1..100 from Seiichi Manyama)
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A000009, A320347.

Cf. A007294, A007862, A048767, A098859, A179269, A320509, A320510, A325324, A325325, A325349, A325367, A325404, A325468.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&UnsameQ@@Differences[Append[#,0]]&]],{n,30}] (* Gus Wiseman, May 04 2019 *)

A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

Original entry on oeis.org

1, 1, 2, 1, 3, 4, 4, 7, 7, 7, 10, 15, 13, 22, 25, 26, 31, 43, 39, 55, 54, 68, 75, 98, 97, 128, 135, 165, 177, 217, 223, 277, 282, 339, 356, 438, 444, 527, 553, 667, 694, 816, 868, 1015, 1054, 1279, 1304, 1538, 1631, 1849, 1958, 2304, 2360, 2701, 2899, 3267

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) (with the last part taken to be 0) are (-3,-2,-1).

The Heinz numbers of these partitions are given by A325367.

			The a(1) = 1 through a(11) = 15 partitions (A = 10, B = 11):
  (1)  (2)   (3)  (4)   (5)    (6)    (7)    (8)    (9)    (A)    (B)
       (11)       (22)  (32)   (33)   (43)   (44)   (54)   (55)   (65)
                  (31)  (41)   (51)   (52)   (53)   (72)   (64)   (74)
                        (311)  (411)  (61)   (62)   (81)   (73)   (83)
                                      (322)  (71)   (441)  (82)   (92)
                                      (331)  (332)  (522)  (91)   (A1)
                                      (511)  (611)  (711)  (433)  (443)
                                                           (622)  (533)
                                                           (631)  (551)
                                                           (811)  (632)
                                                                  (641)
                                                                  (722)
                                                                  (731)
                                                                  (911)
                                                                  (6311)
For example, (6,3,1,1) has differences (-3,-2,0,-1), which are distinct, so (6,3,1,1) is counted under a(11).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400
Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A007862, A049988, A098859, A130091, A240026, A320348, A320466, A320509, A325325, A325349, A325366, A325367, A325368, A325404, A325407.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,0]]&]],{n,0,30}]

A325349 Number of integer partitions of n whose augmented differences are distinct.

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 4, 5, 7, 7, 12, 10, 13, 15, 21, 21, 31, 34, 38, 45, 55, 60, 71, 80, 84, 103, 119, 134, 152, 186, 192, 228, 263, 292, 321, 377, 399, 454, 514, 565, 618, 709, 752, 840, 958, 1050, 1140, 1297, 1402, 1568, 1755, 1901, 2080, 2343, 2524, 2758, 3074

Offset: 0

Gus Wiseman, Apr 23 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 A325366.

			The a(1) = 1 through a(11) = 10 partitions (A = 10, B = 11):
  (1)  (2)  (3)   (4)   (5)   (6)   (7)    (8)    (9)    (A)    (B)
            (21)  (22)  (41)  (33)  (43)   (44)   (54)   (55)   (65)
                  (31)        (42)  (52)   (62)   (63)   (64)   (83)
                              (51)  (61)   (71)   (72)   (73)   (92)
                                    (421)  (422)  (81)   (82)   (A1)
                                           (431)  (522)  (91)   (443)
                                           (521)  (621)  (433)  (641)
                                                         (442)  (722)
                                                         (541)  (731)
                                                         (622)  (821)
                                                         (631)
                                                         (721)
For example, (4,4,3) has augmented differences (1,2,3), which are distinct, so (4,4,3) is counted under a(11).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..440

Cf. A000837, A049988, A098859, A325324, A325325, A325328, A325351, A325366, A325404.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@Differences[Append[#,1]]&]],{n,0,30}]

A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 22, 23, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 55, 57, 58, 59, 61, 62, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 119, 122

Offset: 1

Gus Wiseman, May 02 2019

nonn

First differs from A325388 in lacking 130.
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 A325404.

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   19: {8}
   22: {1,5}
   23: {9}
   26: {1,6}
   29: {10}
   31: {11}
   33: {2,5}
   34: {1,7}
   35: {3,4}

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

A subsequence of A005117.

Cf. A056239, A112798, A279945, A325325, A325366, A325367, A325368, A325397, A325398, A325399, A325400, A325404, A325406, A325467.

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

A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 3, 5, 6, 6, 9, 11, 10, 15, 17, 19, 24, 31, 26, 40, 43, 51, 52, 72, 66, 89, 88, 111, 119, 150, 130, 183, 193, 229, 231, 279, 287, 358, 365, 430, 426, 538, 535, 649, 680, 742, 803, 943, 982, 1136, 1115

Offset: 0

Gus Wiseman, May 03 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 A325467.

			The a(1) = 1 through a(9) = 6 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)
                                    (421)  (431)  (81)
                                           (521)  (621)

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

Cf. A000009, A325324, A325325, A325349, A325353, A325354, A325391, A325393, A325404, A325406, A325467.

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

A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 11, 15, 19, 24, 25, 36, 37, 43, 54, 63, 64, 80, 81, 100, 113, 122, 123, 151, 166, 178, 195, 217, 218, 269, 270, 295, 316, 332, 372, 424, 425, 447, 472, 547, 550, 616, 617, 659, 750, 777, 782, 862, 885, 995, 1032, 1083, 1090, 1176, 1275

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 A325400.

			The a(1) = 1 through a(8) = 15 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (112)   (113)    (33)      (34)       (35)
                    (1111)  (1112)   (114)     (115)      (44)
                            (11111)  (123)     (124)      (116)
                                     (222)     (223)      (125)
                                     (1113)    (1114)     (224)
                                     (11112)   (11113)    (1115)
                                     (111111)  (111112)   (1124)
                                               (1111111)  (2222)
                                                          (11114)
                                                          (111113)
                                                          (1111112)
                                                          (11111111)

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

Cf. A007294, A240026, A325353, A325356, A325360, A325362, A325391, A325393, A325394, A325400, A325404, A325406, A325468.

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

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}]

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}]

cp's OEIS Frontend

A325325 Number of integer partitions of n with distinct differences between successive parts.

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A320348 Number of partition into distinct parts (a_1, a_2, ... , a_m) (a_1 > a_2 > ... > a_m and Sum_{k=1..m} a_k = n) such that a1 - a2, a2 - a_3, ... , a_{m-1} - a_m, a_m are different.

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A325324 Number of integer partitions of n whose differences (with the last part taken to be 0) are distinct.

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A325349 Number of integer partitions of n whose augmented differences are distinct.

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A325405 Heinz numbers of integer partitions y such that the k-th differences of y are distinct for all k >= 0 and are disjoint from the i-th differences for i != k.

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A325468 Number of integer partitions y of n such that the k-th differences of y are distinct (independently) for all k >= 0.

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A325354 Number of reversed integer partitions of n whose k-th differences are weakly increasing for all k.

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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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