A088922 -id:A088922 - cp's OEIS frontend

A325466 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.

1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1, 3, 2, 1, 0, 0, 1, 5, 4, 0, 1, 0, 0, 1, 4, 6, 3, 0, 1, 0, 0, 1, 6, 6, 4, 3, 1, 1, 0, 0, 1, 6, 10, 4, 2, 4, 1, 2, 0, 0, 1, 7, 12, 8, 3, 3, 4, 1, 2, 1, 0, 1, 6, 13, 11, 2, 11, 3, 4, 0, 3, 1, 1, 1, 10, 16, 7, 10, 10

Offset: 0

Gus Wiseman, May 04 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.

			Triangle begins:
  1
  1  0
  1  1  0
  1  2  0  0
  1  3  1  0  0
  1  3  2  1  0  0
  1  5  4  0  1  0  0
  1  4  6  3  0  1  0  0
  1  6  6  4  3  1  1  0  0
  1  6 10  4  2  4  1  2  0  0
  1  7 12  8  3  3  4  1  2  1  0
  1  6 13 11  2 11  3  4  0  3  1  1
  1 10 16  7 10 10  6  6  5  1  1  2  1
  1  7 18 14  7 16 11  6  4  8  0  5  0  1
  1  9 20 18 10 20 13 10 10  4  5  5  2  2  2
  1 10 26 18 10 24 13 19 13 10  6  6  2  8  1  2
  1 11 25 24 16 28 19 24 14 15  9 10  9  5  2  7  1
Row 7 counts the following reversed partitions (empty columns not shown):
  (7)  (16)       (115)     (133)   (11122)
       (25)       (124)     (1123)
       (34)       (223)     (1222)
       (1111111)  (1114)
                  (11113)
                  (111112)
Row 9 counts the following reversed partitions (empty columns not shown):
(9)  (18)         (117)       (126)    (1125)   (1134)    (11223)  (111222)
     (27)         (135)       (144)    (11124)  (1224)             (1111122)
     (36)         (225)       (1233)            (11133)
     (45)         (234)       (12222)           (111123)
     (333)        (1116)
     (111111111)  (2223)
                  (11115)
                  (111114)
                  (1111113)
                  (11111112)

Row sums are A000041. Column k = 1 is A088922.

Cf. A098859, A279945, A325242, A325324, A325325, A325349, A325404, A325405, A325406, A325468.

Table[Length[Select[Reverse/@IntegerPartitions[n],Length[Union@@Table[Differences[#,i],{i,1,Length[#]}]]==k&]],{n,0,16},{k,0,n}]

A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

Original entry on oeis.org

1, 1, 2, 2, 1, 1, 2, 3, 2, 0, 2, 3, 2, 1, 1, 2, 3, 4, 2, 4, 1, 3, 3, 1, 4, 2, 4, 3, 2, 1, 1, 2, 3, 4, 5, 2, 4, 0, 2, 4, 3, 0, 3, 0, 3, 4, 2, 0, 4, 2, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5, 6, 2, 4, 6, 1, 3, 5, 3, 6, 2, 5, 1, 4, 4, 1, 5, 2, 6, 3, 5, 3, 1, 6, 4, 2, 6, 5, 4, 3, 2, 1, 1, 2, 3, 4, 5

Offset: 1

Luca Onnis, Mar 24 2022

Each matrix represents all possible products between the elements of Z_(n+1), where Z_k is the ring of integers mod k.
Those matrices are symmetric.
The first row is equal to the first column which is equal to 1,2,...,n.

			Matrices begin:
  n=1:  1,
  n=2:  1, 2,
        2, 1,
  n=3:  1, 2, 3,
        2, 0, 2,
        3, 2, 1,
  n=4:  1, 2, 3, 4,
        2, 4, 1, 3,
        3, 1, 4, 2,
        4, 3, 2, 1;
For example, the 6 X 6 matrix generated by Z_7 is the following:
  1 2 3 4 5 6
  2 4 6 1 3 5
  3 6 2 5 1 4
  4 1 5 2 6 3
  5 3 1 6 4 2
  6 5 4 3 2 1
The trace of this matrix is 14 = A048153(7).

Onno Cain, Gaussian Integers, Rings, Finite Fields, and the Magic Square of Squares, arXiv:1908.03236 [math.RA], 2019.
Matt Parker and Brady Haran, Finite Fields & Return of The Parker Square, Numberphile video (Oct 7, 2021).

Cf. A048153 (traces), A349099 (permanents), A160255 (sum entries), A088922 (ranks).

Cf. A074930.

Flatten[Table[Table[Mod[k*Table[i, {i, 1, p - 1}], p], {k, 1, p - 1}], {p, 1, 10}]]

A218322 Maillet determinant for prime(n).

Original entry on oeis.org

1, -5, 49, 14641, -371293, -410338673, 16983563041, 124279533640947, -82085029703668817512, 6812495416987166882889, -16890053810563300749953435929, -531714676529925182191868570093681, 98548851401030959947062957685234211, 4247541973383735863138308138153477847255, -62534081783371829558502906501683809565833328077, 1581923629964045589238110056212521488781448927448053161

Offset: 2

Max Alekseyev, Oct 25 2012

sign

Wikipedia, Maillet's determinant

Cf. A055513, A203411, A000927, A088922.

a(n) = p=prime(n); matdet(matrix((p-1)/2,(p-1)/2,i,j,(i/j)%p))

a(n) = (-p)^((p-3)/2) * h^-(p) = (-1)^((p-3)/2) * A203411(n) * A000927(n), where p=A000040(n).

A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

Original entry on oeis.org

1, 1, 5, 32, 1074, 12600, 1525292, 34078720, 4072850100, 263459065600, 106809546673488, 2254519427530752, 3172225081523720416, 210351382651302645760, 45654014718074873700000, 11122845097194072534155264, 18156837198112938091803999360, 795289872611524024920215715840

Offset: 0

Stefano Spezia, Mar 25 2022

nonn

Det(M(n)) = 0 iff n = 4 or n > 5.
Rank(M(n)) = A088922(n+1).
Tr(M(n)) = A048153(n+1).

			See A352620 for the examples of matrix M(n).

Cf. A352620.

Cf. A048153, A074930, A088922, A160255.

a:= n-> `if`(n=0, 1, LinearAlgebra[Permanent](
         Matrix(n, (i, j)-> (i*j) mod (n+1)))):
seq(a(n), n=0..16);  # Alois P. Heinz, Mar 25 2022

Join[{1},Table[Permanent[Table[Mod[j*Table[i, {i, n}], n+1], {j, n}]], {n, 17}]]

a(n) = matpermanent(matrix(n,n,i,j,(i*j)%(n+1))); \\ Michel Marcus, Mar 26 2022

A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 1, 3, 3, 1, 5, 3, 2, 1, 4, 7, 2, 1, 1, 6, 7, 6, 2, 1, 6, 10, 6, 7, 1, 7, 12, 11, 8, 3, 1, 6, 16, 11, 17, 3, 2, 1, 10, 14, 20, 19, 10, 2, 1, 1, 7, 22, 17, 31, 14, 7, 2, 1, 9, 22, 27, 37, 22, 11, 6, 1, 10, 24, 27, 51, 32, 16, 15

Offset: 0

Gus Wiseman, Nov 19 2023

We define a semi-sum of a multiset to be any sum of a 2-element submultiset. This is different from sums of pairs of elements. For example, 2 is the sum of a pair of elements of {1}, but there are no semi-sums.

			Triangle begins:
  1
  1  1
  1  2
  1  3  1
  1  3  3
  1  5  3  2
  1  4  7  2  1
  1  6  7  6  2
  1  6 10  6  7
  1  7 12 11  8  3
  1  6 16 11 17  3  2
  1 10 14 20 19 10  2  1
  1  7 22 17 31 14  7  2
  1  9 22 27 37 22 11  6
  1 10 24 27 51 32 16 15
  1 11 27 39 57 43 27 22  4
  1  9 33 34 79 57 36 39  7  2
  1 13 31 51 86 77 45 62 14  4  1
Row n = 9 counts the following partitions:
  (9)  (81)         (711)       (621)      (5211)
       (72)         (6111)      (531)      (4311)
       (63)         (522)       (432)      (4221)
       (54)         (51111)     (33111)    (42111)
       (333)        (441)       (222111)   (3321)
       (111111111)  (411111)    (2211111)  (32211)
                    (3222)                 (321111)
                    (3111111)
                    (22221)
                    (21111111)

Row sums are A000041.
Column k = 1 is A088922.
The non-binary version (with zeros) is A365658.
The strict non-binary version (with zeros) is A365832.
The corresponding rank statistic is A366739.
A001358 lists semiprimes, squarefree A006881, conjugate A065119.
A126796 counts complete partitions, ranks A325781, strict A188431.
A276024 counts positive subset-sums of partitions, strict A284640.
A365924 counts incomplete partitions, ranks A365830, strict A365831.
A366738 counts semi-sums of partitions, non-binary A304792.
A366741 counts semi-sums of strict partitions, non-binary A365925.
Cf. A046663, A117855, A122768, A238628, A299701, A365543, A366753, A367095.

DeleteCases[Table[Length[Select[IntegerPartitions[n], Length[Union[Total/@Subsets[#, {2}]]]==k&]], {n,10},{k,0,n}],0,2]

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A325466 Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0.

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A352620 Irregular triangle read by rows which are rows of successive n X n matrices M(n) with entries M(n)[i,j] = i*j mod n+1.

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A218322 Maillet determinant for prime(n).

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A349099 a(n) is the permanent of the n X n matrix M(n) defined as M(n)[i,j] = i*j (mod n + 1).

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A367412 Triangle read by rows with all zeros removed where T(n,k) is the number of integer partitions of n with k different semi-sums.

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