A369079 -id:A369079 - cp's OEIS frontend

A375924 Number A(n,k) of partitions of [n] such that the element sum of each block is one more than a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 5, 0, 1, 1, 0, 2, 15, 0, 1, 1, 0, 0, 4, 52, 0, 1, 1, 0, 1, 1, 10, 203, 0, 1, 1, 0, 1, 2, 3, 28, 877, 0, 1, 1, 0, 0, 0, 3, 9, 96, 4140, 0, 1, 1, 0, 0, 0, 0, 1, 17, 320, 21147, 0, 1, 1, 0, 0, 0, 1, 1, 8, 108, 1436, 115975, 0

Offset: 0

Alois P. Heinz, Sep 02 2024

			A(5,2) = 10: 12345, 124|3|5, 12|34|5, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 14|25|3, 1|245|3, 1|25|34.
A(6,3) = 9: 136|25|4, 13|256|4, 13|25|46, 16|235|4, 1|2356|4, 1|235|46, 16|25|34, 1|256|34, 1|25|346.
A(7,4) = 8: 14|23|5|67, 1|234|5|67, 1|23|45|67, 1|23|467|5, 14|27|36|5, 1|247|36|5, 1|27|346|5, 1|27|36|45.
A(8,5) = 1: 12345678.
A(8,8) = 4: 18|27|36|45, 1|278|36|45, 1|27|368|45, 1|27|36|458.
A(9,6) = 87: 123469|58|7, 12349|568|7, 12349|58|67, 123568|49|7, ..., 1|25|346789, 16|289|3457, 1|2689|3457, 1|289|34567.
A(9,8) = 5: 18|27|36|45|9, 1|278|36|45|9, 1|27|368|45|9, 1|27|36|458|9, 1|27|36|45|89.
A(9,10) = 1: 1|29|38|47|56.
Square array A(n,k) begins:
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  1,      1,    1,    1,   1,  1,  1,  1, 1, 1, 1, ...
  0,      2,    1,    0,   0,  0,  0,  0, 0, 0, 0, ...
  0,      5,    2,    0,   1,  1,  0,  0, 0, 0, 0, ...
  0,     15,    4,    1,   2,  0,  0,  0, 1, 1, 0, ...
  0,     52,   10,    3,   3,  0,  1,  1, 0, 0, 0, ...
  0,    203,   28,    9,   1,  1,  3,  0, 0, 1, 1, ...
  0,    877,   96,   17,   8, 15,  4,  0, 1, 1, 0, ...
  0,   4140,  320,  108,  32,  1,  0,  1, 4, 0, 0, ...
  0,  21147, 1436,  324,  51, 10, 87, 72, 5, 0, 1, ...
  0, 115975, 5556, 1409, 621, 50,  1,  0, 0, 1, 5, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A019590(n+1), A000110, A369079, A374692, A375939, A375940, A375941, A375942, A375943, A375944, A375957.
Rows n=1-2 give: A000012, A033322 (for k>=1).
Main diagonal gives A142150 (for n>=2).
A(n+1,n) gives A158416 (for n>=2).
A(n,n+1) gives A135528(n+1).

A374692 Number of partitions of [n] such that the element sum of each block is one more than a multiple of three.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 9, 17, 108, 324, 1409, 3807, 13365, 105764, 347976, 1825416, 11309203, 76330674, 453496320, 2638063039, 19573857765, 121970492469, 939636604957, 5944482289926, 41069772211068, 388012765239587, 2642662404898875, 21056674779654003

Offset: 0

Alois P. Heinz, Aug 31 2024

nonn

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(4) = 1: 1234.
a(5) = 3: 13|25|4, 1|235|4, 1|25|34.
a(6) = 9: 136|25|4, 13|256|4, 13|25|46, 16|235|4, 1|2356|4, 1|235|46, 16|25|34, 1|256|34, 1|25|346.
a(7) = 17: 1234567, 136|25|4|7, 13|256|4|7, 13|25|46|7, 13|25|4|67, 16|235|4|7, 1|2356|4|7, 1|235|46|7, 1|235|4|67, 16|25|34|7, 1|256|34|7, 1|25|346|7, 1|25|34|67, 16|25|37|4, 1|256|37|4, 1|25|367|4, 1|25|37|46.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Wikipedia, Partition of a set

Column k=3 of A375924.

Cf. A369079.

A369080 Number of permutations of [n] such that the element sum of each cycle is odd.

Original entry on oeis.org

1, 1, 1, 2, 6, 36, 180, 1080, 7560, 75600, 680400, 6804000, 74844000, 1047816000, 13621608000, 190702512000, 2860537680000, 51489678240000, 875324530080000, 15755841541440000, 299360989287360000, 6585941764321920000, 138304777050760320000, 3042705095116727040000

Offset: 0

Alois P. Heinz, Jan 12 2024

nonn

Number of permutations of [n] such that each cycle has an odd number of odd elements.

a(n+1)/a(n) is an integer for all n >= 0.

			a(0) = 1: the empty permutation.
a(1) = 1: (1).
a(2) = 1: (12).
a(3) = 2: (12)(3), (1)(23).
a(4) = 6: (124)(3), (142)(3), (12)(34), (14)(23), (1)(234), (1)(243).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Cf. A000142, A369079.

b:= proc(x, y) option remember; `if`(x+y=0, 1, add(
      `if`(j::odd, binomial(x-1, j-1)*add((i+j-1)!*
       b(x-j, y-i)*binomial(y, i), i=0..y), 0), j=1..x))
    end:
a:= n-> (h-> b(n-h, h))(iquo(n, 2)):
seq(a(n), n=0..23);
# second Maple program:
b:= n-> (<<0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>,
          <0|0|0|0|1>, <-1|1|0|0|1>>^n. <<1, 2, 3, 6, 5>>)[1, 1]:
a:= proc(n) option remember; `if`(n<2, 1, a(n-1)*b(n-2)) end:
seq(a(n), n=0..23);

A375099 Number of partitions of [n] into blocks whose element sum is <= n.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 26, 57, 141, 333, 885, 2259, 6391, 17302, 51685, 147937, 460561, 1389093, 4504136, 14127767, 47719998, 155559696, 542178148, 1835105103, 6600158865, 23035501468, 85428655084, 307266398440, 1168951871972, 4331125790382, 16897269822235

Offset: 0

Alois P. Heinz, Jul 29 2024

nonn

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 2: 12|3, 1|2|3.
a(4) = 3: 12|3|4, 13|2|4, 1|2|3|4.
a(5) = 6: 12|3|4|5, 13|2|4|5, 14|23|5, 1|23|4|5, 14|2|3|5, 1|2|3|4|5.
a(6) = 12: 123|4|5|6, 12|3|4|5|6, 13|24|5|6, 13|2|4|5|6, 14|23|5|6, 15|23|4|6, 1|23|4|5|6, 14|2|3|5|6, 15|24|3|6, 1|24|3|5|6, 15|2|3|4|6, 1|2|3|4|5|6.

Wikipedia, Partition of a set

Main diagonal of A374932.
Row sums of A375023.
Cf. A000110, A369079.

cp's OEIS Frontend

A375924 Number A(n,k) of partitions of [n] such that the element sum of each block is one more than a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A374692 Number of partitions of [n] such that the element sum of each block is one more than a multiple of three.

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A369080 Number of permutations of [n] such that the element sum of each cycle is odd.

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A375099 Number of partitions of [n] into blocks whose element sum is <= n.

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