A274547 -id:A274547 - cp's OEIS frontend

A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 5, 7, 4, 1, 0, 1, 7, 14, 12, 5, 1, 0, 1, 11, 30, 33, 19, 6, 1, 0, 1, 15, 57, 84, 62, 27, 7, 1, 0, 1, 23, 119, 222, 204, 108, 37, 8, 1, 0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1, 0, 1, 47, 460, 1425, 2006, 1558, 763, 254, 61, 10, 1

Offset: 0

Alois P. Heinz, Jun 29 2016

			T(5,1) = 1: 12345.
T(5,2) = 5: 1234|5, 123|45, 12|345, 145|23, 1|2345.
T(5,3) = 7: 123|4|5, 12|34|5, 12|3|45, 1|234|5, 145|2|3, 1|2|345, 1|23|45.
T(5,4) = 4: 12|3|4|5, 1|23|4|5, 1|2|34|5, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,  1;
  0, 1,  2,   1;
  0, 1,  3,   3,   1;
  0, 1,  5,   7,   4,   1;
  0, 1,  7,  14,  12,   5,   1;
  0, 1, 11,  30,  33,  19,   6,   1;
  0, 1, 15,  57,  84,  62,  27,   7,  1;
  0, 1, 23, 119, 222, 204, 108,  37,  8, 1;
  0, 1, 31, 224, 545, 627, 409, 169, 48, 9, 1;
  ...

Alois P. Heinz, Rows n = 0..35, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A057427, A052955(n-2) for n>1, A305777, A305778, A305779, A305780, A305781, A305782, A305783, A305784.
Diagonals include A000012, A001477, A077043.
Row sums give A274547.
T(n,ceiling(n/2)) gives A305785.
Cf. A124419, A274310 (parities alternate within blocks), A305823.

b:= proc(l, i, t) option remember; `if`(l=[], x,
     `if`(l[1]=t, 0, expand(x*b(subsop(1=[][], l), 1, 1-t)
       ))+add(`if`(l[j]=t, 0, b(subsop(j=[][], l), j, 1-t)
       ), j=i..nops(l)))
    end:
T:= n-> `if`(n=0, 1, (p-> seq(coeff(p, x, j), j=0..n))(
         b([seq(irem(i, 2), i=2..n)], 1$2))):
seq(T(n), n=0..12);

b[l_, i_, t_] := b[l, i, t] = If[l == {}, x, If[l[[1]] == t, 0, Expand[x*b[Rest[l], 1, 1 - t]]] + Sum[If[l[[j]] == t, 0, b[Delete[l, j], j, 1 - t]], {j, i, Length[l]}]];
T[n_] := If[n==0, {1}, Function[p, Table[Coefficient[p, x, j], {j, 0, n}]][ b[Table[Mod[i, 2], {i, 2, n}], 1, 1]]];
Flatten[Table[T[n], {n, 0, 12}]] (* Jean-François Alcover, May 27 2018, from Maple *)

Sum_{k=0..n} k * T(n,k) = A305823(n).

A274859 Number A(n,k) of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 5, 8, 1, 1, 2, 4, 15, 16, 1, 1, 2, 4, 8, 52, 32, 1, 1, 2, 4, 8, 18, 203, 64, 1, 1, 2, 4, 8, 16, 40, 877, 128, 1, 1, 2, 4, 8, 16, 32, 101, 4140, 256, 1, 1, 2, 4, 8, 16, 32, 68, 254, 21147, 512, 1, 1, 2, 4, 8, 16, 32, 64, 144, 723, 115975, 1024

Offset: 0

Alois P. Heinz, Jul 09 2016

			Square array A(n,k) begins:
:   1,      1,    1,   1,   1,   1,   1, ...
:   1,      1,    1,   1,   1,   1,   1, ...
:   2,      2,    2,   2,   2,   2,   2, ...
:   4,      5,    4,   4,   4,   4,   4, ...
:   8,     15,    8,   8,   8,   8,   8, ...
:  16,     52,   18,  16,  16,  16,  16, ...
:  32,    203,   40,  32,  32,  32,  32, ...
:  64,    877,  101,  68,  64,  64,  64, ...
: 128,   4140,  254, 144, 128, 128, 128, ...
: 256,  21147,  723, 304, 264, 256, 256, ...
: 512, 115975, 2064, 692, 544, 512, 512, ...

Alois P. Heinz, Antidiagonals n = 0..34, flattened

Columns k=0-10 give: A011782, A000110, A274547, A274860, A274861, A274862, A274863, A274864, A274865, A274866, A274867.

b:= proc(l, k, i, t) option remember; `if`(l=[], 1, add(`if`(l[j]=t,
      b(subsop(j=[][], l), k, j, irem(1+t, k)), 0), j=[1, $i..nops(l)]))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k=0, 2^(n-1), b([seq(
            irem(i, k), i=2..n)], k, 1, irem(2, k)))):
seq(seq(A(n, d-n), n=0..d), d=0..15);

b[l_, k_, i_, t_] := b[l, k, i, t] = If[l == {}, 1, Sum[If[l[[j]] == t, b[ReplacePart[l, j -> Nothing], k, j, Mod[1+t, k]], 0], {j, Prepend[ Range[i, Length[l]], 1]}]]; A[n_, k_] := If[n==0, 1, If[k==0, 2^(n-1), b[Flatten[Table[Mod[i, k], {i, 2, n}]], k, 1, Mod[2, k]]]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2017, translated from Maple *)

A274310 Triangle read by rows: T(n,k) = number of parity alternating partitions of [n] into k blocks (1 <= k <= m).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 11, 6, 1, 1, 10, 28, 26, 9, 1, 1, 14, 61, 86, 50, 12, 1, 1, 22, 136, 276, 236, 92, 16, 1, 1, 30, 275, 770, 927, 530, 150, 20, 1, 1, 46, 580, 2200, 3551, 2782, 1130, 240, 25, 1, 1, 62, 1141, 5710, 12160, 12632, 6987, 2130, 355, 30, 1

Offset: 1

N. J. A. Sloane, Jun 23 2016

The first element of any block may be odd or even and then the parity of terms alternates within each block. - Alois P. Heinz, Jun 28 2016

Let a(n,k,i) be the number of parity alternating partitions of n into k blocks, i of which have even maximal elements. Dzhumadil'daev and Yeliussizov, Proposition 5.3, give recurrences for a(n,k,i), which depend on the parity of n. It is easy to verify that the solution to these recurrences is given by a(2*n,k,i) = Stirling2(n,i)*Stirling2(n+1,k+1-i) and a(2*n+1,k,i) = Stirling2(n+1,i+1) * Stirling2(n+1,k-i). The formula below for the table entries T(n,k) follows from this observation. - Peter Bala, Apr 09 2018

			Triangle begins:
  1;
  1,   1;
  1,   2,   1;
  1,   4,   4,   1;
  1,   6,  11,   6,   1;
  1,  10,  28,  26,   9,   1;
  1,  14,  61,  86,  50,  12,   1;
  1,  22, 136, 276, 236,  92,  16,   1;
  ...
From _Alois P. Heinz_, Jun 28 2016: (Start)
T(5,1) = 1: 12345.
T(5,2) = 6: 1234|5, 123|45, 125|34, 12|345, 145|23, 1|2345.
T(5,3) = 11: 123|4|5, 12|34|5, 125|3|4, 12|3|45, 14|23|5, 1|234|5, 1|23|45, 145|2|3, 14|25|3, 1|25|34, 1|2|345.
T(5,4) = 6: 12|3|4|5, 1|23|4|5, 14|2|3|5, 1|2|34|5, 1|25|3|4, 1|2|3|45.
T(5,5) = 1: 1|2|3|4|5. (End)

Alois P. Heinz, Rows n = 1..141, flattened
Askar Dzhumadil'daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

Row sums give A124419(n+1).

Cf. A274547, A274581.

A274310 := proc (n, k) local i;
with(combinat):
   add(Stirling2(floor((1/2)*n+1), i+1)*Stirling2(floor((1/2)*n+1/2), k-i), i = 0..k-1);
end proc:
for n from 1 to 10 do
   seq(A274310(n, k), k = 1..n);
end do; # Peter Bala, Apr 09 2018

T[n_, k_] = Sum[StirlingS2[Floor[(n + 2)/2], i + 1] * StirlingS2[Floor[(n + 1)/2], k - i], {i, 0, k - 1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, May 17 2018, after Peter Bala *)

T(n,k) = Sum_{i = 0..k-1} Stirling2(floor((n+2)/2), i+1) * Stirling2(floor((n+1)/2), k-i). - Peter Bala, Apr 09 2018

More terms from Alois P. Heinz, Jun 26 2016

A363519 Number T(n,k) of partitions of [n] having exactly k parity changes within the partition, n>=0, 0<=k<=max(0,n-1), read by rows.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 4, 0, 3, 4, 8, 0, 2, 18, 14, 18, 0, 7, 27, 87, 42, 40, 0, 5, 102, 162, 360, 147, 101, 0, 20, 179, 866, 931, 1456, 434, 254, 0, 15, 675, 1746, 5836, 4755, 5778, 1619, 723, 0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064, 0, 52, 5216, 19863, 93452, 117172, 206570, 115178, 94210, 20271, 6586

Offset: 0

Alois P. Heinz, Jun 07 2023

The blocks are ordered with increasing least elements.

			T(4,1) = 3: 134|2, 13|24, 13|2|4.
T(4,2) = 4: 124|3, 14|23, 14|2|3, 1|24|3.
T(4,3) = 8: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34, 1|2|3|4.
T(5,2) = 18: 1245|3, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5.
T(5,4) = 18: 12345, 1234|5, 123|45, 123|4|5, 12|345, 12|34|5, 12|3|45, 12|3|4|5, 145|23, 1|2345, 1|234|5, 1|23|45, 1|23|4|5, 145|2|3, 1|2|345, 1|2|34|5, 1|2|3|45, 1|2|3|4|5.
Triangle T(n,k) begins:
  1;
  1;
  0,  2;
  0,  1,    4;
  0,  3,    4,    8;
  0,  2,   18,   14,    18;
  0,  7,   27,   87,    42,    40;
  0,  5,  102,  162,   360,   147,   101;
  0, 20,  179,  866,   931,  1456,   434,   254;
  0, 15,  675, 1746,  5836,  4755,  5778,  1619,  723;
  0, 67, 1321, 9087, 16416, 36031, 22893, 23052, 5044, 2064;
  ...

Alois P. Heinz, Rows n = 0..27, flattened
Wikipedia, Partition of a set

Column k=1 gives A363550.
Row sums give A000110.
T(n,max(0,n-1)) gives A274547.
Cf. A363493, A363549.

b:= proc(l, i, t) option remember; expand(`if`(l=[], 1,
      add((f-> b(subsop(j=[][], l), j, `if`(f, 1-t, t))*
      `if`(f, x, 1))(l[j]=t), j=[1, $i..nops(l)])))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
         b([ seq(irem(i, 2), i=2..n)], 1, 0)):
seq(T(n), n=0..12);

b[l_, i_, t_] := b[l, i, t] = Expand[If[l == {}, 1, Sum[Function[f, b[ReplacePart[l, j -> Nothing], j, If[f, 1 - t, t]]*If[f, x, 1]][l[[j]] == t], {j, Join[{1}, Range[i, Length@l]]}]]];
T[n_] := CoefficientList[b[ Table[Mod[i, 2], {i, 2, n}], 1, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 17 2023, after Alois P. Heinz *)

Sum_{k=0..max(0,n-1)} k * T(n,k) = A363549(n).

cp's OEIS Frontend

A274581 Number T(n,k) of set partitions of [n] with alternating parity of elements and exactly k blocks; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A274859 Number A(n,k) of set partitions of [n] such that the difference between each element and its index (in the partition) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A274310 Triangle read by rows: T(n,k) = number of parity alternating partitions of [n] into k blocks (1 <= k <= m).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A363519 Number T(n,k) of partitions of [n] having exactly k parity changes within the partition, n>=0, 0<=k<=max(0,n-1), read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula