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

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

1,5

Comments

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

Links

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.

Formula

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

Example

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)

Maple

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

Mathematica

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 *)

Crossrefs

Row sums give A124419(n+1).

Cf. A274547, A274581.

Sequence in context: A274643 A172991 A203906 * A096806 A116672 A161126

Adjacent sequences: A274307 A274308 A274309 * A274311 A274312 A274313

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jun 23 2016

Extensions

More terms from Alois P. Heinz, Jun 26 2016

Status

approved