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

%I A274310 #33 May 17 2018 08:30:20
%S A274310 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,
%T A274310 22,136,276,236,92,16,1,1,30,275,770,927,530,150,20,1,1,46,580,2200,
%U A274310 3551,2782,1130,240,25,1,1,62,1141,5710,12160,12632,6987,2130,355,30,1
%N A274310 Triangle read by rows: T(n,k) = number of parity alternating partitions of [n] into k blocks (1 <= k <= m).
%C A274310 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
%C A274310 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
%H A274310 Alois P. Heinz, <a href="/A274310/b274310.txt">Rows n = 1..141, flattened</a>
%H A274310 Askar Dzhumadil'daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%F A274310 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
%e A274310 Triangle begins:
%e A274310   1;
%e A274310   1,   1;
%e A274310   1,   2,   1;
%e A274310   1,   4,   4,   1;
%e A274310   1,   6,  11,   6,   1;
%e A274310   1,  10,  28,  26,   9,   1;
%e A274310   1,  14,  61,  86,  50,  12,   1;
%e A274310   1,  22, 136, 276, 236,  92,  16,   1;
%e A274310   ...
%e A274310 From _Alois P. Heinz_, Jun 28 2016: (Start)
%e A274310 T(5,1) = 1: 12345.
%e A274310 T(5,2) = 6: 1234|5, 123|45, 125|34, 12|345, 145|23, 1|2345.
%e A274310 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.
%e A274310 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.
%e A274310 T(5,5) = 1: 1|2|3|4|5. (End)
%p A274310 A274310 := proc (n, k) local i;
%p A274310 with(combinat):
%p A274310    add(Stirling2(floor((1/2)*n+1), i+1)*Stirling2(floor((1/2)*n+1/2), k-i), i = 0..k-1);
%p A274310 end proc:
%p A274310 for n from 1 to 10 do
%p A274310    seq(A274310(n, k), k = 1..n);
%p A274310 end do; # _Peter Bala_, Apr 09 2018
%t A274310 T[n_, k_] = Sum[StirlingS2[Floor[(n + 2)/2], i + 1] * StirlingS2[Floor[(n + 1)/2], k - i], {i, 0, k - 1}];
%t A274310 Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, May 17 2018, after _Peter Bala_ *)
%Y A274310 Row sums give A124419(n+1).
%Y A274310 Cf. A274547, A274581.
%K A274310 nonn,tabl,easy
%O A274310 1,5
%A A274310 _N. J. A. Sloane_, Jun 23 2016
%E A274310 More terms from _Alois P. Heinz_, Jun 26 2016