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Triangle read by rows: number of 321-avoiding ordered set partitions of [n] into k blocks, n>=1, 1<=k<=n.
0

%I #10 Jul 23 2018 09:53:07

%S 1,1,2,1,6,5,1,14,27,14,1,30,99,112,42,1,62,307,564,450,132,1,126,867,

%T 2284,2895,1782,429,1,254,2307,8124,14485,13992,7007,1430,1,510,5891,

%U 26492,62085,83446,65065,27456,4862,1,1022,14595,81148,239269,418578,450905,294632,107406,16796

%N Triangle read by rows: number of 321-avoiding ordered set partitions of [n] into k blocks, n>=1, 1<=k<=n.

%H Anisse Kasraoui, <a href="http://arxiv.org/abs/1307.0495">Pattern avoidance in ordered set partitions and words</a>, arXiv:1307.0495 [math.CO], 2013.

%e Triangle starts

%e 1

%e 1 2

%e 1 6 5

%e 1 14 27 14

%e 1 30 99 112 42

%e 1 62 307 564 450 132

%e 1 126 867 2284 2895 1782 429

%e 1 254 2307 8124 14485 13992 7007 1430

%e 1 510 5891 26492 62085 83446 65065 27456 4862

%e 1 1022 14595 81148 239269 418578 450905 294632 107406 16796

%t T[n_, k_] := (-1)^(k-1) k + Sum[(-1)^(k-j) Binomial[k, j+2] 2^(n-2j) Sum[(2 j-2i+1)/(j+1) Binomial[2i, i] Binomial[2(j-i), j-i] Binomial[n+2i, 2i], {i, 0, j}], {j, 0, k-2}];

%t Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 23 2018, from Kasraoui theorem 3.5 *)

%K nonn,tabl

%O 1,3

%A _Joerg Arndt_, Jul 03 2013