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A318806 Triangular array read by rows, where T(n,k) is the number of almost distinct partitions of n in which every part is <= k for 1 <= k <= n.

%I A318806 #25 Dec 12 2018 14:24:26
%S A318806 1,1,2,1,2,3,1,2,3,4,1,2,4,5,6,1,2,4,6,7,8,1,2,4,7,9,10,11,1,2,4,7,10,
%T A318806 12,13,14,1,2,4,8,12,15,17,18,19,1,2,4,8,13,17,20,22,23,24,1,2,4,8,14,
%U A318806 20,24,27,29,30,31,1,2,4,8,15,22,28,32,35,37,38,39,1,2,4,8,15,24,32,38,42,45,47
%N A318806 Triangular array read by rows, where T(n,k) is the number of almost distinct partitions of n in which every part is <= k for 1 <= k <= n.
%C A318806 An almost distinct partition of n with parts bounded by k is a decreasing sequence of positive integers (a(1), a(2), ..., a(k)) such that n = a(1) + a(2) +...+ a(k), any a(i) > 1 is distinct from all other values, and all a(i) <= k.
%H A318806 Michael De Vlieger, <a href="/A318806/b318806.txt">Table of n, a(n) for n = 1..3240</a> (rows 1 <= n <= 80, flattened).
%H A318806 Sara Billey, Matjaž Konvalinka, and Joshua P. Swanson, <a href="http://arxiv.org/abs/1809.07386">Tableaux posets and the fake degrees of coinvariant algebras</a>, arXiv:1809.07386 [math.CO], 2018.
%e A318806 There are T(5,6) = 7 almost distinct partitions of 6 in which every part is <= 5: [5,1], [4,2], [4,1,1], [3,2,1], [3,1,1,1], [2,1,1,1,1], [1,1,1,1,1,1].
%e A318806 Triangle starts:
%e A318806 1;
%e A318806 1, 2;
%e A318806 1, 2, 3;
%e A318806 1, 2, 3, 4;
%e A318806 1, 2, 4, 5,  6;
%e A318806 1, 2, 4, 6,  7,  8;
%e A318806 1, 2, 4, 7,  9, 10, 11;
%e A318806 1, 2, 4, 7, 10, 12, 13, 14;
%e A318806 1, 2, 4, 8, 12, 15, 17, 18, 19;
%e A318806 1, 2, 4, 8, 13, 17, 20, 22, 23, 24;
%e A318806 ...
%t A318806 Array[Table[Count[#, _?(# <= k &)], {k, Max@ #}] &@ DeleteCases[Map[Boole[Flatten@ MapAt[Union, TakeDrop[#, LengthWhile[#, # == 1 &]], -1] == # &@ Reverse@ #] Max@ # &, Reverse@ IntegerPartitions[#]], 0] &, 13] // Flatten (* _Michael De Vlieger_, Dec 12 2018 *)
%Y A318806 Cf. A000009, A026820, A008302.
%K A318806 nonn,tabl
%O A318806 1,3
%A A318806 _Sara Billey_, Sep 04 2018