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%I A364971 #32 Oct 27 2023 08:23:22
%S A364971 1,1,2,2,3,5,6,4,2,35,10,5,27,60,95,15,6,2,371,315,161,21,7,142,938,
%T A364971 2002,770,252,28,8,282,4005,9744,5313,1386,372,36,9,1073,16950,50275,
%U A364971 33705,11082,2310,525,45,10,2,74657,283525,217800,78078,20097,3630,715,55,11
%N A364971 Number T(n,k) of partitions of [n] for which the difference between the longest and the shortest block size is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%C A364971 T(0,0) = 1 by convention.
%H A364971 Alois P. Heinz, <a href="/A364971/b364971.txt">Rows n = 0..150, flattened</a>
%H A364971 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A364971 T(4,0) = 5: 1|2|3|4, 12|34, 13|24, 14|23, 1234.
%e A364971 T(4,1) = 6: 1|2|34, 1|23|4, 1|24|3, 12|3|4, 13|2|4, 14|2|3.
%e A364971 T(4,2) = 4: 1|234, 123|4, 124|3, 134|2.
%e A364971 Triangle T(n,k) begins:
%e A364971      1;
%e A364971      1;
%e A364971      2;
%e A364971      2,     3;
%e A364971      5,     6,     4;
%e A364971      2,    35,    10,     5;
%e A364971     27,    60,    95,    15,     6;
%e A364971      2,   371,   315,   161,    21,    7;
%e A364971    142,   938,  2002,   770,   252,   28,   8;
%e A364971    282,  4005,  9744,  5313,  1386,  372,  36,  9;
%e A364971   1073, 16950, 50275, 33705, 11082, 2310, 525, 45, 10;
%e A364971   ...
%p A364971 b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
%p A364971      b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1), j=1..n))
%p A364971     end:
%p A364971 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
%p A364971 seq(T(n), n=0..12);
%t A364971 b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1], {j, 1, n}]];
%t A364971 T[n_] := CoefficientList[b[n, n, 0], x];
%t A364971 Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 27 2023, after _Alois P. Heinz_ *)
%Y A364971 Row sums give A000110.
%Y A364971 Column k=0 gives A038041 (for n>=1).
%Y A364971 T(n,n-2) gives A000027 (for n>=2).
%Y A364971 Cf. A080510, A178979, A364967.
%K A364971 nonn,tabf
%O A364971 0,3
%A A364971 _Alois P. Heinz_, Aug 15 2023

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