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A368338 Number T(n,k) of partitions of [n] whose sum of block maxima minus block minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.

%I A368338 #41 Dec 27 2023 18:00:01
%S A368338 1,1,1,1,1,2,2,1,3,5,4,2,1,4,9,12,12,8,6,1,5,14,25,34,36,36,28,18,6,1,
%T A368338 6,20,44,74,100,122,132,132,108,78,36,24,1,7,27,70,139,224,318,408,
%U A368338 490,534,536,468,378,258,162,96,24,1,8,35,104,237,440,710,1032,1398,1764,2094,2296,2364,2220,1962,1584,1242,816,528,192,120
%N A368338 Number T(n,k) of partitions of [n] whose sum of block maxima minus block minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows.
%H A368338 Alois P. Heinz, <a href="/A368338/b368338.txt">Rows n = 0..33, flattened</a>
%H A368338 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A368338 Sum_{k=0..A002620(n)} k * T(n,k) = A367850(n).
%F A368338 T(n,A002620(n)) = A081123(n+1).
%e A368338 T(4,0) = 1: 1|2|3|4.
%e A368338 T(4,1) = 3: 12|3|4, 1|23|4, 1|2|34.
%e A368338 T(4,2) = 5: 123|4, 12|34, 13|2|4, 1|234, 1|24|3.
%e A368338 T(4,3) = 4: 1234, 124|3, 134|2, 14|2|3.
%e A368338 T(4,4) = 2: 13|24, 14|23.
%e A368338 T(5,5) = 8: 124|35, 125|34, 13|245, 13|25|4, 145|23, 15|23|4, 14|2|35, 15|2|34.
%e A368338 T(5,6) = 6: 134|25, 135|24, 14|235, 15|234, 14|25|3, 15|24|3.
%e A368338 T(6,9) = 6: 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
%e A368338 Triangle T(n,k) begins:
%e A368338   1;
%e A368338   1;
%e A368338   1, 1;
%e A368338   1, 2,  2;
%e A368338   1, 3,  5,  4,  2;
%e A368338   1, 4,  9, 12, 12,   8,   6;
%e A368338   1, 5, 14, 25, 34,  36,  36,  28,  18,   6;
%e A368338   1, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24;
%e A368338   ...
%p A368338 b:= proc(n, m) option remember; `if`(n=0, x^add(-i, i=m), add(
%p A368338       b(n-1, subs(j=n, m)), j=m)+expand(b(n-1, {m[], n})*x^n))
%p A368338     end:
%p A368338 T:= (n, k)-> coeff(b(n, {}), x, k):
%p A368338 seq(seq(T(n, k), k=0..(h-> h*(n-h))(iquo(n, 2))), n=0..10);
%p A368338 # second Maple program:
%p A368338 b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
%p A368338       b(n-1, s)*(k+1), 0)+`if`(n>k+1, b(n-1, {s[], n}), 0)+
%p A368338       add(expand(x^(h-n)*b(n-1, s minus {h})), h=s))(nops(s)))
%p A368338     end:
%p A368338 T:= (n, k)-> coeff(b(n, {}), x, k):
%p A368338 seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);
%Y A368338 Columns k=0..3 give: A000012, A001477(n-1), A000096(n-2), A000297(n-4).
%Y A368338 Row sums give A000110.
%Y A368338 Cf. A002620, A081123, A367450, A367850, A368401.
%K A368338 nonn,tabf
%O A368338 0,6
%A A368338 _Alois P. Heinz_, Dec 21 2023