%I #41 Dec 27 2023 18:00:01

%S 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 6,20,44,74,100,122,132,132,108,78,36,24,1,7,27,70,139,224,318,408,

%U 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 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 Alois P. Heinz, <a href="/A368338/b368338.txt">Rows n = 0..33, flattened</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F Sum_{k=0..A002620(n)} k * T(n,k) = A367850(n).

%F T(n,A002620(n)) = A081123(n+1).

%e T(4,0) = 1: 1|2|3|4.

%e T(4,1) = 3: 12|3|4, 1|23|4, 1|2|34.

%e T(4,2) = 5: 123|4, 12|34, 13|2|4, 1|234, 1|24|3.

%e T(4,3) = 4: 1234, 124|3, 134|2, 14|2|3.

%e T(4,4) = 2: 13|24, 14|23.

%e 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 T(5,6) = 6: 134|25, 135|24, 14|235, 15|234, 14|25|3, 15|24|3.

%e T(6,9) = 6: 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 2, 2;

%e 1, 3, 5, 4, 2;

%e 1, 4, 9, 12, 12, 8, 6;

%e 1, 5, 14, 25, 34, 36, 36, 28, 18, 6;

%e 1, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24;

%e ...

%p b:= proc(n, m) option remember; `if`(n=0, x^add(-i, i=m), add(

%p b(n-1, subs(j=n, m)), j=m)+expand(b(n-1, {m[], n})*x^n))

%p end:

%p T:= (n, k)-> coeff(b(n, {}), x, k):

%p seq(seq(T(n, k), k=0..(h-> h*(n-h))(iquo(n, 2))), n=0..10);

%p # second Maple program:

%p b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,

%p b(n-1, s)*(k+1), 0)+`if`(n>k+1, b(n-1, {s[], n}), 0)+

%p add(expand(x^(h-n)*b(n-1, s minus {h})), h=s))(nops(s)))

%p end:

%p T:= (n, k)-> coeff(b(n, {}), x, k):

%p seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);

%Y Columns k=0..3 give: A000012, A001477(n-1), A000096(n-2), A000297(n-4).

%Y Row sums give A000110.

%Y Cf. A002620, A081123, A367450, A367850, A368401.

%K nonn,tabf

%O 0,6

%A _Alois P. Heinz_, Dec 21 2023