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.
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, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24, 1, 7, 27, 70, 139, 224, 318, 408, 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
Offset: 0
Examples
T(4,0) = 1: 1|2|3|4. T(4,1) = 3: 12|3|4, 1|23|4, 1|2|34. T(4,2) = 5: 123|4, 12|34, 13|2|4, 1|234, 1|24|3. T(4,3) = 4: 1234, 124|3, 134|2, 14|2|3. T(4,4) = 2: 13|24, 14|23. T(5,5) = 8: 124|35, 125|34, 13|245, 13|25|4, 145|23, 15|23|4, 14|2|35, 15|2|34. T(5,6) = 6: 134|25, 135|24, 14|235, 15|234, 14|25|3, 15|24|3. T(6,9) = 6: 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34. Triangle T(n,k) begins: 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, 6, 20, 44, 74, 100, 122, 132, 132, 108, 78, 36, 24; ...
Links
- Alois P. Heinz, Rows n = 0..33, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, x^add(-i, i=m), add( b(n-1, subs(j=n, m)), j=m)+expand(b(n-1, {m[], n})*x^n)) end: T:= (n, k)-> coeff(b(n, {}), x, k): seq(seq(T(n, k), k=0..(h-> h*(n-h))(iquo(n, 2))), n=0..10); # second Maple program: b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k, b(n-1, s)*(k+1), 0)+`if`(n>k+1, b(n-1, {s[], n}), 0)+ add(expand(x^(h-n)*b(n-1, s minus {h})), h=s))(nops(s))) end: T:= (n, k)-> coeff(b(n, {}), x, k): seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);