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%I #39 Jan 08 2015 06:01:26
%S 1,0,1,0,1,1,0,2,0,1,0,3,1,0,1,0,6,0,0,0,1,0,7,2,1,0,0,1,0,13,1,0,0,0,
%T 0,1,0,16,4,0,1,0,0,0,1,0,25,2,2,0,0,0,0,0,1,0,33,6,1,0,1,0,0,0,0,1,0,
%U 49,4,2,0,0,0,0,0,0,0,1,0,61,9,3,2,0,1,0,0,0,0,0,1,0,90,6,3,1,0,0,0,0,0,0,0,0,1,0,113,16,2,2,0,0,1,0,0,0,0,0,0,1,0,156,9,7,1,2,0,0,0,0,0,0,0,0,0,1
%N Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.
%C T(0,0) = 1 by convention.
%C Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.
%C Row sums are A000041.
%H Joerg Arndt and Alois P. Heinz, <a href="/A243978/b243978.txt">Table of n, a(n) for n = 0..10010</a> (rows 0..140, flattened)
%e Triangle starts:
%e 00: 1;
%e 01: 0, 1;
%e 02: 0, 1, 1;
%e 03: 0, 2, 0, 1;
%e 04: 0, 3, 1, 0, 1;
%e 05: 0, 6, 0, 0, 0, 1;
%e 06: 0, 7, 2, 1, 0, 0, 1;
%e 07: 0, 13, 1, 0, 0, 0, 0, 1;
%e 08: 0, 16, 4, 0, 1, 0, 0, 0, 1;
%e 09: 0, 25, 2, 2, 0, 0, 0, 0, 0, 1;
%e 10: 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1;
%e 11: 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1;
%e 12: 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
%e 13: 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 14: 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
%e 15: 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 16: 0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
%e 17: 0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 18: 0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 19: 0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e 20: 0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
%e ...
%e The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are:
%e 01: [ 1 1 1 1 1 1 1 1 1 ] 9
%e 02: [ 1 1 1 1 1 1 1 2 ] 1
%e 03: [ 1 1 1 1 1 1 3 ] 1
%e 04: [ 1 1 1 1 1 2 2 ] 2
%e 05: [ 1 1 1 1 1 4 ] 1
%e 06: [ 1 1 1 1 2 3 ] 1
%e 07: [ 1 1 1 1 5 ] 1
%e 08: [ 1 1 1 2 2 2 ] 3
%e 09: [ 1 1 1 2 4 ] 1
%e 10: [ 1 1 1 3 3 ] 2
%e 11: [ 1 1 1 6 ] 1
%e 12: [ 1 1 2 2 3 ] 1
%e 13: [ 1 1 2 5 ] 1
%e 14: [ 1 1 3 4 ] 1
%e 15: [ 1 1 7 ] 1
%e 16: [ 1 2 2 2 2 ] 1
%e 17: [ 1 2 2 4 ] 1
%e 18: [ 1 2 3 3 ] 1
%e 19: [ 1 2 6 ] 1
%e 20: [ 1 3 5 ] 1
%e 21: [ 1 4 4 ] 1
%e 22: [ 1 8 ] 1
%e 23: [ 2 2 2 3 ] 1
%e 24: [ 2 2 5 ] 1
%e 25: [ 2 3 4 ] 1
%e 26: [ 2 7 ] 1
%e 27: [ 3 3 3 ] 3
%e 28: [ 3 6 ] 1
%e 29: [ 4 5 ] 1
%e 30: [ 9 ] 1
%e Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1].
%p b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
%p end:
%p T:= (n, k)-> b(n$2, k) -`if`(n=0 and k=0, 0, b(n$2, k+1)):
%p seq(seq(T(n, k), k=0..n), n=0..14);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 08 2015, translated from Maple *)
%Y Cf. A183568, A242451 (the same for compositions).
%Y Cf. A091602 (partitions by max multiplicity of any part).
%K nonn,tabl
%O 0,8
%A _Joerg Arndt_ and _Alois P. Heinz_, Jun 28 2014
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