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A233321
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Triangle read by rows: T(n,k) = number of palindromic partitions of n in which the largest part is equal to k, 1 <= k <= n.
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2
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1, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 1, 1, 3, 1, 1, 0, 1, 1, 1, 3, 0, 1, 0, 1, 1, 4, 2, 3, 0, 1, 0, 1, 1, 2, 4, 1, 2, 0, 1, 0, 1, 1, 5, 3, 5, 1, 2, 0, 1, 0, 1, 1, 2, 6, 2, 4, 0, 2, 0, 1, 0, 1, 1, 6, 5, 8, 2, 4, 0, 2, 0, 1, 0, 1, 1, 3, 8, 3, 7, 1, 3, 0, 2, 0, 1, 0, 1, 1, 7, 7, 11, 4, 7, 1, 3, 0, 2, 0, 1, 0, 1
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OFFSET
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1,8
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COMMENTS
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A partition of n is said to be "palindromic" if its parts can be arranged to form a palindrome in at least one way (cf. A025065).
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LINKS
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EXAMPLE
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Triangle begins:
1;
1, 1;
1, 0, 1;
1, 2, 0, 1;
1, 1, 1, 0, 1;
1, 3, 1, 1, 0, 1;
1, 1, 3, 0, 1, 0, 1;
1, 4, 2, 3, 0, 1, 0, 1;
1, 2, 4, 1, 2, 0, 1, 0, 1;
1, 5, 3, 5, 1, 2, 0, 1, 0, 1;
1, 2, 6, 2, 4, 0, 2, 0, 1, 0, 1;
...
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MATHEMATICA
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(* run this first: *)
Needs["Combinatorica`"];
(* run the following in a different cell: *)
a233321[n_] := {}; Do[Do[a = Partitions[n]; count = 0; Do[If[Max[a[[j]]] == k, x = Permutations[a[[j]]]; Do[If[x[[m]] == Reverse[x[[m]]], count++; Break[]], {m, Length[x]}]], {j, Length[a]}]; AppendTo[a233321[n], count], {k, n}], {n, nmax}]; Table[a233321[n], {n, nmax}](* L. Edson Jeffery, Oct 09 2017 *)
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PROG
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PartitionCount(n, maxpartsize)={my(t=0); forpart(p=n, t++, maxpartsize); t}
V(n, k)=sum(i=0, (k-n%2)\2, PartitionCount(n\2-i, k));
T(n, k)=V(n, k)-V(n, k-1);
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Oct 09 2017
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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