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A091298
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Triangle read by rows: T(n,k) is the number of plane partitions of n containing exactly k parts.
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15
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1, 1, 2, 1, 2, 3, 1, 4, 3, 5, 1, 4, 7, 5, 7, 1, 6, 10, 13, 7, 11, 1, 6, 14, 20, 19, 11, 15, 1, 8, 18, 33, 32, 31, 15, 22, 1, 8, 25, 43, 56, 54, 43, 22, 30, 1, 10, 29, 66, 81, 99, 78, 64, 30, 42, 1, 10, 37, 83, 126, 150, 148, 118, 88, 42, 56, 1, 12, 44, 114, 174, 246, 235, 230, 166, 124, 56, 77
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OFFSET
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1,3
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COMMENTS
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First column is 1, representing the single-part {{n}}, last column is P(n), since the all-ones plane partitions form the Ferrers-Young plots of the (linear) partitions of n.
A plane partition of n is a two-dimensional table (or matrix) with nonnegative elements summing up to n, and nonincreasing rows and columns. (Zero rows and columns are ignored.) - M. F. Hasler, Sep 22 2018
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LINKS
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EXAMPLE
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This plane partition of n=7: {{3,1,1},{2}} contains 4 parts: 3,1,1,2.
Triangle T(n,k) begins:
1;
1, 2;
1, 2, 3;
1, 4, 3, 5;
1, 4, 7, 5, 7;
1, 6, 10, 13, 7, 11;
1, 6, 14, 20, 19, 11, 15;
1, 8, 18, 33, 32, 31, 15, 22;
1, 8, 25, 43, 56, 54, 43, 22, 30;
1, 10, 29, 66, 81, 99, 78, 64, 30, 42;
...
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MATHEMATICA
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(* see A089924 for "planepartition" *) Table[Length /@ Split[Sort[Length /@ Flatten /@ planepartitions[n]]], {n, 16}]
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PROG
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(PARI) A091298(n, k)=sum(i=1, #n=PlanePartitions(n), sum(j=1, #n[i], #n[i][j])==k)
PlanePartitions(n, L=0, PP=List())={ n<2&&return([if(n, [[1]], [])]); for(N=1, n, my(P=apply(Vecrev, if(L, select(p->vecmin(L-Vecrev(p, #L))>=0, partitions(N, L[1], #L)), partitions(N)))); if(N<n, for(i=1, #P, my(pp = PlanePartitions(n-N, P[i])); for(j=1, #pp, listput(PP, concat([P[i]], pp[j])))), for(i=1, #P, listput(PP, [P[i]])))); Set(PP)} \\ M. F. Hasler, Sep 24 2018
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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