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A124496
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Triangle read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the size of the last block is k, 1<=k<=n; the blocks are ordered with increasing least elements.
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10
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1, 1, 1, 3, 1, 1, 9, 4, 1, 1, 31, 14, 5, 1, 1, 121, 54, 20, 6, 1, 1, 523, 233, 85, 27, 7, 1, 1, 2469, 1101, 400, 125, 35, 8, 1, 1, 12611, 5625, 2046, 635, 175, 44, 9, 1, 1, 69161, 30846, 11226, 3488, 952, 236, 54, 10, 1, 1, 404663, 180474, 65676, 20425, 5579, 1366, 309, 65, 11, 1, 1
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OFFSET
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1,4
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COMMENTS
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Number of restricted growth functions of length n with a multiplicity k of the maximum value. RGF's are here defined as f(1)=1, f(i) <= 1+max_{1<=j<i} f(j). - R. J. Mathar, Mar 18 2016
This is table 9.2 in the Gould-Quaintance reference. - Peter Luschny, Apr 25 2016
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LINKS
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FORMULA
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The row enumerating polynomial P[n](t)=Q[n](t,1), where Q[1](t,s)=ts and Q[n](t,s)=s*dQ[n-1](t,s)/ds +(t-1)Q[n-1](t,s)+tsQ[n-1](1,s) for n>=2.
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EXAMPLE
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T(4,2) = 4 because we have 13|24, 14|23, 12|34 and 1|2|34.
Triangle starts:
1;
1,1;
3,1,1;
9,4,1,1;
31,14,5,1,1;
121,54,20,6,1,1;
523,233,85,27,7,1,1;
2469,1101,400,125,35,8,1,1;
12611,5625,2046,635,175,44,9,1,1;
69161,30846,11226,3488,952,236,54,10,1,1;
404663,180474,65676,20425,5579,1366,309,65,11,1,1;
2512769,1120666,407787,126817,34685,8494,1893,395,77,12,1,1;
...
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MAPLE
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Q[1]:=t*s: for n from 2 to 12 do Q[n]:=expand(t*s*subs(t=1, Q[n-1])+s*diff(Q[n-1], s)+t*Q[n-1]-Q[n-1]) od:for n from 1 to 12 do P[n]:=sort(subs(s=1, Q[n])) od: for n from 1 to 12 do seq(coeff(P[n], t, j), j=1..n) od;
# second Maple program:
T:= proc(n, k) option remember; `if`(n=k, 1,
add(T(n-j, k)*binomial(n-1, j-1), j=1..n-k))
end:
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MATHEMATICA
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T[n_, k_] := T[n, k] = If[n == k, 1, Sum[T[n-j, k]*Binomial[n-1, j-1], {j, 1, n-k}]];
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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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