Search: a219273 -id:a219273
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A219272
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Number A(n,k) of standard Young tableaux for partitions of n into distinct parts with largest part <= k; triangle A(n,k), k>=0, 0<=n<=k*(k+1)/2, read by columns.
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+10
7
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1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 5, 16, 1, 1, 1, 3, 4, 9, 25, 49, 70, 168, 768, 1, 1, 1, 3, 4, 10, 30, 63, 162, 372, 1506, 3300, 7887, 15015, 48048, 292864, 1, 1, 1, 3, 4, 10, 31, 69, 182, 525, 1911, 5115, 17347, 43758, 149721, 626769, 1946516, 4934930
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OFFSET
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0,7
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COMMENTS
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A(n,k) is defined for n,k >= 0. A(n,k) = 0 iff n > k*(k+1)/2 = A000217(k). The triangle contains only the nonzero terms. A(n,k) = A(n,n) for k>=n.
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LINKS
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FORMULA
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T(n,k) = Sum_{i=0..k} A219274(n,i).
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EXAMPLE
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A(3,2) = 2:
+------+ +------+
| 1 2 | | 1 3 |
| 3 .--+ | 2 .--+
+---+ +---+
A(3,3) = 3:
+------+ +------+ +---------+
| 1 2 | | 1 3 | | 1 2 3 |
| 3 .--+ | 2 .--+ +---------+
+---+ +---+
Triangle A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
. 1, 1, 1, 1, 1, 1, 1, 1, ...
. 1, 1, 1, 1, 1, 1, 1, ...
. 2, 3, 3, 3, 3, 3, 3, ...
. 3, 4, 4, 4, 4, 4, ...
. 5, 9, 10, 10, 10, 10, ...
. 16, 25, 30, 31, 31, 31, ...
. 49, 63, 69, 70, 70, ...
. 70, 162, 182, 189, 190, ...
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MAPLE
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h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) local s; s:=i*(i+1)/2;
`if`(n=s, h([l[], seq(i-j, j=0..i-1)]), `if`(n>s, 0,
g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i-1, [l[], i]))))
end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, k), n=0..k*(k+1)/2), k=0..7);
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MATHEMATICA
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h[l_] := With[{n=Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[If[ l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := g[n, i, l] = With[{s=i*(i+1)/2}, If[n==s, h[Join[l, Table[ i-j, {j, 0, i-1}]]], If[n>s, 0, g[n, i-1, l] + If[i>n, 0, g[n-i, i-1, Append[l, i]]]]]];
A[n_, k_] := g[n, k, {}];
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CROSSREFS
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Leftmost nonzero elements give A219339.
Column of leftmost nonzero element is A002024(n) for n>0.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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A219275
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Number of standard Young tableaux for partitions of nonnegative integers into distinct parts with largest part n.
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+10
3
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1, 1, 3, 25, 1069, 368168, 1299366501, 55208013380403, 32401197537296758130, 297072961835477978342245712, 47538199827835784548062928051198402, 146779873623344672821145371965795071455181183, 9581411392319396646028223743176779937161862866453789852
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OFFSET
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0,3
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LINKS
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EXAMPLE
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a(2) = 3:
+------+ +------+ +------+
| 1 2 | | 1 3 | | 1 2 |
| 3 .--+ | 2 .--+ +------+
+---+ +---+
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MAPLE
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h:= proc(l) local n; n:=nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
b:= (n, l)-> `if`(n<1, h(l), b(n-1, l) +b(n-1, [l[], n])):
a:= n-> `if`(n=0, 1, b(n-1, [n])):
seq(a(n), n=0..12);
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MATHEMATICA
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h[l_] := With[{n = Length[l]}, Total[l]!/Product[ Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];
a[n_] := If[n == 0, 1, b[n - 1, {n}]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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