Search: a368204 -id:a368204
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A124327
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Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).
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+10
8
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1, 1, 0, 1, 1, 0, 2, 1, 0, 1, 1, 0, 4, 2, 1, 3, 2, 1, 0, 1, 1, 0, 8, 4, 2, 10, 6, 7, 2, 5, 3, 2, 1, 0, 1, 1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1, 1, 0, 32, 16, 8, 85, 56, 64, 42, 101, 62, 75, 69, 47, 54, 38, 38, 24, 23, 10, 13, 7, 5, 3, 2, 1, 0, 1, 1, 0, 64, 32, 16
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OFFSET
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1,7
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COMMENTS
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Row n has n(n+1)/2 terms. Row sums yield the Bell numbers (A000110). T(n,1)=1; T(n,2)=0; T(n,3)=2^(n-2). for n>=2; T(n,4)=2^(n-3) for n>=3; T(n,5)=2^(n-4) for n>=4.
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LINKS
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FORMULA
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The generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(n,t,s)=s*dQ(n-1,t,s)/ds + st^n*Q(n-1,t,s); Q(1,t,s)=ts.
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EXAMPLE
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T(4,7) = 2 because we have 13|2|4 and 1|23|4.
Triangle starts:
1;
1, 0, 1;
1, 0, 2, 1, 0, 1;
1, 0, 4, 2, 1, 3, 2, 1, 0, 1;
1, 0, 8, 4, 2, 10, 6, 7, 2, 5, 3, 2, 1, 0, 1;
1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1;
...
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MAPLE
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Q[1]:=t*s: for n from 2 to 8 do Q[n]:=expand(s*diff(Q[n-1], s)+t^n*s*Q[n-1]) od: for n from 1 to 8 do P[n]:=sort(subs(s=1, Q[n])) od: for n from 1 to 8 do seq(coeff(P[n], t, k), k=1..n*(n+1)/2) od; # yields sequence in triangular form
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MATHEMATICA
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Q[1, t_, s_] := t s;
Q[n_, t_, s_] := Q[n, t, s] = s D[Q[n-1, t, s], s] + s t^n Q[n-1, t, s] // Expand;
P[n_, t_] := Q[n, t, s] /. s -> 1;
T[n_] := Rest@CoefficientList[P[n, t], t];
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CROSSREFS
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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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A365441
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Number of partitions of [2n] whose block maxima sum to 3n.
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+10
5
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1, 1, 2, 5, 10, 23, 47, 103, 209, 449, 908, 1909, 3864, 8011, 16165, 33244, 66933, 136628, 274876, 558107, 1121160, 2268526, 4552291, 9183569, 18417449, 37073504, 74300048, 149334422, 299134695, 600481001, 1202436958, 2411536369, 4827532935, 9675363921, 19364235775
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OFFSET
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0,3
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COMMENTS
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Rows of A367955 and the reversed columns of A367955 converge to this sequence.
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LINKS
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FORMULA
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EXAMPLE
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a(0) = 1: the empty partition.
a(1) = 1: 1|2.
a(2) = 2: 12|34, 134|2.
a(3) = 5: 123|456, 12456|3, 13|2456, 1456|23, 1|2|3456.
a(4) = 10: 1234|5678, 1235678|4, 124|35678, 125678|34, 134|25678, 135678|24, 14|235678, 15678|234, 1|23|45678, 1|245678|3.
a(5) = 23: 12345|6789(10), 12346789(10)|5, 1235|46789(10), 1236789(10)|45, 1245|36789(10), 1246789(10)|35, 125|346789(10), 126789(10)|345, 12|3|456789(10), 1345|26789(10), 1346789(10)|25, 135|246789(10), 136789(10)|245, 13|2|456789(10), 145|236789(10), 146789(10)|235, 15|2346789(10), 16789(10)|2345, 1|234|56789(10), 1|2356789(10)|4, 1456789(10)|2|3, 1|24|356789(10), 1|256789(10)|34.
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MAPLE
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b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
end:
a:= n-> coeff(b(2*n, 0), x, 3*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
`if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
(t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
end:
a:= n-> b(3*n, 2*n, 0):
seq(a(n), n=0..42);
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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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A368246
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Number of permutations of [n] whose cycle minima sum to n.
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+10
4
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1, 1, 0, 2, 3, 8, 90, 384, 2940, 18864, 232848, 1919520, 23364000, 261282240, 3486637440, 48900116160, 746747164800, 11559784320000, 201817271416320, 3580457619916800, 68121866659875840, 1366946563510886400, 28802183294533017600, 627950275273991577600
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OFFSET
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0,4
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COMMENTS
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Also the number of permutations of [n] for which the sum of the positions of the left-to-right maxima is n: a(4) = 3: 2143, 3142, 3241; a(5) = 8: 31254, 32154, 41253, 41352, 42153, 42351, 43152, 43251.
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LINKS
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FORMULA
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a(n) ~ c * (n-1)!, where c = 0.561459..., conjecture: c = exp(-gamma) = A080130, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 29 2023
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EXAMPLE
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a(0) = 1: the empty permutation.
a(1) = 1: (1).
a(2) = 0.
a(3) = 2: (13)(2), (1)(23).
a(4) = 3: (124)(3), (142)(3), (12)(34).
a(5) = 8: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4), (123)(45), (132)(45).
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MAPLE
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b:= proc(n) option remember; `if`(n=0, 1,
expand(b(n-1)*(x^n+n-1)))
end:
a:= n-> coeff(b(n), x, n):
seq(a(n), n=0..23);
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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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