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Search: a124327 -id:a124327
Displaying 1-7 of 7 results found. page 1
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A367955 Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows. +10
11
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 2, 3, 1, 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1, 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1, 1, 1, 2, 5, 10, 23, 47, 39, 49, 81, 84, 129, 74, 78, 70, 87, 33, 23, 29, 5, 6, 1, 1, 1, 2, 5, 10, 23, 47, 103, 81, 129, 172, 261, 304, 431, 299, 325, 376, 317, 424, 196, 183, 144, 165, 52, 34, 41, 6, 7, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
COMMENTS
Rows and also columns reversed converge to A365441.
T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Partition of a set
FORMULA
Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n+1) for n>=1.
Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
T(n,n) = T(n,n*(n+1)/2) = 1.
EXAMPLE
T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
T(5,15) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
1;
. 1;
. . 1, 1;
. . . 1, 1, 2, 1;
. . . . 1, 1, 2, 5, 2, 3, 1;
. . . . . 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1;
. . . . . . 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
...
MAPLE
b:= proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
end:
T:= (n, k)-> coeff(b(n, 0), x, k):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
# 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:
T:= (n, k)-> b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
CROSSREFS
Row sums give A000110.
Column sums give A204856.
Antidiagonal sums give A368102.
T(2n,3n) gives A365441.
T(n,2n) gives A368675.
Row maxima give A367969.
Row n has A000124(n-1) terms (for n>=1).
Cf. A000217, A124327 (the same for block minima), A200660, A278677.
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Dec 05 2023
STATUS
approved
A124325 Number of blocks of size >1 in all partitions of an n-set. +10
6
0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Sum of the first entries in all blocks of all set partitions of [n-1]. a(4) = 17 because the sum of the first entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 1+4+3+3+6 = 17. - Alois P. Heinz, Apr 24 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
FORMULA
a(n) = B(n+1)-B(n)-n*B(n-1), where B(q) are the Bell numbers (A000110).
E.g.f.: (exp(z)-1-z)*exp(exp(z)-1).
a(n) = Sum_{k=0..floor(n/2)} k*A124324(n,k).
a(n) = A285595(n-1,1). - Alois P. Heinz, Apr 24 2017
a(n) = Sum_{k=1..n*(n-1)/2} k * A124327(n-1,k) for n>1. - Alois P. Heinz, Dec 05 2023
EXAMPLE
a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.
MAPLE
with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n), n=0..23);
MATHEMATICA
nn=22; Range[0, nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1], {x, 0, nn}], x] (* Geoffrey Critzer, Mar 28 2013 *)
PROG
(PARI)
N = 66; x = 'x + O('x^N);
egf = (exp(x)-1-x)*exp(exp(x)-1) + 'c0;
gf = serlaplace(egf);
v = Vec(gf); v[1]-='c0; v
/* Joerg Arndt, Mar 29 2013 */
CROSSREFS
Cf. A000110, A123324, A124327, A285595.
Column k=2 of A283424.
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 28 2006
STATUS
approved
A368204 Number of partitions of [n] whose block minima sum to n. +10
4
1, 1, 0, 2, 2, 2, 29, 56, 191, 380, 5097, 14288, 74359, 283884, 1106529, 13588409, 53640963, 350573155, 1867738775, 10770352150, 50050737949, 744605446778, 3615378756421, 29368052533243, 195027586980839, 1442227919200245, 8964685271444243, 61478734886319324 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..677
Wikipedia, Partition of a set
FORMULA
a(n) = A124327(n,n).
EXAMPLE
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 0.
a(3) = 2: 13|2, 1|23.
a(4) = 2: 124|3, 12|34.
a(5) = 2: 1235|4, 123|45.
a(6) = 29: 12346|5, 1234|56, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 156|24|3, 15|246|3, 15|24|36, 16|245|3, 1|2456|3, 1|245|36, 16|24|35, 1|246|35, 1|24|356, 156|2|34, 15|26|34, 15|2|346, 16|25|34, 1|256|34, 1|25|346, 16|2|345, 1|26|345, 1|2|3456.
MAPLE
b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> b(n, 1, 0, n):
seq(a(n), n=0..42);
MATHEMATICA
b[n_, i_, t_, m_] := b[n, i, t, m] = If[n == 0, t^(m - i + 1),
If[(i + m)*(m + 1 - i)/2 < n || i > n, 0, If[t == 0, 0,
t*b[n, i + 1, t, m]] + b[n - i, i + 1, t + 1, m]]];
a[n_] := If[n == 0, 1, b[n, 1, 0, n]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Jun 10 2024, after Alois P. Heinz *)
CROSSREFS
Main diagonal of A124327.
Cf. A365441, A368246.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 16 2023
STATUS
approved
A369596 Number T(n,k) of permutations of [n] whose fixed points sum to k; triangle T(n,k), n>=0, 0<=k<=A000217(n), read by rows. +10
4
1, 0, 1, 1, 0, 0, 1, 2, 1, 1, 1, 0, 0, 1, 9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1, 44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1, 265, 44, 44, 53, 53, 62, 64, 29, 22, 24, 16, 16, 8, 6, 5, 4, 2, 1, 1, 0, 0, 1, 1854, 265, 265, 309, 309, 353, 362, 406, 150, 159, 126, 126, 93, 86, 44, 36, 29, 19, 19, 9, 7, 5, 4, 2, 1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
LINKS
Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Permutation
FORMULA
Sum_{k=0..A000217(n)} k * T(n,k) = A001710(n+1) for n >= 1.
Sum_{k=0..A000217(n)} (1+k) * T(n,k) = A038720(n) for n >= 1.
Sum_{k=0..A000217(n)} (n*(n+1)/2-k) * T(n,k) = A317527(n+1).
T(n,A161680(n)) = A331518(n).
T(n,A000217(n)) = 1.
EXAMPLE
T(3,0) = 2: 231, 312.
T(3,1) = 1: 132.
T(3,2) = 1: 321.
T(3,3) = 1: 213.
T(3,6) = 1: 123.
T(4,0) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Triangle T(n,k) begins:
1;
0, 1;
1, 0, 0, 1;
2, 1, 1, 1, 0, 0, 1;
9, 2, 2, 3, 3, 2, 1, 1, 0, 0, 1;
44, 9, 9, 11, 11, 13, 5, 5, 4, 4, 2, 1, 1, 0, 0, 1;
...
MAPLE
b:= proc(s) option remember; (n-> `if`(n=0, 1, add(expand(
`if`(j=n, x^j, 1)*b(s minus {j})), j=s)))(nops(s))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n})):
seq(T(n), n=0..7);
# second Maple program:
g:= proc(n) option remember; `if`(n=0, 1, n*g(n-1)+(-1)^n) end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
`if`(n=0, g(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m-1)))
end:
T:= (n, k)-> b(k, min(n, k), n):
seq(seq(T(n, k), k=0..n*(n+1)/2), n=0..7);
MATHEMATICA
g[n_] := g[n] = If[n == 0, 1, n*g[n - 1] + (-1)^n];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0,
If[n == 0, g[m], b[n, i-1, m] + b[n-i, Min[n-i, i-1], m-1]]];
T[n_, k_] := b[k, Min[n, k], n];
Table[Table[T[n, k], {k, 0, n*(n + 1)/2}], {n, 0, 7}] // Flatten (* Jean-François Alcover, May 24 2024, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives A000166.
Column k=3 gives A000255(n-2) for n>=2.
Row sums give A000142.
Row lengths give A000124.
Reversed rows converge to A331518.
T(n,n) gives A369796.
Cf. A000217, A001710, A008289, A008290, A038720, A053632, A062869, A124327, A143946, A143947, A161680, A263753, A263756, A317527, A367955, A368338.
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Mar 02 2024
STATUS
approved
A365821 Total number of partitions of [n-s] whose block minima sum to s, summed over all s. +10
3
1, 0, 1, 1, 1, 2, 3, 6, 11, 23, 46, 97, 205, 448, 1004, 2339, 5661, 14291, 37507, 101962, 285386, 817772, 2386946, 7069893, 21195092, 64225525, 196636559, 608551084, 1905848637, 6049696252, 19501015441, 63960251538, 213822965681, 729536174204, 2541833303563 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..350
Wikipedia, Partition of a set
FORMULA
a(7) = 6: 12|3, 134|2, 13|24, 14|23, 1|234, 123456.
a(8) = 11: 124|3, 12|34, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1234567.
MAPLE
b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> add(b(k, 1, 0, n-k), k=0..n):
seq(a(n), n=0..42);
CROSSREFS
Antidiagonal sums of A124327.
Cf. A368102.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 14 2023
STATUS
approved
A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number. +10
3
1, 1, 1, 2, 4, 10, 29, 101, 367, 1562, 6891, 37871, 197930, 1121634, 6888085, 46190282, 323250987, 2349020516, 17897285514, 142512956148, 1178963284732, 10248806222398, 91421283039658, 847666112839362, 8100455404172267, 79925567946537362, 814508927747776069 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Partition of a set
MAPLE
b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
`if`((i+m)*(m+1-i)/2<n or i>n, 0, `if`(t=0, 0,
t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
end:
a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
seq(a(n), n=0..26);
# second Maple program:
a:= proc(h) option remember; local b; b:=
proc(n, m) option remember; `if`(n=0, 1,
b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
end: forget(b); max(coeffs(b(h, 0)))
end:
seq(a(n), n=0..26);
CROSSREFS
Row maxima of A124327.
Cf. A367969.
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Dec 14 2023
STATUS
approved
A368401 Number T(n,k) of permutations of [n] whose sum of cycle maxima minus cycle minima gives k, triangle T(n,k), n>=0, 0<=k<=A002620(n), read by rows. +10
2
1, 1, 1, 1, 1, 2, 3, 1, 3, 7, 11, 2, 1, 4, 12, 28, 53, 12, 10, 1, 5, 18, 52, 135, 289, 84, 72, 58, 6, 1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42, 1, 7, 33, 125, 429, 1407, 4545, 12983, 3520, 3976, 4292, 3950, 3422, 790, 486, 330, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
Alois P. Heinz, Rows n = 0..21, flattened
Wikipedia, Permutation
FORMULA
Sum_{k=0..A002620(n)} k * T(n,k) = A002538(n-1) for n >= 1.
EXAMPLE
T(3,0) = 1: (1)(2)(3).
T(3,1) = 2: (12)(3), (1)(23).
T(3,2) = 3: (123), (132), (13)(2).
Triangle T(n,k) begins:
1;
1;
1, 1;
1, 2, 3;
1, 3, 7, 11, 2;
1, 4, 12, 28, 53, 12, 10;
1, 5, 18, 52, 135, 289, 84, 72, 58, 6;
1, 6, 25, 84, 257, 734, 1825, 524, 564, 496, 422, 60, 42;
...
MAPLE
b:= proc(n, s) option remember; `if`(n=0, 1, (k-> `if`(n>k,
b(n-1, s)+add(b(n-1, subs(h=h+[0, 1], s)), h=s), 0)+
`if`(n>k+1, b(n-1, {s[], [n, 1]}), 0)+add(h[2]!*expand(
x^(h[1]-n)*b(n-1, s minus {h})), h=s))(nops(s)))
end:
T:= (n, k)-> coeff(b(n, {}), x, k):
seq(seq(T(n, k), k=0..floor(n^2/4)), n=0..10);
CROSSREFS
Row sums give A000142.
Cf. A002538, A002620, A124327, A143946, A367955, A368338.
KEYWORD
nonn,look,tabf
AUTHOR
Alois P. Heinz, Dec 22 2023
STATUS
approved
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Last modified August 29 18:55 EDT 2024. Contains 375518 sequences. (Running on oeis4.)