A124327 -id:A124327 - cp's OEIS frontend

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.

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

Offset: 0

Alois P. Heinz, Dec 05 2023

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.

			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;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Partition of a set

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.

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 b(k, n, 0):
seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);

b[n_, i_, t_] := b[n, i, t] = 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]]];
T[0, 0] = 1; T[n_, k_] := b[k, n, 0];
Table[Table[T[n, k], {k, n, n*(n + 1)/2}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz's second Maple program *)

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.

A124325 Number of blocks of size >1 in all partitions of an n-set.

Original entry on oeis.org

0, 0, 1, 4, 17, 76, 362, 1842, 9991, 57568, 351125, 2259302, 15288000, 108478124, 805037105, 6233693772, 50257390937, 421049519856, 3659097742426, 32931956713294, 306490813820239, 2945638599347760, 29198154161188501

Offset: 0

Emeric Deutsch, Oct 28 2006

nonn

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

			a(3) = 4 because in the partitions 123, 12|3, 13|2, 1|23, 1|2|3 we have four blocks of size >1.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A000110, A123324, A124327, A285595.

Column k=2 of A283424.

with(combinat): c:=n->bell(n+1)-bell(n)-n*bell(n-1): seq(c(n),n=0..23);

nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x]-1-x)Exp[Exp[x]-1],{x,0,nn}],x]  (* Geoffrey Critzer, Mar 28 2013 *)

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 */

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

A368204 Number of partitions of [n] whose block minima sum to n.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Dec 16 2023

nonn

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..677
Wikipedia, Partition of a set

Main diagonal of A124327.

Cf. A365441, A368246.

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 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);

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 *)

a(n) = A124327(n,n).

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.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Mar 02 2024

			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;
  ...

Alois P. Heinz, Rows n = 0..50, flattened
Wikipedia, Permutation

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.

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);

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 *)

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.

A365821 Total number of partitions of [n-s] whose block minima sum to s, summed over all s.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Dec 14 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..350
Wikipedia, Partition of a set

Antidiagonal sums of A124327.

Cf. A368102.

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 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);

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.

A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Dec 14 2023

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..100
Wikipedia, Partition of a set

Row maxima of A124327.

Cf. A367969.

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 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);

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_] := Module[{s}, Q[n, t, s] /. s -> 1];
a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[P[n, t], t] // Max]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 03 2024 *)

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.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Dec 22 2023

			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;
  ...

Alois P. Heinz, Rows n = 0..21, flattened
Wikipedia, Permutation

Row sums give A000142.

Cf. A002538, A002620, A124327, A143946, A367955, A368338.

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);

Sum_{k=0..A002620(n)} k * T(n,k) = A002538(n-1) for n >= 1.

cp's OEIS Frontend

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.

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A124325 Number of blocks of size >1 in all partitions of an n-set.

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A368204 Number of partitions of [n] whose block minima sum to n.

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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.

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A365821 Total number of partitions of [n-s] whose block minima sum to s, summed over all s.

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A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.

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Author

Keywords

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Programs

Maple

Mathematica

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.

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