A276727 -id:A276727 - cp's OEIS frontend

A276891 Number T(n,k) of ordered set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 6, 4, 3, 0, 24, 20, 18, 13, 0, 120, 114, 118, 114, 75, 0, 720, 750, 878, 924, 870, 541, 0, 5040, 5616, 7224, 8152, 8760, 7818, 4683, 0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293, 0, 362880, 443400, 652446, 845874, 998560, 1135776, 1148016, 954474, 545835

Offset: 0

Alois P. Heinz, Sep 21 2016

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2,     1;
  0,     6,     4,     3;
  0,    24,    20,    18,    13;
  0,   120,   114,   118,   114,    75;
  0,   720,   750,   878,   924,   870,   541;
  0,  5040,  5616,  7224,  8152,  8760,  7818,  4683;
  0, 40320, 47304, 65514, 79682, 90084, 94560, 81078, 47293;
  ...

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

Columns k=0-10 give: A000007, A000142 (for n>0), A320615, A320616, A320617, A320618, A320619, A320620, A320621, A320622, A320623.
Row sums give: A000670.
Main diagonal gives A000670(n-1) for n>0.
T(2n,n) gives A276892.
Cf. A276727, A276890.

b:= proc(n, m, l) option remember; `if`(n=0, m!,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0),
             `if`(k=1, n!, b(n, 0, [0$(k-1)]))):
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, m!, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[k == 0, If[n == 0, 1, 0], If[k == 1, n!, b[n, 0, Array[0 &, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *)

T(n,k) = A276890(n,k) - A276890(n,k-1) for k>0, T(n,0) = A000007(n).

A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 5, 5, 1, 0, 1, 1, 2, 5, 10, 8, 1, 0, 1, 1, 2, 5, 15, 20, 13, 1, 0, 1, 1, 2, 5, 15, 37, 42, 21, 1, 0, 1, 1, 2, 5, 15, 52, 87, 87, 34, 1, 0, 1, 1, 2, 5, 15, 52, 151, 208, 179, 55, 1, 0, 1, 1, 2, 5, 15, 52, 203, 409, 515, 370, 89, 1, 0

Offset: 0

Alois P. Heinz, Sep 16 2016

The sequence of column k satisfies a linear recurrence with constant coefficients of order 2^(k-1) for k>0.

			A(3,2) = 3: 12|3, 1|23, 1|2|3.
A(4,3) = 10: 123|4, 12|34, 12|3|4, 13|24, 13|2|4, 1|234, 1|23|4, 1|24|3, 1|2|34, 1|2|3|4.
A(5,4) = 37: 1234|5, 123|45, 123|4|5, 124|35, 124|3|5, 12|345, 12|34|5, 12|35|4, 12|3|45, 12|3|4|5, 134|25, 134|2|5, 13|245, 13|24|5, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 1|2345, 1|234|5, 1|235|4, 1|23|45, 1|23|4|5, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|24|35, 1|24|3|5, 1|25|34, 1|2|345, 1|2|34|5, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.
Square array A(n,k) begins:
  1, 1,  1,   1,   1,    1,    1,    1,    1, ...
  0, 1,  1,   1,   1,    1,    1,    1,    1, ...
  0, 1,  2,   2,   2,    2,    2,    2,    2, ...
  0, 1,  3,   5,   5,    5,    5,    5,    5, ...
  0, 1,  5,  10,  15,   15,   15,   15,   15, ...
  0, 1,  8,  20,  37,   52,   52,   52,   52, ...
  0, 1, 13,  42,  87,  151,  203,  203,  203, ...
  0, 1, 21,  87, 208,  409,  674,  877,  877, ...
  0, 1, 34, 179, 515, 1100, 2066, 3263, 4140, ...

Alois P. Heinz, Antidiagonals n = 0..40, flattened
Pierpaolo Natalini, Paolo Emilio Ricci, New Bell-Sheffer Polynomial Sets, Axioms 2018, 7(4), 71.
Wikipedia, Partition of a set

Columns k=0..10 give: A000007, A000012, A000045(n+1), A129847, A276720, A276721, A276722, A276723, A276724, A276725, A276726.
Main diagonal gives A000110.
A(n+1,n) gives A005493(n-1) for n>0.
Cf. A276727, A276837.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
seq(seq(A(n, d-n), n=0..d), d=0..14);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k-1]]]]; Table[A[n, d-n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 06 2017, after Alois P. Heinz *)

A(n,k) = Sum_{i=0..k} A276727(n,i).

A263757 Triangle read by rows: T(n,k) (n>=1, 0<=k

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 4, 7, 12, 1, 7, 17, 35, 60, 1, 12, 44, 93, 210, 360, 1, 20, 103, 275, 651, 1470, 2520, 1, 33, 234, 877, 2047, 5208, 11760, 20160, 1, 54, 533, 2544, 7173, 18423, 46872, 105840, 181440, 1, 88, 1196, 7135, 27085, 67545, 184230, 468720, 1058400, 1814400

Offset: 1

Christian Stump, Oct 25 2015

Row sums give A000142, n >= 1.

Main diagonal gives A001710. - Alois P. Heinz, Sep 20 2016

			Triangle begins:
  1;
  1,  1;
  1,  2,  3;
  1,  4,  7, 12;
  1,  7, 17, 35,  60;
  1, 12, 44, 93, 210, 360;
  ...

Alois P. Heinz, Rows n = 1..15, flattened
FindStat - Combinatorial Statistic Finder, Maximum difference of elements in cycles.

Cf. A000071, A000142, A001710, A276727, A276837, A276974, A277031.

T(n,k) = A276837(n,k+1) - A276837(n,k). - Alois P. Heinz, Sep 20 2016

More terms (rows n=7-10) from Alois P. Heinz, Sep 20 2016

A320553 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most three elements and for at least one block c the smallest integer interval containing c has exactly three elements.

Original entry on oeis.org

2, 5, 12, 29, 66, 145, 315, 676, 1436, 3031, 6367, 13323, 27798, 57873, 120281, 249657, 517663, 1072520, 2220724, 4595938, 9508022, 19664296, 40659943, 84057614, 173750589, 359110196, 742150185, 1533651213, 3169118648, 6548358736, 13530454573, 27956404705

Offset: 3

Alois P. Heinz, Oct 15 2018

			a(4) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A276727.

Cf. A000045, A129847, A320616.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(3):
seq(a(n), n=3..35);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
a[n_] := With[{k = 3}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
a /@ Range[3, 35] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

G.f.: (x+2)*x^3/((x^2+x-1)*(x^4+2*x^3+x^2+x-1)).

a(n) = A129847(n) - A000045(n+1).

A320554 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most four elements and for at least one block c the smallest integer interval containing c has exactly four elements.

Original entry on oeis.org

5, 17, 45, 121, 336, 901, 2347, 6014, 15314, 38766, 97531, 244054, 608339, 1511919, 3748379, 9273353, 22901665, 56477538, 139114445, 342325451, 841676972, 2067997764, 5078117000, 12463618356, 30577931115, 74993361731, 183870516407, 450708620604, 1104563863868

Offset: 4

Alois P. Heinz, Oct 15 2018

			a(5) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.

Alois P. Heinz, Table of n, a(n) for n = 4..1000

Column k=4 of A276727.

Cf. A129847, A276720, A320617.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(4):
seq(a(n), n=4..35);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
a[n_] := With[{k = 4}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
a /@ Range[4, 35] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

G.f.: -(-x^8 -3*x^7 +x^6 +7*x^5 +5*x^4) / (x^12 +5*x^11 +7*x^10 -x^9 -13*x^8 -19*x^7 -14*x^6 -6*x^5 +x^4 +2*x^2 +2*x-1).

a(n) = A276720(n) - A129847(n).

A320555 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most five elements and for at least one block c the smallest integer interval containing c has exactly five elements.

Original entry on oeis.org

15, 64, 201, 585, 1741, 5375, 16355, 48601, 141921, 410425, 1182828, 3398411, 9728692, 27745449, 78861484, 223573925, 632578393, 1786856056, 5039984789, 14197033194, 39945491361, 112282665839, 315352029653, 885048266680, 2482371076351, 6958712870273

Offset: 5

Alois P. Heinz, Oct 15 2018

Alois P. Heinz, Table of n, a(n) for n = 5..1000

Column k=5 of A276727.

Cf. A276720, A276721, A320618.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(5):
seq(a(n), n=5..50);

b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1]~Complement~{0}}]];
A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0 &, k - 1]]]];
a[n_] := With[{k = 5}, A[n, k] - If[k == 0, 0, A[n, k - 1]]];
a /@ Range[5, 35] (* Jean-François Alcover, Dec 08 2020, after Alois P. Heinz *)

a(n) = A276721(n) - A276720(n).

A320556 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most six elements and for at least one block c the smallest integer interval containing c has exactly six elements.

Original entry on oeis.org

52, 265, 966, 3172, 10100, 32918, 111138, 373888, 1238236, 4034221, 12991481, 41567855, 132719006, 423099220, 1346053178, 4271656023, 13520858094, 42696919677, 134582517515, 423599583268, 1331701708711, 4182193622677, 13121508724973, 41131777789545

Offset: 6

Alois P. Heinz, Oct 15 2018

Alois P. Heinz, Table of n, a(n) for n = 6..1000

Column k=6 of A276727.

Cf. A276721, A276722, A320619.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(6):
seq(a(n), n=6..50);

a(n) = A276722(n) - A276721(n).

A320557 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most seven elements and for at least one block c the smallest integer interval containing c has exactly seven elements.

Original entry on oeis.org

203, 1197, 4971, 18223, 63768, 220419, 779242, 2845864, 10418560, 37768970, 135153976, 477964329, 1676343822, 5852483376, 20403590238, 71080014610, 247360604490, 859493636214, 2980904955378, 10318666659192, 35656973487023, 123042978647274, 424121272321296

Offset: 7

Alois P. Heinz, Oct 15 2018

Alois P. Heinz, Table of n, a(n) for n = 7..1000

Column k=7 of A276727.

Cf. A276722, A276723, A320620.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(7):
seq(a(n), n=7..50);

a(n) = A276723(n) - A276722(n).

A320558 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most eight elements and for at least one block c the smallest integer interval containing c has exactly eight elements.

Original entry on oeis.org

877, 5852, 27267, 110545, 422396, 1578192, 5877165, 22355618, 87597223, 345223398, 1352883364, 5249340393, 20158426185, 76729396494, 290259302392, 1094289866107, 4121529511428, 15518374075986, 58402401729381, 219602989556557, 824720185307142, 3092742982300231

Offset: 8

Alois P. Heinz, Oct 15 2018

Alois P. Heinz, Table of n, a(n) for n = 8..1000

Column k=8 of A276727.

Cf. A276723, A276724, A320621.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(8):
seq(a(n), n=8..50);

a(n) = A276724(n) - A276723(n).

A320559 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most nine elements and for at least one block c the smallest integer interval containing c has exactly nine elements.

Original entry on oeis.org

4140, 30751, 158766, 705926, 2928164, 11774145, 46852653, 186723275, 759062433, 3170429794, 13343960839, 56013146481, 233387096649, 963938933894, 3948441860748, 16062919807404, 65036845178255, 262641546675463, 1059920408695467, 4277149345637299

Offset: 9

Alois P. Heinz, Oct 15 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..512

Column k=9 of A276727.

Cf. A276724, A276725, A320622.

b:= proc(n, m, l) option remember; `if`(n=0, 1,
      add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
      `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
    end:
A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
a:= n-> (k-> A(n, k) -`if`(k=0, 0, A(n, k-1)))(9):
seq(a(n), n=9..40);

a(n) = A276725(n) - A276724(n).

cp's OEIS Frontend

A276891 Number T(n,k) of ordered set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A276719 Number A(n,k) of set partitions of [n] such that for each block b the smallest integer interval containing b has at most k elements; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A263757 Triangle read by rows: T(n,k) (n>=1, 0<=k

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A320553 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most three elements and for at least one block c the smallest integer interval containing c has exactly three elements.

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A320554 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most four elements and for at least one block c the smallest integer interval containing c has exactly four elements.

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A320555 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most five elements and for at least one block c the smallest integer interval containing c has exactly five elements.

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A320556 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most six elements and for at least one block c the smallest integer interval containing c has exactly six elements.

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A320557 Number of set partitions of [n] such that for each block b the smallest integer interval containing b has at most seven elements and for at least one block c the smallest integer interval containing c has exactly seven elements.

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