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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Abdullahi Umar

Abdullahi Umar's wiki page.

Abdullahi Umar has authored 48 sequences. Here are the ten most recent ones:

A221879 Triangle T(n,k) read by rows: Number of order-reversing full contraction mappings (of an n-chain) with 1 fixed point and height exactly k.

Original entry on oeis.org

1, 2, 0, 3, 2, 1, 4, 6, 4, 0, 5, 12, 12, 4, 1, 6, 20, 28, 18, 6, 0, 7, 30, 55, 52, 27, 6, 1, 8, 42, 96, 120, 88, 36, 8, 0, 9, 56, 154, 240, 230, 136, 48, 8, 1, 10, 72, 232, 434, 516, 400, 200, 60, 10, 0, 11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

Row sums are A059570.

Examples

			T (4,6) = 6 because there are exactly 6 order-reversing full contraction mappings (of a 4-chain) with 1 fixed point and of height exactly 2, namely: (3222), (2221), (2211), (4433), (4333), (3332).
Triangle starts:
  1,
  2, 0,
  3, 2, 1,
  4, 6, 4, 0,
  5, 12, 12, 4, 1,
  6, 20, 28, 18, 6, 0,
  7, 30, 55, 52, 27, 6, 1,
  8, 42, 96, 120, 88, 36, 8, 0,
  9, 56, 154, 240, 230, 136, 48, 8, 1,
  10, 72, 232, 434, 516, 400, 200, 60, 10, 0,
  11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1
  ...

Links

Crossrefs

Cf. A059570 (row sums), A221876, A221877, A221878, A221880, A221881, A221882.

Programs

  • Maple
    A221879 := proc(n,k)
    option remember ;
    if n<1 then
        0 ;
    elif n=1 then
        if k = 1 then
            1;
        else
            0 ;
        end if;
    else
        if n = 2 and k=2 then
            0;
        else
            (n-k+1)*binomial(n-2,k-1)+procname(n-2,k-2) ;
        end if;
    end if;
end proc:
seq(seq( A221879(n,k),k=1..n),n=1..20) ; # R. J. Mathar, Aug 15 2025

Formula

T(n, 1) = 1, T(2,2) = 0 and T(n,k) = (n-k+1)*C(n-2,k-1) + T(n-2,k-2) for k > 0.
Sum_{k=1..n} T(n,k) = A059570(n).

A221878 Number of order-preserving or order-reversing full contraction mappings (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 2, 8, 2, 1, 6, 22, 5, 2, 1, 14, 57, 12, 5, 2, 1, 34, 136, 28, 12, 5, 2, 1, 78, 315, 64, 28, 12, 5, 2, 1, 178, 710, 144, 64, 28, 12, 5, 2, 1, 398, 1577, 320, 144, 64, 28, 12, 5, 2, 1, 882, 3460, 704, 320, 144, 64, 28, 12, 5, 2, 1
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

Its row sum is A221882.

Examples

			T (4,0) = 6 because there are exactly 6 order-preserving or order-reversing full contraction mappings (of a 4-chain) with no fixed point, namely: (2111), (3321), (3322), (4321), (4322), (4443).
Triangle:
1,
0, 1,
1, 2, 1,
2, 8, 2, 1,
6, 22, 5, 2, 1,
14, 57, 12, 5, 2, 1,
34, 136, 28, 12, 5, 2, 1,
78, 315, 64, 28, 12, 5, 2, 1,
178, 710, 144, 64, 28, 12, 5, 2, 1,
398, 1577, 320, 144, 64, 28, 12, 5, 2, 1,
882, 3460, 704, 320, 144, 64, 28, 12, 5, 2, 1

Links

Crossrefs

Cf. A221876, A221877, A221879, A221880, A221881, A221882.

Formula

T(n,0) = T(n-1,1), T(n,1) = A059570(n) + A221876(n,1) - n and T(n,k) = A221876 if k > 1.

A221881 Number of order-preserving or order-reversing full contraction mappings (of an n-chain) with (right) waist exactly k.

Original entry on oeis.org

1, 1, 3, 1, 5, 7, 1, 7, 13, 15, 1, 9, 21, 29, 31, 1, 11, 31, 51, 61, 63, 1, 13, 43, 83, 113, 125, 127, 1, 15, 57, 127, 197, 239, 253, 255, 1, 17, 73, 185, 325, 437, 493, 509, 511, 1, 19, 91, 259, 511, 763, 931, 1003, 1021, 1023
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

Row sums are A221882.

Examples

			T(5,2) = 9 because there are exactly 9 order-preserving or order-reversing full contraction mappings (of a 5-chain) with (right) waist exactly 2, namely: (11112), (11122), (11222), (12222), (21111), (22111), (22211), (22221), (22222).

Links

Crossrefs

Cf. A221876, A221877, A221878, A221879, A221880, A221882.

Formula

T(n,k) = 2*Sum_{p=1..k} C(n-1,p-1) - 1 for k >=1.

A221877 Triangle read by rows: T(n,k) = number of order-preserving or order-reversing full contraction mappings (of an n-chain) with height exactly k.

Original entry on oeis.org

1, 2, 2, 3, 8, 2, 4, 18, 12, 2, 5, 32, 36, 16, 2, 6, 50, 80, 60, 20, 2, 7, 72, 150, 160, 90, 24, 2, 8, 98, 252, 350, 280, 126, 28, 2, 9, 128, 392, 672, 700, 448, 168, 32, 2, 10, 162, 576, 1176, 1512, 1260, 672, 216, 36, 2
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

Row sums are A221882.

Examples

			T(3,2) = 8 because there are exactly 8 order-preserving full contraction mappings (of a 3-chain) with exactly height 2, namely: (112), (122), (211), (221), (223), (233), (322), (332).
From _Paolo Xausa_, Aug 18 2025: (Start)
Triangle begins:
   1;
   2,   2;
   3,   8,   2;
   4,  18,  12,    2;
   5,  32,  36,   16,    2;
   6,  50,  80,   60,   20,    2;
   7,  72, 150,  160,   90,   24,   2;
   8,  98, 252,  350,  280,  126,  28,   2;
   9, 128, 392,  672,  700,  448, 168,  32,  2;
  10, 162, 576, 1176, 1512, 1260, 672, 216, 36, 2;
  ... (End)

Links

Crossrefs

Cf. A221876, A221878, A221879, A221880, A221881, A221882.

Programs

  • Mathematica
    A221877[n_, k_] := If[k == 1, n, 2*(n-k+1)*Binomial[n-1, k-1]];
Table[A221877[n, k], {n, 15}, {k, n}] (* Paolo Xausa, Aug 18 2025 *)

Formula

T(n,1) = n and T(n,k) = 2(n-k+1)*C(n-1,k-1) if k > 1.

Extensions

Name edited by Paolo Xausa, Aug 18 2025

A221876 T(n,k) is the number of order-preserving full contraction mappings (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 12, 5, 2, 1, 28, 12, 5, 2, 1, 64, 28, 12, 5, 2, 1, 144, 64, 28, 12, 5, 2, 1, 320, 144, 64, 28, 12, 5, 2, 1, 704, 320, 144, 64, 28, 12, 5, 2, 1, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

Row sum is A001792(n-1).
The matrix inverse starts
1;
-2,1;
-1,-2,1;
0,-1,-2,1;
1,0,-1,-2,1;
2,1,0,-1,-2,1;
3,2,1,0,-1,-2,1;
4,3,2,1,0,-1,-2,1;
5,4,3,2,1,0,-1,-2,1;
6,5,4,3,2,1,0,-1,-2,1;
7,6,5,4,3,2,1,0,-1,-2,1; - R. J. Mathar, Apr 12 2013
...
T(n,k) is also the total number of occurrences of parts k in all compositions (ordered partitions) of n, see example. The equivalent sequence for partitions is A066633. Omar E. Pol, Aug 26 2013

Examples

			T(5,3) = 5 because there are exactly 5 order-preserving full contraction mappings (of a 5-chain) with exactly 3 fixed points, namely: (12333), (12334), (22344), (23345), (33345).
Triangle begins:
1,
2, 1,
5, 2, 1,
12, 5, 2, 1,
28, 12, 5, 2, 1,
64, 28, 12, 5, 2, 1,
144, 64, 28, 12, 5, 2, 1,
320, 144, 64, 28, 12, 5, 2, 1,
704, 320, 144, 64, 28, 12, 5, 2, 1,
1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1,
...
Note that column k is column 1 shifted down by k positions.
Row 4 is [12, 5, 2, 1]: in the compositions of 4
[ 1]  [ 1 1 1 1 ]
[ 2]  [ 1 1 2 ]
[ 3]  [ 1 2 1 ]
[ 4]  [ 1 3 ]
[ 5]  [ 2 1 1 ]
[ 6]  [ 2 2 ]
[ 7]  [ 3 1 ]
[ 8]  [ 4 ]
there are 12 parts=1, 5 parts=2, 2 part=3, and 1 part=4.
- _Joerg Arndt_, Sep 01 2013

Links

Crossrefs

Cf. A001792, A221877, A221878, A221879, A221880, A221881, A221882.

Programs

  • Mathematica
    T[n_, n_] = 1; T[n_, k_] := (n - k + 3)*2^(n - k - 2);
Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 21 2018 *)

Formula

T(n,n) = 1, T(n,k) = (n-k+3)*2^(n-k-2) for n>=2 and n > k > 0.
T(2*n+1,n+1) = T(n+1,1) = A045623(n) for n>=0.
T(n,k) = A045623(n-k), n>=1, 1<=k<=n. - Omar E. Pol, Sep 01 2013

A221882 Number of order-preserving or order-reversing full contraction mappings of an n-chain.

Original entry on oeis.org

1, 4, 13, 36, 91, 218, 505, 1144, 2551, 5622, 12277, 26612, 57331, 122866, 262129, 557040, 1179631, 2490350, 5242861, 11010028, 23068651, 48234474, 100663273, 209715176, 436207591, 905969638, 1879048165, 3892314084, 8053063651, 16642998242, 34359738337
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Comments

a(n) is the order of the semigroup (monoid) of order-preserving or order-reversing full contraction mappings (of an n-chain).

Examples

			a(3) = 13 because there are exactly 13 order-preserving or order-reversing full contraction mappings of a 3-chain, namely: (111), (112), (211), (122), (221), (123), (321), (222), (223), (233), (322), (332), (333).

Links

Crossrefs

Cf. A045992, A221876, A221877, A221878, A221880, A221881.

Programs

Formula

a(n) = (n+1)*2^(n-1) - n.
a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4).
a(n) = Sum_{k=1..n} A221877(n,k) = Sum_{k=0..n-1} A221878(n,k) = Sum_{k=1..n} A221881(n,k). [Edited by Paolo Xausa, Aug 18 2025]
G.f.: x*(1-2*x+2*x^2-2*x^3)/(1-3*x+2*x^2)^2. [Bruno Berselli, Mar 01 2013]

Extensions

More terms from Joerg Arndt, Mar 01 2013

A221880 Number of order-preserving or order-reversing full contraction mappings (of an n-chain) with exactly 1 fixed point.

Original entry on oeis.org

1, 2, 8, 22, 57, 136, 315, 710, 1577, 3460, 7527, 16258, 34917, 74624, 158819, 336766, 711777, 1500028, 3152991, 6611834, 13835357, 28894072, 60234843, 125363062, 260512857, 540599156, 1120345175, 2318984050, 4794555477, 9902285680, 20430920787, 42114540398
Offset: 1

Views

Author

Abdullahi Umar, Feb 28 2013

Keywords

Examples

			a(3) = 8 because there are exactly 8 order-preserving or order-reversing full contraction mappings (of a 3-chain) with exactly 1 fixed point, namely: (111), (112), (222), (233), (333), (321), (322), (221).

Links

Crossrefs

Cf. A221876, A221877, A221878, A221879, A221881, A221882.

Formula

a(n) = A221878(n,1).
a(n) = A059570(n) + A221876(n,1) - n.
G.f.: x*(1-3*x+5*x^2-3*x^3-3*x^4+x^5)/((1+x)*(1-3*x+2*x^2)^2). [Bruno Berselli, Mar 01 2013]
a(n) = -n+(2^(n-1)*(21*n+34)-8*(-1)^n)/36 for n>1, a(1)=1. [Bruno Berselli, Mar 01 2013]

Extensions

More terms from Bruno Berselli, Mar 01 2013

A184050 T(n,k) is the number of order-preserving and order-decreasing partial isometries (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 3, 3, 1, 12, 4, 6, 4, 1, 27, 5, 10, 10, 5, 1, 58, 6, 15, 20, 15, 6, 1, 121, 7, 21, 35, 35, 21, 7, 1
Offset: 0

Views

Author

Abdullahi Umar, Jan 12 2011

Keywords

Examples

			T (4,2) = 6 because there are exactly 6 order-preserving and order-decreasing partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise

Links

Crossrefs

Row sums are A000325 for n >= 0

Formula

T(n,0)= A000325(n-1) and T(n,k)=C(n,k), (k>0)

A184051 T(n,k) is the number of order-decreasing partial isometries (of an n-chain) with exactly k fixed points.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 5, 4, 3, 1, 13, 6, 6, 4, 1, 30, 10, 10, 10, 5, 1, 66, 14, 15, 20, 15, 6, 1, 137, 22, 21, 35, 35, 21, 7, 1
Offset: 0

Views

Author

Abdullahi Umar, Jan 12 2011

Keywords

Examples

			T (4,2) = 6 because there are exactly 6 order-decreasing partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise

Links

Crossrefs

Row sums are A184052 for n >= 0

Formula

T(n,0)= A184052(n) and T(n,k)=C(n,k), (k>0)

A184049 T(n,k) is the number of order-preserving and order-decreasing partial isometries (of an n-chain) of height k (height of alpha = |Im(alpha)|).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 1, 1, 10, 10, 5, 1, 1, 15, 20, 15, 6, 1, 1, 21, 35, 35, 21, 7, 1, 1, 28, 56, 70, 56, 28, 8, 1, 1, 36, 84, 126, 126, 84, 36, 9, 1, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 1, 66, 220
Offset: 0

Views

Author

Abdullahi Umar, Jan 12 2011

Keywords

Comments

Row n gives the coefficients of the polynomial p(n,x) = (x + 1)*p(n-1,x) + (n - 1)*x, where p(0,x) = 1. - Clark Kimberling, Dec 02 2014

Examples

			T (4,2) = 10 because there are exactly 10 order-preserving and order-decreasing partial isometries (on a 4-chain) of height 2, namely: (1,2)-->(1,2); (2,3)-->(1,2); (2,3)-->(2,3); (3,4)-->(1,2); (3,4)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(1,3); (2,4)-->(2,4);
    (1,4)-->(1,4) - the mappings are coordinate-wise
1,
1, 1,
1, 3, 1,
1, 6, 4, 1,
1, 10, 10, 5, 1,
1, 15, 20, 15, 6, 1,
1, 21, 35, 35, 21, 7, 1,
1, 28, 56, 70

Links

Crossrefs

Cf. A007318; Row sums are A000325 for n >= 0.

Programs

  • Mathematica
    z = 14; p[n_, x_] := (x + 1) p[n - 1, x] + (n - 1)*x; p[0, x_] = 1;
t = Table[Factor[p[n, x]], {n, 0, z}]
TableForm[Rest[Table[CoefficientList[t[[n]], x], {n, 0, z}]]] (* A184049 array *)
Flatten[CoefficientList[t, x]] (* A184049 sequence *)
(* Clark Kimberling, Dec 02 2014 *)

Formula

T(n;0)=1 and T(n,k)=C(n+1,k+1), (k>0)

Extensions

More terms from Clark Kimberling, Dec 02 2014

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