Mohammad K. Azarian - cp's OEIS frontend

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

0, 0, 2, 0, 3, 21, 0, 4, 40, 232, 0, 5, 65, 505, 3005, 0, 6, 96, 936, 7056, 45936, 0, 7, 133, 1561, 14287, 112609, 818503, 0, 8, 176, 2416, 26048, 241984, 2056832, 16736896, 0, 9, 225, 3537, 43929, 470961, 4601529, 42683841, 387057609, 0, 10, 280, 4960, 69760, 848800

Offset: 1

Mohammad K. Azarian, Aug 28 2021

The formula for this sequence is Theorem 2.2(iv) of the author's paper, p. 131 (see the link).

			For T(2,3): the number of functions is 3^2 and the number of one-to-one functions is 6, so 3^2 - 6 = 3 and thus T(2,3) = 3.
Triangle T(n,k) begins:
       k=1  k=2   k=3   k=4    k=5     k=6
  n=1:  0    0    0     0      0       0
  n=2:       2    3     4      5       6
  n=3:            21    40     65      96
  n=4:                  232    505     936
  n=5:                         3005    7056
  n=6:                                 45936

Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A089072, A101030, A199656, A344110, A344112, A344113, A344114, A344115, A344116.

A347034 := proc(n,k)
    k^n-k!/(k-n)! ;
end proc:
seq(seq(A347034(n,k),n=1..k),k=1..12) ; # R. J. Mathar, Jan 12 2023

Table[k^n - k!/(k - n)!, {k, 12}, {n, k}] // Flatten

T(n,k) = k^n - k!/(k - n)!;
row(k) = vector(k, i, T(i, k)); \\ Michel Marcus, Oct 01 2021

T(n,k) = k^n - k!/(k - n)!, k>=n.

T(n,n) = A036679(n).

A344118 Triangle of numbers T(n,k) = floor((A089072(n,k)-A019538(n,k))/A019538(n,k)) read by rows, n>=1, 1<=k<=n.

Original entry on oeis.org

0, 0, 1, 0, 0, 3, 0, 0, 1, 9, 0, 0, 0, 3, 25, 0, 0, 0, 1, 7, 63, 0, 0, 0, 0, 3, 17, 162, 0, 0, 0, 0, 2, 7, 39, 415, 0, 0, 0, 0, 1, 4, 16, 91, 1066, 0, 0, 0, 0, 0, 2, 8, 34, 212, 2754, 0, 0, 0, 0, 0, 1, 5, 16, 73, 500, 7146, 0, 0, 0, 0, 0, 1, 3, 9, 33, 160, 1190, 18612, 0

Offset: 1

Mohammad K. Azarian, Jul 28 2021

			T(3, 3) = floor((27-6)/6) = 3.
Triangle begins:
  0
  0    1
  0    0    3
  0    0    1    9
  0    0    0    3    25
  0    0    0    1    7    63
  0    0    0    0    3    17    162

Cf. A008277, A008292, A028246, A089072, A019538.

Floor[Table[(k^n - k!*StirlingS2[n, k])/(k!*StirlingS2[n, k]), {n, 15}, {k, n}]] // Flatten

T(n, k) = floor((k^n - k!*Stirling2(n, k))/(k!*Stirling2(n, k))).

A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

Original entry on oeis.org

1, 2, 14, 5, 58, 506, 12, 244, 4072, 65512, 27, 1004, 32708, 1048456, 33554312, 58, 4066, 262024, 16776856, 1073741104, 68719476016, 121, 16342, 2096942, 268434616, 34359735848, 4398046506064, 562949953416272, 248, 65480, 16776880, 4294965616, 1099511621056

Offset: 1

Mohammad K. Azarian, Jun 06 2021

If n=k, then T(n,k) = 2^(n^2) - n!, which is A344114, and if kA344110.

			For T(2,2): the number of relations is 2^4 and the number of one-to-one functions is 2, so 2^4 - 2 = 14 and thus T(2,2) = 14.
Triangle T(n,k) begins:
   1;
   2,   14;
   5,   58,   506;
  12,  244,  4072,   65512;
  27, 1004, 32708, 1048456, 33554312;

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A068424, A101030, A199656, A344110, A344112, A344113, A344114.

Table[2^(n*k) - k!/(k - n)!, {k, 10}, {n, k}] // Flatten

T(n,k) = 2^(n*k) - k!/(k-n)!, k >= n.

A344116 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.

Original entry on oeis.org

1, 3, 14, 7, 58, 506, 15, 242, 4060, 65512, 31, 994, 32618, 1048336, 33554312, 63, 4034, 261604, 16775656, 1073740024, 68719476016, 127, 16258, 2095346, 268427056, 34359721568, 4398046495984, 562949953416272, 255, 65282, 16771420, 4294926472, 1099511501776, 281474976519136, 72057594037786816, 18446744073709511296

Offset: 1

Mohammad K. Azarian, Jun 07 2021

			For T(2,2), the number of relations is 2^4 and the number of onto functions is 2, so 2^4 - 2 = 14.
Triangle T(n,k) begins:
   1
   3     14
   7     58      506
  15    242     4060      65512
  31    994    32618    1048336    33554312

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A101030, A131689, A199656, A344110, A344112, A344113, A344115.

TableForm[Table[2^(n*k) - Sum[Binomial[k, k - i] (k - i)^n*(-1)^i, {i, 0, k}], {n, 5}, {k, n}]]

T(n,k) = 2^(n*k) - k!*stirling(n, k, 2); \\ Michel Marcus, Jun 26 2021

T(n,k) = 2^(n*k) - k!*Stirling2(n,k).

T(n,k) = A344110(n,k) - A131689(n,k).

A344114 a(n) = 2^(n^2) - n!.

Original entry on oeis.org

1, 14, 506, 65512, 33554312, 68719476016, 562949953416272, 18446744073709511296, 2417851639229258349049472, 1267650600228229401496699576576, 2658455991569831745807614120520772352, 22300745198530623141535718272648361026978816, 748288838313422294120286634350736906063831234982912

Offset: 1

Mohammad K. Azarian, Jun 04 2021

a(n) is the number of relations on a set with n elements that are not one-to-one functions.

			a(1) = 2^(1^2) - 1! =   1;
a(2) = 2^(2^2) - 2! =  14;
a(3) = 2^(3^2) - 3! = 506.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A101030, A199656, A344110, A344112, A344113.

Mathematica
```
Table[2^(n^2) - n!, {n, 16}] // Flatten
```

A344113 a(n) = 2^(n^2) - n^n.

Original entry on oeis.org

1, 12, 485, 65280, 33551307, 68719430080, 562949952597769, 18446744073692774400, 2417851639229257961991863, 1267650600228229401486703205376, 2658455991569831745807613835249018541, 22300745198530623141535718272639445405532160

Offset: 1

Mohammad K. Azarian, May 14 2021

a(n) is the number of relations on a set with n elements that are not functions.

			a(1) = 2^(1^2) - 1^1 = 1.
a(2) = 2^(2^2) - 2^2 = 12.
a(3) = 2^(3^2) - 3^3 = 485.

Michael De Vlieger, Table of n, a(n) for n = 1..57
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000312, A002416, A036679, A344110, A344112, A344114.

Table[2^(n^2) - n^n, {n, 12}] // Flatten

A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

Original entry on oeis.org

1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151

Offset: 1

Mohammad K. Azarian, May 10 2021

			T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
   1;
   3,  12;
   7,  56,   485;
  15, 240,  4015,   65280;
  31, 992, 32525, 1047552, 33551307;

Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A101030, A199656, A036679, A344110.

Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]

T(n,k) = 2^(n*k) - k^n, n,k >= 1.

A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 1, 8, 64, 512, 1, 16, 256, 4096, 65536, 1, 32, 1024, 32768, 1048576, 33554432, 1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 1, 128, 16384, 2097152, 268435456, 34359738368, 4398046511104, 562949953421312

Offset: 0

Mohammad K. Azarian, May 10 2021

T(n, k) is the number of relations from an n-element set into a k-element set, n >= 0, 0 <= k <= n.

T(n,k) is the size of the right principal ideal generated by A where A is an n X n matrix over GF(2) having rank k. The right principal ideal of A contains precisely the matrices whose image is contained in the image of A. - Geoffrey Critzer, Sep 25 2022

			T(3,3) = number of relations from a 3-element set into a 3-element set=2^(3*3)=512.
Triangle begins:
   1
   1   2
   1   4      16
   1   8      64      512
   1  16     256     4096      65536
   1  32    1024    32768    1048576    33554432
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..1325 (rows n = 0..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000079, A089072, A008292, A031971, A117401, A344260 (row sums).

Cf. A022166, A288853.

Mathematica
```
Table[2^(n*k), {n, 0, 10}, {k, 0, n}]
```

T(n,k) = 2^(n*k).

T(n,k) = Sum_{j=0..k} A288853(n,j)*A022166(n,j). - Geoffrey Critzer, Jan 02 2023

A280190 Base-60 (Babylonian or sexagesimal) expansion of sine of 3 degrees.

Original entry on oeis.org

3, 8, 24, 33, 59, 34, 28, 14, 50, 5, 28, 29, 38, 47, 51, 57, 24, 25, 56, 50, 24, 30, 4, 12, 1, 50, 22, 37, 14, 19, 57, 37, 6, 18, 54, 32, 55, 20, 41, 45, 16, 27, 55, 52, 52, 7, 9, 20, 25, 8, 58, 18, 22, 58, 32, 34, 3, 2, 15, 27, 36, 33, 23, 19, 12, 48, 0, 33, 42, 3, 6, 1, 37, 1, 19, 21, 55, 46, 56

Offset: 1

Mohammad K. Azarian, Jan 14 2017

The Fifteenth Century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) from sine of 3 degrees in his Risala al-Watar wa'l Jaib.

Mohammad K. Azarian, A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.

Cf. A019810, A019812, A049469, A110937, A280188, A280189.

RealDigits[Sin[3 Degrees], 60, 200][[1]]

A280189 Version of sexagesimal expansion of sine of one degree given by the Persian mathematician Al-Kashi in the 15th Century.

Original entry on oeis.org

1, 2, 49, 43, 11, 14, 44, 16, 26, 17

Offset: 1

Mohammad K. Azarian, Dec 28 2016

The fifteenth century Persian mathematician Jamshid Al-Kashi was the first to calculate the value of sine of one degree correct to ten sexagesimal places (17 decimal digits) in his Risala al-Watar wa'l Jaib.

Mohammad K. Azarian, A Study of Risala al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"), Forum Geometricorum, Volume 15 (2015) 229-242. Mathematical Reviews, MR 3418854 (Reviewed), Zentralblatt MATH, Zbl 1328.01015.

Cf. A019810, A019812, A049469, A110937, A280188.

cp's OEIS Frontend

User: Mohammad K. Azarian

A347034 Triangle read by columns: T(n,k) is the number of functions from an n-element set to a k-element set that are not one-to-one, k>=n>=1.

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A344118 Triangle of numbers T(n,k) = floor((A089072(n,k)-A019538(n,k))/A019538(n,k)) read by rows, n>=1, 1<=k<=n.

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A344115 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not one-to-one functions.

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A344116 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not onto functions.

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A344114 a(n) = 2^(n^2) - n!.

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A344113 a(n) = 2^(n^2) - n^n.

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A344112 Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.

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A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

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