Martin Fuller - cp's OEIS frontend

A254529 a(n) = n! * (number of mapping patterns on n).

1, 1, 6, 42, 456, 5640, 93600, 1728720, 38344320, 948931200, 26555558400, 817935148800, 27735629644800, 1020596255078400, 40642432179148800, 1737890081351424000, 79498734605402112000, 3871319396080840704000, 200017645344178421760000, 10925549584125028909056000

Offset: 0

Martin Fuller, Feb 01 2015

a(n) is the number of ordered pairs (p, f) such that p f = f p, where p is a permutation and f is an endofunction.

Alois P. Heinz, Table of n, a(n) for n = 0..382
Math Overflow, What is the probability two random maps on n symbols commute?, 2013.

Cf. A001372, A053529, A181162.

a(n) = n! * A001372(n). - Joerg Arndt, Feb 01 2015

A138482 Mutually-praising pairs excluding autobiographical numbers A138480. Version 2: numbers may have more than 10 digits.

Original entry on oeis.org

130, 230, 430, 530, 630, 730, 830, 930, 1101, 2210, 10110, 11200, 23100, 43100, 53100, 63100, 73100, 83100, 93100, 211100, 230100, 311100, 411100, 430100, 511100, 530100, 611100, 630100, 711100, 730100, 811100, 830100, 911100, 930100

Offset: 1

Martin Fuller, Mar 20 2008

			130 and 1101 are a mutually-praising pair: 130 specifies a number with 1 digit equal to "0", 3 digit "1" and nothing else. 1101 specifies 1 digit "0", 1 digit "1", no digits "2" and 1 digit "3". 73100 and 21010001000 is the first pair with a number of more than 10 digits.

Martin Fuller, Table of n, a(n) for n = 1..172
Tanya Khovanova, Autobiographical Numbers

Cf. A046043, A138480, A138481.

A138480 Autobiographical numbers: the first digit specifies how many 0's in the number, the next digit specifies how many 1's, etc. This version is not limited to 10 digits.

Original entry on oeis.org

1210, 2020, 21200, 3211000, 42101000, 521001000, 6210001000, 72100001000, 821000001000, 9210000001000

Offset: 1

Martin Fuller, Mar 20 2008

			72100001000 has 7 digits equal to "0", 2 digits equal to "1", 1 digit each "2" and "7" and no other digits.

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Shyam Sunder Gupta, On Some Special Numbers, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 22, 527-565.
Tanya Khovanova, Can you solve the Leonardo da Vinci riddle?, TED-Ed YouTube channel.

Autobiographical numbers up to 10 digits: A046043. Mutually-praising pairs: A138481 and A138482.

A124003 Triangle T(n,k) of the number of unlabeled graphs on n nodes with universal reconstruction number k, 3<=k<=n. URN(G) is the minimum size for which all multisubsets of vertex-deleted subgraphs of G can uniquely reconstruct G up to isomorphism.

Original entry on oeis.org

3, 2, 9, 7, 19, 8, 8, 56, 90, 2, 16, 496, 520, 12, 0, 266, 8308, 3584, 284, 4, 0, 45186, 199247, 28781, 1434, 20, 0, 0, 6054148, 5637886, 301530, 10686, 914, 4, 0, 0

Offset: 3

Martin Fuller, Dec 08 2006

The (vertex) Reconstruction Conjecture, due to Kelly and Ulam, states that every graph with three or more vertices is reconstructible up to isomorphism given the multiset of vertex deleted subgraphs. Equivalently, every graph has an URN and so sum(k=3,n,T(n,k))==A000088(n) for all n>=3.

			Triangle begins
        3
        2       9
        7      19      8
        8      56     90     2
       16     496    520    12   0
      266    8308   3584   284   4 0
    45186  199247  28781  1434  20 0 0
  6054148 5637886 301530 10686 914 4 0 0

P. J. Kelly, A congruence theorem for trees, Pacific J. Math., 7 (1957), 961-968.
B. McMullen, Graph reconstruction numbers.
Wikipedia, Reconstruction conjecture.

Cf. A124002, A000088, A006652-A006655.

A124002 Triangle T(n,k) of the number of unlabeled graphs on n nodes with existential reconstruction number k, 3<=k<=n. ERN(G) is the minimum number of vertex-deleted subgraphs of G required to uniquely reconstruct G up to isomorphism.

Original entry on oeis.org

4, 8, 3, 34, 0, 0, 150, 4, 2, 0, 1044, 0, 0, 0, 0, 12334, 8, 2, 2, 0, 0, 274666, 0, 2, 0, 0, 0, 0, 12005156, 6, 4, 0, 2, 0, 0, 0

Offset: 3

Martin Fuller, Dec 08 2006

The (vertex) Reconstruction Conjecture, due to Kelly and Ulam, states that every graph with three or more vertices is reconstructible up to isomorphism given the multiset of vertex deleted subgraphs. Equivalently, every graph has an ERN and so sum(k=3,n,T(n,k))==A000088(n) for all n>=3.

			Triangle begins
         4
         8, 3
        34, 0, 0
       150, 4, 2, 0
      1044, 0, 0, 0, 0
     12334, 8, 2, 2, 0, 0
    274666, 0, 2, 0, 0, 0, 0
  12005156, 6, 4, 0, 2, 0, 0, 0

P. J. Kelly, A congruence theorem for trees, Pacific J. Math., 7 (1957), 961-968.
B. McMullen, Graph reconstruction numbers.
Wikipedia, Reconstruction conjecture.

Cf. A124003, A000088, A006652-A006655.

A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

Original entry on oeis.org

0, 1, 1, 3, 1, 2, 11, 1, 1, 4, 1, 9, 2, 2, 1, 1, 4, 4, 2, 2, 2, 1, 14, 1, 2, 2, 2, 7, 2, 2, 1, 1, 4, 2, 4, 1, 11, 7, 2, 8, 32, 2, 1, 293, 2, 145, 1, 2, 1, 21, 1, 1, 3, 1, 1, 8, 8, 5, 2, 3, 4, 3, 1, 3, 1, 1, 1, 1, 3, 2, 1, 3, 1, 2, 2, 1, 2, 19, 3, 2, 1, 15, 1, 2, 1, 2, 5, 3, 1, 1, 1, 38, 1, 10, 1, 2, 1, 80, 1

Offset: 0

Martin Fuller, Dec 20 2006

This might be interpreted as the expected value of phi(n)/n for very large n. - David W. Wilson, Dec 05 2006

			0.55986561693237348...

Cf. A124175, A085548, A085541, A085964, A085965, A085966, A085967, A085968, A085969.

contfrac(exp(-suminf(m=2,log(zeta(m))*sumdiv(m,k,if(k

A125031 Total number of highest scorers in all 2^(n(n-1)/2) tournaments with n players.

Original entry on oeis.org

1, 2, 12, 104, 1560, 53184, 3422384, 430790144, 111823251840, 56741417927680, 57729973360342272, 118195918779085344768, 479770203506298422135808, 3914602958361039682677710848, 63809077054456699374663196416000, 2076906726499655025703507210668998656

Offset: 1

Martin Fuller, Nov 16 2006

nonn

All highest scorers are also king chickens, A123553.

			With 4 players there are 32 tournaments with 1 highest scorer, 24 tournaments with 2 highest scorers and 8 tournaments with 3 highest scorers. Therefore a(4)=32*1+24*2+8*3=104.

Andrew Howroyd, Table of n, a(n) for n = 1..20
Index entries for sequences related to tournaments

Cf. A006125, A013976, A123553, A125032, A123903.

\\ Requires Winners from A013976.
a(n)={my(M=Winners(n)); sum(i=1, matsize(M)[1], pollead(M[i, 1])*M[i, 2])} \\ Andrew Howroyd, Feb 29 2020

a(5)-a(10) also computed by Gordon Royle, Nov 16 2006

Terms a(12) and beyond from Andrew Howroyd, Feb 28 2020

A125032 Triangle read by rows: T(n,k) = number of tournaments with n players which have the k-th score sequence. The score sequences are in the same order as A068029 and start with the empty score sequence.

Original entry on oeis.org

1, 1, 2, 6, 2, 24, 8, 8, 24, 120, 40, 40, 120, 40, 120, 240, 280, 24, 720, 240, 240, 720, 240, 720, 1440, 1680, 144, 240, 80, 720, 1440, 2880, 1680, 1680, 1680, 8640, 2400, 144, 2400, 2640, 5040, 1680, 1680, 5040, 1680, 5040, 10080, 11760, 1008, 1680, 560

Offset: 1

Martin Fuller, Nov 16 2006

The score sequences are sorted by number of players and then lexicographically.

There are A000571(m) score sequences for m players. The sum of all the a(n) for m players is A006125(m)=2^(m(m-1)/2).

			There are two score sequences with 3 players: [0,1,2] from 6 tournaments and [1,1,1] from 2 tournaments. These score sequences come 4th and 5th respectively, so a(4)=6 and a(5)=2.

Martin Fuller, Table of n, a(n) for n = 1..2242
Eric Weisstein's World of Mathematics, Score Sequence
Index entries for sequences related to tournaments

Cf. A000571, A006125, A068029, A125031 (number of highest scorers), A123553.

Other sequences that can be calculated using this one: A013976, A125031.

A119415 Steps until an EKG sequence starting 1,2,3,...,n merges with the standard EKG sequence A064413, or 0 if the sequences never merge.

Original entry on oeis.org

45, 45, 11, 11, 45, 45, 45, 45, 30, 30, 45, 45, 45, 45, 59, 59, 59, 59, 59, 59, 63, 63, 63, 63, 63, 63, 105, 105, 91, 91, 91, 91, 91, 91, 102, 102, 102, 102, 117, 117, 127, 127, 127, 127, 136, 136, 136, 136, 136, 136, 149, 149

Offset: 3

Martin Fuller, Jul 26 2006

nonn

Are there any zeros?

			a(5)=11 because that EKG sequence starts 1,2,3,4,5,10,6,8,12,9,15 and from the 11th term onwards it is the same as A064413.

Cf. A064413, A169857.

A120066 (n-1)! divided by (product phi(d)! ; d divides n).

Original entry on oeis.org

1, 1, 1, 3, 1, 30, 1, 105, 28, 630, 1, 207900, 1, 12012, 45045, 675675, 1, 171531360, 1, 2618916300, 3527160, 3879876, 1, 139143023019000, 10626, 67603900, 43743700, 21925567263600, 1, 2360561385517335000, 1, 203067496256625, 14902327440

Offset: 1

Martin Fuller, Jun 06 2006

nonn

Cf. A120065.

a(n) = (n-1)! / prod(i=1, n, if(n%i==0, eulerphi(i)!, 1))

cp's OEIS Frontend

User: Martin Fuller

A254529 a(n) = n! * (number of mapping patterns on n).

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A138482 Mutually-praising pairs excluding autobiographical numbers A138480. Version 2: numbers may have more than 10 digits.

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A138480 Autobiographical numbers: the first digit specifies how many 0's in the number, the next digit specifies how many 1's, etc. This version is not limited to 10 digits.

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A124003 Triangle T(n,k) of the number of unlabeled graphs on n nodes with universal reconstruction number k, 3<=k<=n. URN(G) is the minimum size for which all multisubsets of vertex-deleted subgraphs of G can uniquely reconstruct G up to isomorphism.

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A124002 Triangle T(n,k) of the number of unlabeled graphs on n nodes with existential reconstruction number k, 3<=k<=n. ERN(G) is the minimum number of vertex-deleted subgraphs of G required to uniquely reconstruct G up to isomorphism.

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A126226 Continued fraction of Product_{primes p} ((p-1)/p)^(1/p).

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PARI

A125031 Total number of highest scorers in all 2^(n(n-1)/2) tournaments with n players.

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A125032 Triangle read by rows: T(n,k) = number of tournaments with n players which have the k-th score sequence. The score sequences are in the same order as A068029 and start with the empty score sequence.

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A119415 Steps until an EKG sequence starting 1,2,3,...,n merges with the standard EKG sequence A064413, or 0 if the sequences never merge.

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A120066 (n-1)! divided by (product phi(d)! ; d divides n).

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