Noah A Rosenberg - cp's OEIS frontend

A381811 The largest nonnegative integer j for which each integer n,n+2,...,n+2j can be written as the sum of the squares for some partition of n.

0, 1, 1, 3, 4, 4, 7, 13, 13, 18, 25, 25, 32, 32, 40, 49, 52, 62, 73, 85, 102, 112, 127, 133, 160, 166, 166, 184, 203, 208, 228, 249, 271, 294, 322, 343, 373, 376, 376, 403, 431, 490, 521, 521, 553, 592, 620, 655, 662, 662, 735, 735, 773, 812, 852, 893, 901, 943, 986

Offset: 1

Noah A Rosenberg, May 05 2025

nonn

a(n) has an asymptotic equivalence with (1/2)*n^2-sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).

			a(3) = 1, because n, n+2 (3 and 5) can be written as the sum of the squares for some partition of n; 3=1^2+1^2+1^2 and 5=2^2+1^2. However, 7 cannot be written as the sum of squares of the parts of a partition of 3, so a(3) = 1.
a(4) = 3, because n, n+2, n+4 and n+6 (4, 6, 8 and 10) can be written as the sum of the squares for some partition of n; 4=1^2+1^2+1^2+1^2, 6=2^2+1^2+1^2, 8=2^2+2^2, and 10=3^2+1^2. However, 12 cannot be written as the sum of squares of the parts of a partition of 4, so a(4) = 3.

B. Reznick, The sum of the squares of the parts of a partition, and some related questions, J. Number Theory 33 (1989), 199-208.
P. Winkler, Mean distance in a tree, Discr. Appl. Math. (1990), 179-185.

Cf. A069999 (a(n) provides a lower bound for A069999(n)).

Cf. A000041, A383682.

a(n) = (A383682(n) - n) / 2.

A383682 The largest nonnegative integer value of j for which each integer n, n+2, ..., j-4, j-2, j can be written as the sum of the squares of the elements of a partition of n.

Original entry on oeis.org

1, 4, 5, 10, 13, 14, 21, 34, 35, 46, 61, 62, 77, 78, 95, 114, 121, 142, 165, 190, 225, 246, 277, 290, 345, 358, 359, 396, 435, 446, 487, 530, 575, 622, 679, 722, 783, 790, 791, 846, 903, 1022, 1085, 1086, 1151, 1230, 1287, 1358, 1373, 1374, 1521, 1522, 1599

Offset: 1

Noah A Rosenberg, May 05 2025

nonn

			Consider n=3: 3 and 5 can be written as sums of squares of partitions of 3, as 3=1^2+1^2+1^2 and 5=2^2+1^2, but 7 cannot be written as a sum of squares of a partition of 3, so a(3)=5.
Consider n=4: 4, 6, 8, and 10 can be written as sums of squares of partitions of 4, as 4=1^2+1^2+1^2+1^2, 6=2^2+1^2+1^2, 8=2^2+2^2, and 10=3^2+1^2, but 12 cannot be written as a sum of squares of a partition of 4, so a(4)=10.

B. Reznick, The sum of the squares of the parts of a partition, and some related questions, J. Number Theory 33 (1989), 199-208.
P. Winkler, Mean distance in a tree, Discr. Appl. Math. (1990), 179-185.

Cf. A381811.

a(n) ~ n^2-2*sqrt(2)*n^(3/2)+O(n^(5/4)) (Reznick 1989, p. 201).

A383683 The number of possible values that can be obtained for the Shannon diversity index across all partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 39, 52, 68, 89, 117, 150, 192, 244, 309, 387, 485, 603, 749, 922, 1130, 1384, 1680, 2035, 2440, 2922, 3478, 4118, 4867, 5728, 6740, 7879, 9206, 10741, 12502, 14516, 16846, 19533, 22620, 26164, 30252, 34967, 40450, 46786

Offset: 0

Noah A Rosenberg, May 05 2025

nonn

For a partition P of n into parts (n_1, n_2, ..., n_k), the Shannon diversity index is S(P) = -Sum_{i=1..k} (n_i/n)*log(n_i/n). a(n) is the number of distinct values that S(P) obtains across all possible partitions P of n.

			For n=0 through 7, each partition of n produces a distinct value of the Shannon diversity index, so that a(n) is equal to the number of partitions, A000041(n).
For n=8, partitions (2,2,2,2) and (4,1,1,1,1) both have the same Shannon diversity index, 2*log(2), so that a(8) = 21, one less than A000041(8).

A000607 provides a lower bound for a(n).

Cf. A000041.

A381948 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 4^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 4 players.

Original entry on oeis.org

1, 1, 75, 3016718788056802445, 940214577272785072764883853635996915471902343186386048409875362373502134253520788722829230121857323681047351543536731036815

Offset: 0

Noah A Rosenberg, Mar 10 2025

nonn

a(n) is also the number of tie-permitting labeled histories for a fully symmetric strictly quadfurcating labeled topology with 4^n leaves.

			Two of the 75 cases with n=4 and 4^2=16 players are: (1) (A,B,C,D) play, then (E,F,G,H) play, then (I,J,K,L) play, then (M,N,O,P) play, then the winners of the four matches play; (2) (A,B,C,D) play simultaneously with (E,F,G,H) and (I,J,K,L), then the winners of these three matches play against M, then the winner plays against N, O, and P.

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 15 with r=4)

Cf. A273725 (if matches must be non-simultaneous), A379758 (if matches involve only two players at a time), A381865 (if matches involve only three players at a time).

A381865 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 3^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 3 players.

Original entry on oeis.org

1, 1, 13, 308682013, 20447648974223714249697186722386536049691073

Offset: 0

Noah A Rosenberg, Mar 08 2025

a(n) is also the number of tie-permitting labeled histories for a fully symmetric strictly trifurcating labeled topology with 3^n leaves.

			Two of the 13 cases with n=2 and 3^2=9 players are: (1) (A,B,C) play, then (D,E,F) play, then (G,H,I) play, then the winners of the three matches play; (2) (A,B,C) play simultaneously with (D,E,F), then the winners of these two matches play against G, then the winner plays against H and I.

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307. (see Theorem 15 with r=3)

Cf. A273723 (if matches must be non-simultaneous), A379758 (if matches involve only two players at a time).

A381866 Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are not allowed.

Original entry on oeis.org

1, 1, 126, 162162, 1003458456, 20419376121144, 1084881453316380720, 128835096988586792403600, 30577206578883234961900809600, 13328512616115465470187677202211200, 9988360697491697592427704919982668857600, 12203369577406758958826880335333105520792518400

Offset: 0

Noah A Rosenberg, Mar 08 2025

nonn

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307 (see Table 1).

Cf. A006472, A339411, A381536 for bifurcating, trifurcating, and quadfurcating trees; A381533 if simultaneity is allowed.

Cf. A007696.

a[n_]:=((4*n+1)!/120^n)*Product[(4*i-3),{i,n}]; Array[a,11,0] (* Stefano Spezia, Mar 09 2025 *)

a(n) = ((4*n+1)!/120^n) * Product_{i=1..n} (4*i-3).
a(n) = Gamma(4*n+2)*Gamma(n+1/4)/(30^n*Gamma(1/4)). - Stefano Spezia, Mar 09 2025
a(n) = A007696(n)*(4*n+1)!/120^n. - Alois P. Heinz, Mar 10 2025

A381533 Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are allowed.

Original entry on oeis.org

1, 1, 126, 198198, 1552358808, 41269930621920, 2917021792126858416, 466738566750935966462976, 150642168106131265276308435840, 89930728809765858827345682838905216, 92814015425659158860323886440105229380608, 156870775305420194841270876582071899442900414976, 415352074564676036635314305973768435826840253066044416

Offset: 0

Noah A Rosenberg, Feb 26 2025

nonn

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307 (see Table 4).

Cf. A317059, A381486, A381523.

a:= proc(n) option remember; `if`(n=0, 1, add((4*n+1)!/
      (i!*120^i*(4*n+1-5*i)!)*a(n-i), i=1..(4*n+1)/5))
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Feb 26 2025

a(n) = Y(4n+1), where Y(n) = Sum_{i=1..floor(n/5)} (n!/(i!*120^i*(n-5*i)!)) * Y(n-4*i), with Y(1)=1.

A381536 Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are not allowed.

Original entry on oeis.org

1, 1, 35, 7350, 5255250, 9564555000, 37072215180000, 271183254041700000, 3430468163627505000000, 70238835650273164875000000, 2210064963735845132791875000000, 102493972758213553878355995000000000, 6769214430816214165896021689775000000000, 618638506832293812621237422228537250000000000

Offset: 0

Noah A Rosenberg, Feb 26 2025

nonn

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307 (see Table 1).

Cf. A006472, A339411, A381523.

a:= n-> (3*n+1)!/24^n*mul(3*i-2, i=1..n):
seq(a(n), n=0..13);  # Alois P. Heinz, Feb 26 2025

a(n) = ((3*n+1)!/24^n) * Product_{i=1..n} (3*i-2).

A378855 Triangle read by rows: T(n,k) is the number of sequences in which the games of a single-elimination tournament with n teams can be played if arbitrarily many arenas are available, the tournament bracket is chosen to the bracket with the largest such number of sequences, and the number of distinct times at which games are played is k, log_2(n) <= k <= n-1.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 0, 2, 3, 0, 0, 2, 9, 8, 0, 0, 1, 12, 30, 20, 0, 0, 1, 22, 102, 160, 80, 0, 0, 0, 10, 114, 380, 485, 210, 0, 0, 0, 10, 198, 1100, 2495, 2478, 896, 0, 0, 0, 5, 204, 1930, 7260, 12810, 10640, 3360, 0, 0, 0, 5, 344, 4890, 27110, 72702

Offset: 2

Noah A Rosenberg, Feb 10 2025

T(n,k) is also the number of tie-permitting labeled histories for a labeled topology with n leaves and exactly k times at which events take place, when the labeled topology is chosen to be the labeled topology with the largest number of tie-permitting labeled histories across all labeled topologies with n leaves.

The first row has n=2. Terms for n=2 to 8 appear in Tables 2 and 3 of King & Rosenberg (2023); terms for n=9 to 16 are supplied by Emily H. Dickey.

			Triangle begins:
  1;
  0, 1;
  0, 1, 2;
  0, 0, 2,  3;
  0, 0, 2,  9,   8;
  0, 0, 1, 12,  30,    20;
  0, 0, 1, 22, 102,   160,     80;
  0, 0, 0, 10, 114,   380,    485,     210;
  0, 0, 0, 10, 198,  1100,   2495,    2478,      896;
  0, 0, 0,  5, 204,  1930,   7260,   12810,    10640,      3360;
  0, 0, 0,  5, 344,  4890,  27110,   72702,   101024,     70080,     19200;
  0, 0, 0,  2, 278,  6360,  53000,  211365,   451164,    529116,    321600,     79200;

Matthew C. King and Noah A. Rosenberg, A mathematical connection between single-elimination sports tournaments and evolutionary trees, Math. Mag. 96 (2023), 484-497.

Row sums are A380767.
Cf. A380166 for the triangle if n is a power of 2.
Entries T(n,n-1) follow A056971.

The maximum is computed over unlabeled binary rooted trees T with n leaves (trees in the set enumerated by A001190) of the quantity computed for tree T in eq. 3 of King & Rosenberg (2023). This maximum gives the row sum, tabulated in A380767. For the tree that generates the maximum, the row entries are computed as the specific terms described in Theorem 3 of King & Rosenberg (2023) (and summed in eq. 3).

A381486 Number of labeled histories for rooted ternary trees with 2n+1 leaves if simultaneous trifurcations are allowed.

Original entry on oeis.org

1, 1, 10, 420, 43960, 9347800, 3513910400, 2131249120000, 1952028782704000, 2568150610833808000, 4666919676058159520000, 11351087418588355518080000, 36008099327884173922432000000, 145785514242304854141480256000000, 739598808823839440680777500928000000, 4627885522642342503645368137231360000000

Offset: 0

Noah A Rosenberg, Feb 25 2025

nonn

a(n) is also the number of single-elimination sports tournament schedules possible for 2n+1 teams if matches involve three teams, arbitrarily many arenas are available, and labeled teams have been specified, but the bracket of matches has not been specified.

			Consider 7 named players in a sport in which players compete 3 at a time (e.g. the television gameshow "Jeopardy!"). The number of ways a single-elimination tournament can be arranged, if simultaneous matches can take place, is a(3)=420. Three of these 420 are: (1) A, B, and C play; the winner plays against D and E; the winner plays against F and G. (2) D, E, and F play; the winner plays against A and B; the winner plays against C and G. (3) A, B, and C play simultaneous with D, E, and F; the winners of these matches play against G.

Emily H. Dickey and Noah A. Rosenberg, Labelled histories with multifurcation and simultaneity, Phil. Trans. R. Soc. B 380 (2025), 20230307.

Cf. A317059 for binary rather than ternary trees, A339411 if simultaneity is disallowed.

a:= proc(n) option remember; `if`(n=0, 1, add((2*n+1)!/
      (i!*6^i*(2*n+1-3*i)!)*a(n-i), i=1..(2*n+1)/3))
    end:
seq(a(n), n=0..15);  # Alois P. Heinz, Feb 25 2025

a(n) = Y(2n+1), where Y(n) = Sum_{i=1..floor(n/3)} (n!/(i!*6^i*(n-3*i)!))*Y(n-2*i), with Y(1)=1.

cp's OEIS Frontend

User: Noah A Rosenberg

A381811 The largest nonnegative integer j for which each integer n,n+2,...,n+2j can be written as the sum of the squares for some partition of n.

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A383682 The largest nonnegative integer value of j for which each integer n, n+2, ..., j-4, j-2, j can be written as the sum of the squares of the elements of a partition of n.

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A383683 The number of possible values that can be obtained for the Shannon diversity index across all partitions of n.

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A381948 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 4^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 4 players.

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A381865 Number of sequences in which the matches of a fully symmetric single-elimination tournament with 3^n players can be played if arbitrarily many matches can occur simultaneously and each match involves 3 players.

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A381866 Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are not allowed.

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A381533 Number of labeled histories for rooted 5-furcating trees with 4n+1 leaves if simultaneous 5-furcations are allowed.

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A381536 Number of labeled histories for rooted 4-furcating trees with 3n+1 leaves if simultaneous 4-furcations are not allowed.

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A381486 Number of labeled histories for rooted ternary trees with 2n+1 leaves if simultaneous trifurcations are allowed.

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